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Single photon emission characteristics of on-chip integrable ordered single quantum dots: towards scalable quantum optical circuits

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Single photon emission characteristics of on-chip integrable ordered single quantum dots: towards scalable quantum optical circuits

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Content Single Photon Emission Characteristics of On-chip Integrable Ordered Single Quantum Dots: Towards Scalable Quantum Optical Circuits by Jiefei Zhang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHEN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) December 2017 Copyright 2017 Jiefei Zhang ii Dedication To my teachers and parents iii Acknowledgements I would like to express my deepest respect and gratitude to all of my teachers for their contributions to my education. Among these, the three teachers of the greatest influence and contribution are my parents, Xiaolong Zhang and Yinglin Wei, who devoted over a decade of their life to my intellectual development in early ages, and my advisor Prof. Anupam Madhukar who inspired me and guided my intellectual growth process toward being a professional in science. His scientific enthusiasm and passion will always be my inspiration for excellence in my future professional life. His constant encouragement for better work, his great care about the students and their needs and our numerous discussions related to either science or life in general have always helped me to raise self-confidence and inspired me to conquer new challenges. Prof. Madhukar’s devotion to education and science will be a moral example throughout my future professional life. I am also thankful to all the colleagues in our research group, present and past, including Dr. Siyuan Lu, Dr. Zachary Lingley and Swarnabha Chattaraj. I am thankful to them not only because of what I learned from them during the course of this work, but also for their friendship. Without these colleagues, this dissertation work will not be possible. At the beginning of my Ph.D., I had the great luck of working with Dr. Siyuan Lu, an incredible dedicated responsible researcher, who has always helped me to pass through difficult times. I want to express my greatest thanks to him for all his help and guidance throughout my Ph.D work here at USC. I also want to express iv my thanks to Dr. Zachary Lingley and Swarnabha Chattaraj for their help in the lab and help in conducting structure and simulation based studies assisting my dissertation work. I would like to thank Prof. Todd Brun, Prof. Vitaly Kresin, Prof. Grace Lu, and Prof. Hossein Hashemi, who kindly served on my dissertation committee. Finally, I acknowledge with gratitude the funding agency AFOSR and ARO. I want to express special thanks to funding agency ARO (Program manager Dr. John Prater) for providing financial support that make this dissertation work continue and finish with great achievements. v Table of Contents Dedication ii Acknowledgement iii List of Figures viii List of Tables xxiii Abbreviations xxiv Abstract xxvii Chapter 1. Introduction 1 §1.1 Motivation and objective 1 §1.2 MTSQD arrays and on-chip integrated optical circuits 10 §1.3 Organization of the dissertation 14 Chapter 1 References 16 Chapter 2. Mesa-top single quantum dot synthesis: Substrate-encoded 20 size-reducing epitaxy §2.1 Current status of ordered quantum dot synthesis 21 §2.2 Substrate-encoded size-reducing epitaxy 27 §2.2.1 Continuum solid model: capillary effect 28 §2.2.2 Atomistic model: significance of step orientation and density 30 §2.2.3 Substrate-encoded size-reducing epitaxy 34 §2.3 Mesa-top single quantum dot: growth 41 §2.3.1 SESRE implementation using MBE 42 §2.3.2 Substrate preparation 46 §2.3.3 Growth evolution on square mesa with <100> edges 51 §2.3.4 Inter-facet atom migration 65 §2.4 Mesa-top single quantum dot arrays 77 Chapter 2 References 88 Chapter 3. Optical properties of MTSQDs 97 §3.1 Origin of single particle electronic states in bulk and QD structures 98 §3.1.1 Bulk crystalline solids and semiconductors 99 §3.1.2 Semiconductor quantum dot 106 §3.2 Optical transitions in bulk and QD structures 117 §3.3 Photoluminescence (PL) and Micro-PL instrumentation 136 §3.3.1 Photoluminescence process in bulk and QDs 137 §3.3.2 Micro-PL instrumentation 142 §3.4 Photoluminescence study of MTSQDs: emission uniformity and vi efficiency 147 §3.4.1 MTSQD emission: identification of single neutral exciton 147 §3.4.2 MTSQD emission: spectral uniformity and efficiency 158 §3.5 MTSQD excitonic emission types and their identification 164 §3.5.1 Multi-peak emission from a MTSQD 165 §3.5.2 Power dependent PL studies: excitonic complexes in MTSQDs 168 §3.5.3 Polarization dependent PL studies of MTSQD: excitonic complexes in MTSQDs 179 §3.6 Dynamics of excitonic decay in MTSQDs: Time-resolved PL studies 197 §3.6.1 MTSQD neutral exciton PL decay dynamics 199 §3.6.2 Dynamics of excitonic decays: effect of carrier capture in MTSQD 202 §3.7 Photoluminescence excitation spectroscopy: electronic structure of MTSQDs 210 §3.8 Summary 220 Chapter 3 References 222 Chapter 4. Single photon emission from MTSQDs 230 §4.1 Photon statistics, photon detection, and photon correlation 231 §4.1.1 Light sources and photon emission statistic 231 §4.1.2 Photon detection and photon correlation functions 237 §4.1.3 Measurement of g (2) (τ): Hanbury Brown and Twiss Approach 242 §4.2 Physics of spontaneous emission and polarization entangled photon pair emission from QDs 245 §4.2.1 QD interaction with quantized radiation field 246 §4.2.2 Biexciton-exciton cascade: polarization-entangled photon pair emission 255 §4.3 Single photon emission characteristics of the MTSQD array 259 §4.3.1 Hanbury Brown and Twiss Instrumentation 260 §4.3.2 Physical representation of measured second-order correlation histogram 264 §4.3.3 Single photon emission from MTSQDs: measurements and purity 271 §4.3.4 Single photon emission rate and collection efficiency 276 §4.3.5 Enhancement of emission rate and collection efficiency 281 §4.4 Physical processes affecting single photon emission from MTSQDs 284 Chapter 4 References 299 Chapter 5. Conclusion and future work 304 §5.1 Realizing spatially regular array of spectrally uniform single quantum dots 306 §5.2 Ordered spectrally uniform mesa top single quantum dot array 308 §5.3 Emission characteristics and implications for MTSQD electronic states 310 §5.4 Single photon emission from MTSQDs 312 §5.5 Potential of MTSQDs acting as EPSs 314 §5.6 Outlook: MTSQD based integrated optical circuits for information vii processing 316 Chapter 5 References 321 Bibliography 323 Appendix A: Quantum dot basics and self-assembled quantum dots 340 Appendix B: Equipment and experimental techniques 361 Appendix C: Instrumentation of micro-PL setup 413 Appendix D: Quantization of the electromagnetic field 427 viii List of Figures Figure 1.1 Schematic of confinement potential arising from the alignment of the Brillouin zone center conduction and valence band edges of the QD (A) and barrier (B). The discrete electron and hole energy states and the corresponding wavefunctions due to quantum confinement are schematically shown along with the discrete δ form of density of states. 2 Figure 1.2 Panel (a) and (b) shows schematically the charge configuration of exciton and biexciton states of QDs, respectively. Panel (c) shows the schematic energy diagram of biexciton (|XX>), exciton (|X>) and ground state (|0>). The biexciton-exciton cascade emission of photons of wavelength λ 1 and λ 2 are polarization entangled. Fine structure splitting is excluded in this energy diagram. 5 Figure 1.3 Panel (a) shows a 60° tilted SEM image of a part of 5 × 8 array of single quantum dots on mesas tops. Inset is a magnified SEM image of a single MTSQD. Panel (b) shows the color map of spectral emission distribution from all 40 MTSQDs in the array. Pairs of like-color circles indicated the MTSQDs with emission within the instrument resolution limit of 0.2nm. Panel (c) shows PL from the pair of such MTSQDs marked with yellow circles. Panel (d) shows a typical coincident photon counting revealing a highly efficient single photon emission. 9 Figure 1.4 Schematic of the planarization of the MTSQD surface morphology through the growth of an overlayer on the MTSQD (pyramids) array (panel a) resulting in the buried ordered array of MTSQDs (panel b) around which light manipulating units can be lithographically created within the designed footprint (broken white lines). 11 Figure 1.5 Schematic of the conventional 2D photonic crystal based (left panel) and the new paradigm of sub-wavelength sized dielectric building block (DBBs, blue blocks in the right panel) based implementation of light manipulating functions such as SQD emission rate enhancement via a resonant cavity or nanoantenna-waveguiding structure, etc.. 12 Figure 2.1 (a) AFM image of InAs islands showing the inherent variation in size and random spatial positions along with the resulting broad photoluminescence and extremely sharp emission lines (yellow line) from individual islands when capped with a protective layer. (b) Measured Purcell factor, collection efficiency and β-factor (coupling efficiency) of SAQDs within the 2D phonic crystal cavity with different spectral mismatch with respect to cavity mode, from Ref. [2.28] (c) Calculated ix SAQDs photon emission rate into the 2D photonic crystal waveguide mode as a function of SAQDs position in the waveguide, from Ref. [2.32]. 24 Figure 2.2 Schematic of the surface profile of a patterned mesa. SW represents “side wall”. 29 Figure 2.3 Schematic of the surface profile of a patterned mesa. SW represents “side wall”. Schematic of the ridge mea with edges along (a) [1-10] and (b) [110] direction. The interaction between steps and the orientation of arsenic dangling orbitals with respect to the mesa edges determines the atom migration direction marked with red arrows. 31 Figure 2.4 Schematic picture representing the most critical characteristic length scales to control the growth on patterned substrate and the growing layer morphology on the patterned structure. L M is the size of mesa, l s is the elastic strain field propagation length, l m is the atom migration length. The superscripts T, B and SW represents “top”, “bottom” and “side wall”. 34 Figure 2.5 Schematic of the growth profile on mesa top representing mesa top reduction (red line), mesa top constant (black dashed line) and mesa top expansion (blue line) with the corresponding required growth condition of growth rate on mesa top ( τ T )and side wall ( τ SW ). 35 Figure 2.6 Shows an SEM image (panel c) of an array of pyramidal structures resulting from directed adatom migration from sidewalls to the mesa top (panel b) during GaAs/AlGaAs heteroepitaxial growth on starting square mesas with edges oriented such that As dangling orbitals are at 45° degrees with respect to the edge (panel a), thus maintain four- fold symmetry. Inset in panel (c) shows the vertically-stacked GaAs quantum dots grown near the mesa top. (From Ref. 2.76). 36 Figure 2.7 Shows an SEM image (panel b) of an array of triangular pyramidal structures resulting from directed adatom migration from sidewalls to the mesa top during GaAs/AlGaAs heteroepitaxial growth on starting triangular mesas with edges along <1-10> direction (panel a). Inset in panel (b) shows the vertically-stacked GaAs quantum dots grown near the mesa top. (From Ref. 2.37) 37 Figure 2.8 (a) Cross-section TEM image of a 12ML thick InAs film on a 46nm square GaAs (001) mesa with edge orientation along <100> direction from Ref. 2.37. (b) Multimillion atom molecular dynamics simulation of hydrostatic stress distribution in a 12ML InAs film on a 40.7nm square GaAs (001) mesa from Ref. 2.79. 39 x Figure 2.9 Ultra high vacuum inter-connected semiconductor growth, processing, and in-situ characterization system in the Madhukar group at USC. 42 Figure 2.10 Schematic picture of RHEED with geometry of sample, diffracted beams and RHEED screen. The incident beam is at glancing angle with respect to the substrate surface. The diffracted electron beam is projected on fluorescent screen. 44 Figure 2.11 Schematic drawing of the layout of the pattern design containing non-patterned and patterned regions with nanomesas array for the synthesis of mesa top quantum dot. In the patterned region there are two types of mesa patterns depicted in blue designed for structural characterization using cross-sectional TEM and red designed for optical (micro-PL) and (AFM and SEM) structural characterization. The layout of nanomesa in each type of pattern is shown in the zoomed out images on the right side of the figure. 46 Figure 2.12 Schematics of the major preparation steps needed for the preparation of patterned substrate. 50 Figure 2.13 (a) Top view SEM image of part of an as-etched 5 × 8 GaAs(001) nanomesa array and (b) 45 ˚ tilted view of a typical individual as-etched nanomesa inside the array. 51 Figure 2.14 Top view SEM image of the mesa profile after 700ML growth of mesa with as-etched sizes (a) 3842nm, (b) 2850nm, (c) 1885nm and (d) 780nm and 30° tilted SEM image of the mesa profile of mesas sizes (e) 1885nm and (f) 780nm from sample RG111105. 54 Figure 2.15 Top view SEM image of the growth front profile of mesas with as-etched sizes of (a) 550nm, (b) 480nm, (c) 430nm mesa after 417ML of growth on top. Panel (d) shows a cross-section (87°) tilted SEM image of the growth front profile of a 310nm as-etched mesa projected along [1 -1 0] direction. All these results are from sample RG130321. The dashed white line marked on panel (d) represents the angle between side facet and (001) surface. 57 Figure 2.16 Top view SEM image of the growth front profile of mesas with as-etched sizes of (a) 530nm, (b) 430nm after 529ML amount of growth on top. Panel (c) and (d) shows a cross-section (87°) tilted SEM image of the growth front profile of the mesa shown in panel (a) and (c) projected along [1 -1 0] direction respectively. All these results are from xi sample RG130629. The dashed white line marked on panel (d) represents the angle between side facet and (001) surface. 60 Figure 2.17 Panel (a) shows the TEM imaging geometry. Panel (b) shows the cross-section g(002) dark field image of the mesa with as-etched size of 400nm with {101} plane dominated mesa-top pinch-off after 529ML material growth taken with experiment geometry shown in panel (a). The {101} plane dominated pinch-off occurs after the first {103} plane (marked with white dashed line) dominated pinch-off. The dark white fringes arise from strains in the mesa and thickness contrast. Panel (c) is the zoomed in image of the corner of the mesa, indicating the presence of side facets around corner. [Images courtesy of Zachary Lingley of our group] 62 Figure 2.18 Schematic of growth evolution on square mesa with edges along <100> direction under employed growth condition. 63 Figure 2.19 Schematic of (a) the rhombus (001) top surface before {103} pinch-off (b) the square (001) top surface before {101} pinch-off with the mirror plane and arsenic dangling orbital orientation along edges shown. 64 Figure 2.20 Cross-section dark field TEM image with g=(0,0,2) of a 264nm as-etched mesa from sample RG130321 after 417ML material growth. [Image courtesy of Zachary Lingley of our group] 67 Figure 2.21 Data on (a) the actual and (b) normalized thickness, normalized with respect to deposition amounts, of materials with different deposition amount grown on mesa top from four different mesa sizes obtained from TEM studies on nanomesas. The mesa of size 264nm, 300nm, and 392nm are from sample RG130321 while the mesa of size 780nm is from sample RG111105. 68 Figure 2.22 Schematic of the two phases of (001) top surface reduction with (001) top surface of (a) near square shape and (b) rhombus shape. Angle θ and α used in defining the projected area of {101} and top (001) facets are labeled in red where θ=15°±2° and α~22°±4°. 72 Figure 2.23 Fitted results (dark line) of the measured thickness of material (dark dot) grown on (a) 300nm and (b) 780nm mesa top as a function of deposition using analytical model captured in Eq.2.7. (c) atom migration rate from {101} facet to {103} facet (black dot) and from {103} facet to (001) facet (red dot) obtained from the fitted results on four different mesas of size 264nm, 300nm, 392nm and 780nm. 74 xii Figure 2.24 Schematic of the grown structure (panel a) on mesa of size 125nm with targeted InAs QD bounded with {101} side walls (panel b) on the mesa top of sample RG130625. 79 Figure 2.25 PL collected at 77K using micro-PL setup from (a) the planar region and (b) a single mesa of size of 125nm from sample RG130625 containing, respectively, InAs SAQDs and mesa-top single quantum dot. Excitation is CW at 780nm with power 375nW (power density ~30W/cm 2 ) and spectral resolution of 1nm. Panel (c) shows the power dependence of the observed 1122 nm PL peak intensity at 77.4 K. 82 Figure 2.26 Panels (a) and (b) show AFM images of the 4.25ML (panel (a)) and 4.5ML (panel (b)) In 0.5 Ga 0.5 As SAQDs. A low density of <5/μm 2 in the 4.25ML In 0.5 Ga 0.5 As SAQDs sample is seen. Panel (c) shows the large area PL spectra collected at 77K with 863nm excitation (excitation power density ~17W/cm 2 ) from three In 0.5 Ga 0.5 As SAQD samples with 4.25ML (RG130307-3-F2), 4.5ML (RG130215-2-F2) and 6.0ML (RG121115-2-F2) deposition. The PL intensity ratio of emission from 4.25ML and 4.5ML In 0.5 Ga 0.5 As SAQDs. 85 Figure 2.27 (a) Schematic of the grown structure (panel a) on mesa of size 430nm with the targeted 4.25ML In 0.5 Ga 0.5 As QD bounded with {103} side walls on the mesa top of sample RG130916. (b) Schematic of the formed QD size and shape bounded with {103} side walls (upper one). The QD base shape and edge orientation is shown in the lower panel. 87 Figure 2.28 A (a) top view and (b) a 60° tilted SEM image of the SESRE- grown 4.25ML In 0.5 Ga 0.5 As MTSQD array with the MTSQD residing on the top of each nanomesa. The inset is a magnified image of a MTSQD- bearing single nanomesa (scale bar of 300 nm). Panel (c) shows the schematic of the cross-section of the mesa along (1-10) facet indicating the geometry and position of QD (in red) on mesa top. The dot dashed black line represents the growth front of the capping layer on top. 88 Figure 3.1 Schematic of the band structure near k=0 in a direct bandgap semiconductor of zinc-blende structure: (a) without strain. The blue line depicts the lowest conduction band while the red and magenta lines depict the lowest HH and LH valance band, respectively. The purple line represents the spin orbit split off band. Panel (b) shows the conduction, HH, and LH valance bands of the semiconductor under biaxial compressive strain. 104 Figure 3.2 (a) Schematic of a truncated pyramidal MTSQD structure. The horizontal axis marked as z in the figure is the growth direction of the xiii mesa-top quantum dots grown and studied in this work. (b) Schematic of the change in the crystal potential across materials, aligned energetically assuming a common (flat) vacuum level (E vac ) as the zero of energy. Under such an assumption, shown also is the alignment of the bulk conduction (E CB (k)) and valance (E VB (k)) bands of semiconductors A and B. The bulk band states of the QD region (red) in the energy range shown find no matching states in the band gap of the barrier region (blue) and become quantized for dimensions smaller than the bulk de Broglie wavelength. What at the atomic level is a change in the crystal potential across material interfaces can, in a simplified continuum solid approximation, be modelled by the discontinuity arising from the alignment of the Brillouin zone center conduction (E CB ( Γ )) and valence (E VB ( Γ )) band edges of the QD (A) and barrier (B). 108 Figure 3.3 Schematic of confinement potential arising from the alignment of the measured zone center conduction and valence band edges of the QD (superscript A) and barrier (superscript B) bulk materials. The confinement potentials for electron and hole are marked with black double arrows. With the QD size smaller than the bulk de Broglie wavelength, the electron and hole state energies become discrete (black lines). 112 Figure 3.4 Schematic of the optical transition between electron (e) and hole (HH and LH) states with the transition polarization and transition strength labeled. The electron and the hole state is labeled by the projected value of total angular momentum, M j . All allowed optical transitions between different electron and hole states are marked with arrows. The arrows marked with same color have the same emitted light polarization property (i.e. circular polarized (σ ± ) or linear polarized (π z )). The number listed besides the arrow indicates the relative transition strength. 120 Figure 3.5 (a) Schematic of QD envelop function and Bloch function symmetry for allowed optical transition. Panel (b) and (c) shows the schematic energy diagram of the HH and LH related excitons in QDs with D 2d (panel (b)) and C 2v (panel (c)) symmetry of the confinement potential. The energy separation in between different exciton transitions is only shown as schematic representation without imposed ordering and energy separation amount. The exciton states and the polarization of the transition (in parenthesis) are labeled in the schematic drawing. 129 Figure 3.6 Schematic of electron and hole configuration and corresponding emission polarization properties of (a) trions and (b) biexcitons in QDs with D 2d and C 2v potential symmetry. 135 xiv Figure 3.7 Schematic showing the 3-step model of the PL process: (1) photon generation, (2) carrier relaxation, (3) radiative recombination. 137 Figure 3.8 Schematic drawing of micro-PL setup. The red line with arrows represents the MTSQD emission detection light path. The Blue line with arrows and the blue excitation fibers represents the excitation light path used for PL/TRPL measurement. The purple line with arrows and the purple dash curved line (fiber) represent the light path for PLE measurement. The black line and black dash line represent the electrical control and processing signal path for the setup. 144 Figure 3.9 Measured instrument response function (IRF) of the TRPL setup. The FWHM, rise time and decay time of the IRF are, respectively, 200ps, 150ps and 190ps. 147 Figure 3.10 (a) Size distribution of the uncapped 4.25ML In 0.5 Ga 0.5 As SAQDs sample (RG130307) showing a bimodal size distribution with one around base ~45nm, height ~3nm and the other around base ~60nm, height ~5nm from AFM data (b) PL data of the planar region of the 4.25ML In 0.5 Ga 0.5 As MTSQD sample (RG130916) collected with 780nm, CW excitation and spectra resolution of 2.5nm (3.5meV) at 77K. The excitation power is 591.3nW on mesa (power density ~47.1W/cm 2 ). (c) The plot of the summation of the four PL spectra collected at four different planar regions of the 4.25ML In 0.5 Ga 0.5 As MTSQD sample (RG130916). The red line shows the fit of two Gaussian peaks corresponding to the two observed bimodal QD size distribution in panel (a). 149 Figure 3.11 (a) Schematic of measurement geometry of the MTSQD sample. (b) PL data collected on MTSQD (3,5) with varying beam spot and mesa separation D (excitation beam moving along [110] direction). The QD is collected at 77.4K with 780nm CW excitation power 235nW (power density ~18.8W/cm 2 ) and spectral resolution of 1nm. (c) 930nm PL peak intensity as a function of D with excitation beam moving along [110] (black dot and line) and [1-10] (red dot and line) direction. 151 Figure 3.12 (a) Power dependent PL from MTSQD (3,5) collected at 77.4 K with spectra resolution of 0.2nm and three different excitation power at ~4 nW (~0.32 W/cm 2 , ~160 photon per pulse), ~10.5nW (0.87W/cm 2 , ~420 photon per pulse) and ~25.8nW (2.06W/cm 2 , ~1030 photon per pulse) from 640nm 80MHz pulsed laser. (b) Power dependence of the MTSQD’s PL peak X 1 integrated intensity at 77.4 K. A fit (black line) to the data shows I~P 0.93 . The peak integrated intensity saturates around xv 18nW (1.44W/cm 2 , ~720 photon per pulse). (c) Schematic of the electron and hole configuration in the QD for the single neutral exciton. 154 Figure 3.13 Intensity of the X 1 peak of MTSQD (3,5) at 929.2nm as a function of D for excitation beam moving along (a) [110] and (b) [1-10] direction. Measurements were done at 77K with spectral resolution of 0.2nm and excitation power ~15.8nW (power density ~1.26W/cm 2 , ~630 photon per pulse). 156 Figure 3.14 Shift in the energy of PL peak X 1 of MTSQD (3,5) with temperature from 77.4K to 150K. The shift follows closely the calculated (see text) temperature dependence of the In 0.5 Ga 0.5 As bandgap change (black line). 157 Figure 3.15 (a) the single exciton PL peak wavelength of each MTSQD in the 5×8 array shown as color coded blocks. The two non-emitting MTSQDs are marked as black blocks with white outlines. Pairs of like- color circles identify MTSQDs with emission within the instrument resolution limit of 0.2nm. (b) PL spectra of MTSQD (3,5), (2,4) and (4,4) collected at 77K with spectral resolution of 0.2nm and excitation power P/P sat ~23% under 640nm 80MHz excitation. The primary peaks shown are the single exciton transition from MTSQDs. (c) Histogram of PL peak wavelengths from the 38 emitting MTSQDs in the 5×8 array covering an area of 1000 μm 2 . The standard deviation of the wavelength is 8.3 nm. The red line represents the Gaussian fit to the distribution. 159 Figure 3.16 The normalized PL spectra of the MTSQD (3,5) (black dot and line) and the MTSQD (3,1) (red dot and line), the two MTSQDs marked by the red circles in Fig. 3.14(b), collected at (a) 77K and (b) 8K with spectral resolution of 0.2nm with 640nm, 80MHz excitation at excitation power P/P sat ~23%. The normalized PL is obtained by scaling the spectrum with respect to the peak value. The difference between the single exciton emission wavelengths of the two shown QDs is limited by the spectral resolution of the optical setup. 161 Figure 3.17 (a) PL peak intensity of each MTSQD in the 5×8 array shown as color coded blocks. The two non-emitting MTSQDs are marked as black blocks with white outlines. Pairs of like-color circles identify MTSQDs with emission within the instrument resolution limit of 0.2nm as in Fig. 3.14 (b). (b) The saturation power of MTSQD single exciton transition obtained from 8 MTSQDs at 77K with 640nm, 80MHz excitation. The saturation power is ~28.9±11.8 nW (~1150 photons per pulse) for the shown MTSQDs. 164 xvi Figure 3.18 PL peak intensity of MTSQD (3,5)’s emission peak X 2 (at 927.4nm), peak X 3 (at 931.4nm), and peak X 4 (at 933.4nm) as a function of D with excitation beam moving along (a) [110] and (b) [1-10] direction measured at 77K with spectral resolution of 0.2nm and excitation power ~15.8nW (power density ~1.26W/cm 2 , ~630 photons per pulse). 166 Figure 3.19 Power dependent PL from (a) MTSQD (2,4) and (b) MTSQD (4,4) collected at 77.4 K with spectral resolution of 0.2nm at three different excitation powers at ~ P/P sat ~23%, P/P sat ~60% and P/P sat ~200% from a 640nm 80MHz pulsed laser. The saturation power, P sat , for MTSQD (2,4) and MTSQD (4,4) are respectively 20nW (~1.6W/cm 2 , ~800 photons per pulse) and 16nW (~1.28W/cm 2 ,~640 photons per pulse). 167 Figure 3.20 (a) Multi-peak Gaussian fitting (red line) of the PL spectra of MTSQD (3,5) at excitation power of 4nW, 10.5nW and 25.8nW with each fitted individual Gaussian peak shown in green lines. (b) Power dependence of the MTSQD’s PL peak X 1 (black dot), X 2 (blue dot), X 3 (red dot) and X 4 (magenta dot) integrated intensity at 77.4 K obtained from multi-peak Gaussian fitting shown in panel (a). The fit of peak X 1 , X 2 , X 3 and X 4 shows I~P 0.93 (black line), I~P 1.19 (blue line), I~P 1.35 (red line) and I~P 2.27 (magenta line) respectively. 172 Figure 3.21 The PL peak wavelength (black dot line) and FWHM (full- width at half maximum, red dot line) of peak X 1 (bottom panel), X 2 (middle panel) and X 3 (upper panel) from MTSQD (3,5) as a function of excitation power. All data are taken at 77K. 175 Figure 3.22 (a) PL spectra from MTSQD (3,5) collected at 77K with excitation power at 4nW (640nm 80MHz excitation, black line) and at 37.5nW (850nm 76MHz excitation, red line) shown in the lower panel and the PL collected with excitation power 37.5nW (640nm excitation, black line) and at 337nW (850nm excitation, red line) shown in the upper panel. (b) The ratio of the integrated PL intensity of peak X 2 and X 1 as a function of P/P sat with 640nm (black dot line) and 850 (red dot line) excitation where P sat is the saturation power for peak X 1 . 177 Figure 3.23 (a) A schematic and (b) detailed instrumentation of the micro- PL setup with the added linear polarizer (marked in red) that can be rotated in the x-y plane. 180 Figure 3.24 The schematic (left most image) of the MTSQD sample with patterned mesa arrays labeled Si, i=0 to15, and the triangular markers on the sample to indicate cryptographic orientation. The MTSQD array under optical investigation in this dissertation (top middle image) is from the xvii S11 region. A detailed geometry of a mesa after GaAs buffer (just before InGaAs) growth is shown in the lower middle schematic image. The schematic of the mesa geometry and QD base shape follows the obtained SEM image of the mesa before the {103} plane dominated pinch-off. 181 Figure 3.25 Polar plot of the polarization dependent PL peak intensity (black dot) of (a) peak X 1 and (b) peak X 4 collected at 77K with non- resonant excitation (640nm, 80MHz, P=10.5nW, power density 0.87W/cm 2 ) and spectral resolution of 0.2nm. The black line represents the fit of the measured data using Eq. 3.50 with 0.125   for peak X 1 and 0.108   for peak X 4 . The base shape of the QD (red line) is captured here for easy reference to the QD structure. The QD edge [3 -1 0] marked as red dotted line is 22±4° (angle α) with respect to the [1 -1 0] crystallographic direction. 183 Figure 3.26 Schematic of the GaAs nanomesa (solid black lines) enclosing the quantum dot (red) used in finite element based simulations. Panels (a) and (b) show, respectively, the 60° tilted and top view of the nanomesa. Panel (c) captures the geometry of the calculation. The blue circle, ~3 μm away from the point dipole, represents the integration surface for the calculation of the overall Poynting vector. The red circle represents the instrument objective lens collection cone. 188 Figure 3.27 Simulated results showing the polar plot of the polarization dependent collection efficiency of the emitted photons into the collection cone of the optical microscope for a dipole in the x-y plane with equal magnitude of P x and P y component but with (a) π/2 phase difference (resulting in a TM 11 type radiation) and (b) –π/2 phase difference (resulting in a TM 1-1 type radiation) mimicking respectively the exciton state |1, +1> and |1, -1> . Panel (c) shows the summation of the two transitions in (a) and (b). 190 Figure 3.28 (a) PL of MTSQD (3,5) with polarizer aligned at 30° (black line) and 300° (red line) with respect to the [1-10] direction in the x-y plane and (b) the Polar plot of the polarization dependent PL peak intensity (black dot) of peak X 1 * collected at 77K with non-resonant excitation (640nm, 80MHz, P=10.5nW, power density 0.87W/cm 2 ) and spectral resolution of 0.2nm. The black line represents the fit of the measured data using Eq. 3.50 with 0.11   . 193 Figure 3.29 Polar plot of the polarization dependent PL peak intensity (black dot) of (a) peak X 2 and (b) peak X 3 collected at 77K with non- resonant excitation (640nm, 80MHz, P=10.5nW, power density 0.87W/cm 2 ) and spectral resolution of 0.2nm. The black line represents xviii the fit of the measured data using Eq. 3.50 with 0.08   and 0.04   for peak X 2 and X 3 respectively. The base shape of the QD (red line) is capture here for easy reference to the QD structure. The QD edge [3 -1 0] marked as red dotted line is 22±4° (angle α) with respect to the [1 -1 0] crystallographic direction. 195 Figure 3.30 TRPL of MTSQD (3,5) peak X 1 measured at (a) 77K and (c) 8K with spectral resolution of 0.4nm. Data shown in panel (a) are collected with 640n 80MHz excitation and excitation power P=4nW, power density 0.32W/cm 2 Data in panel (c) are collected with 780nm,76MHz excitation and excitation power P~12nW, power density 1.28W/cm 2 . Panel (b) shows the PL decay time of peak X 1 measured at excitation powers from 23% to 215% of the saturation power of peak X 1 . 200 Figure 3.31 The decay time of the X 1 peak obtained from the TRPL spectra of different MTSQDs studied at 77K with spectral resolution of 0.4nm and low excitation power P~30% of the saturation power of the transition of each MTSQD. The optical excitation used is the 640nm 80MHz excitation. The peak X 1 is identified with the neutral exciton (X 0 ) decay (see text below). 202 Figure 3.32 The measured TRPL spectra of (a) neutral exciton X 0 , (b) positively charged trion X + , (c) negatively charged trion X - and (d) biexciton XX from MTSQD (3,5) marked as open black circle collected at 77K with spectra resolution of 0.2nm under optical excitation power P~30nW (2.4W/cm 2 , P/P sat =60% ). The red lines show the reconvoluted fitted curve of the data. 203 Figure 3.33 Scheme of energy levels and optical transitions considered in the model. The parameters α ( t ), β ( t ) and x(t) are the time dependent capture rates of electron, hole and exciton respectively. Constants x t , cx t and xx t are the decay time of respective states. The probability of finding the QD in i th state at time t is i p . 206 Figure 3.34 Shows the five level rate equation model based fitting (solid lines) to the measured data (same as in Fig.3.20(b)) of neutral exciton X 0 decay peak X 1 (black dots), negatively charged exaction X - decay peak X 3 (red dots) and biexciton XX decay peak X 4 (magenta dots) as a function of power. 209 Figure 3.35 Schematic showing the PLE process in QDs with tunable excitation across QD excited electron and hole states and detection at QD ground state transition. 210 xix Figure 3.36 PLE data from MTSQD (3,5) neutral exciton X 0 (peak X 1 ) collected at 77K with spectral resolution of 0.2nm as a function of the energy difference between excitation and detected photon energy at two different detection wavelength: (a) peak value of peak X 1 at 929.2nm and (b) higher energy end of peak X 1 at 928.6nm. 212 Figure 3.37 (a) Power dependent PL data on MTSQD (3,5) collected at 77K with spectral resolution of 1nm and excitation power from 128nW to 1341.3nW (780nm, CW excitation) (b) Temperature dependence of this MTSQD’s integrated PL intensity from 77.4 K to 150 K. The fit (black line) to the data reveals an exponential dependence on inverse temperature with an activation energy of 40±2 meV representing carrier (electron) escape from the first excited electron state of the QD. 215 Figure 3.38 The PLE data from MTSQD (3,5) peak X 1 collected at 90K with spectral resolution of 0.2nm as a function of the energy difference between excitation and detected photon energy at peak X 1 . 217 Figure 3.39 Temperature dependence of MTSQD (2,4)’s ground state neutral exciton (peak X 1 at 926.2 in Fig. 3.14(a)) integrated PL intensity from 77.4 K to 120 K. The fit (black line) to the data reveals an exponential dependence on inverse temperature with an activation energy of 40.5±2 meV representing carrier escape from the QD. 218 Figure 3.40 The schematic of the MTSQD geometry (from Chapter 2) and the schematic of QD electronic structure based on the PL/PLE spectroscopies results for the synthesized MTSQDs. 219 Figure 4.1 Photon number probability distribution for single mode emission of (a) a Fock state, (b) a coherent state, and (c) a thermal state for two different mean photon numbers <n>=1 (lower panel) and 10 (upper panel) respectively . 237 Figure 4.2 The schematic of HBT setup with a 50/50 beam splitter and two detectors on the two output arm of beam splitter. 243 Figure 4.3 The schematic of the photon interference at a beam splitter with incident light of mode represented by a and b from two input arms of the beam splitter. The mode of outgoing light is represented by c and d. 244 xx Figure 4.4 Calculated second order correlation function (2) () g  plotted as a function of dimensionless number   for cases (a) 1 20 R    and (b) 20 R    . 255 Figure 4.5 Schematic energy-level diagram for the cascaded decay of a biexciton from a QD with C 2v confinement potential symmetry. 257 Figure 4.6 The schematic of the HBT setup used for measuring second order correlation function of emitted photons from individual MTSQDs. 262 Figure 4.7 A photon correlation histogram of 640nm 80MHz pulsed laser light. The numbers above the peak indicate the normalized peak areas with respect to the average peak areas of the shown peaks. 264 Figure 4.8 Color coded plot of (a) PL peak intensity and (b) PL peak wavelength of each MTSQD in the 5×8 array shown as blocks. The two non-emitting MTSQDs are marked as black blocks with white outlines. MTSQDs marked by green circles are those examined for g (2) (τ). 272 Figure 4.9 Single Photon emission from MTSQDs. The 8 K as-measured coincidence count histogram of (a) MTSQD (3,5), (b) MTSQD (2,4) and (c) MTSQD (4,4) measured with 640nm, 80MHz laser at excitation power / ~ 45% sat PP . The intensity autocorrelation g (2) (0) values shown are extracted from the as-measured data and the values in parenthesis are extracted from data after detector dark count subtraction. 273 Figure 4.10 Single Photon emission from MTSQDs. The 77 K as- measured coincidence count histogram of (a) MTSQD (3,5), (b) MTSQD (2,4), (c) MTSQD (4,4) and (d) MTSQD (3,8) measured with 640nm, 80MHZ laser at excitation power / ~ 23% sat PP . The intensity autocorrelation g (2) (0) values shown are extracted from the as-measured data and the values in parenthesis are extracted from data after detector dark count subtraction. 275 Figure 4.11 MTSQD lifetime extracted from photon correlation histogram data at (a) 77K and (b) 8K. The numbers in the parenthesis indicate the MTSQD location in the 5 × 8 array. 277 Figure 4.12 The 3D polar plot of the calculated Poynting vector (panel b) of the electromagnetic field energy flux from the QD with QD represented as in-plane dipole with dipole moment of 1 Debye strength along [1-10]. The schematic representation of the calculation geometry is shown in xxi panel (a). The corresponding ratio of emitted photons from QD into the collection cone (red circle) of the objective and the overall emitted photon (integrated over blue circle) is shown in panel (c) as a function of QD emission wavelength. 281 Figure 4.13 Shows the interacting dielectric building blocks (spheres) based structure whose collective single Mie resonance (here the magnetic dipole mode at 980nm arising from GaAs spheres) provides the shown spatial distribution (lower three panels) of the E y component in the x-y plane as a function of position along the x axis. The directivity induced by the nanoantenna effect is unmistakable in the second panel. The lossless propagation is manifest in the constant electric field in the propagation region. Not visible is the enhancement of the Purcell factor at the SQD (pyramid) location by ~7. Note that all three spatial-region-dependent functions arise from a single (collective) mode of the structure made of the interacting dielectric building blocks which we therefore dub light manipulation unit (LMU). 283 Figure 4.14 The 77 K as-measured coincidence count histogram of MTSQD (3,5), measured with 850nm, 76MHz laser with excitation power / ~ 23% sat PP . The intensity autocorrelation g (2) (0) values shown are extracted from the as-measured data and the values in parenthesis are extracted from data after detector dark count subtraction. 287 Figure 4.15 Photoluminescence (PL) spectra of (a) MTSQD (3,5) , (b) MTSQD (2,4) and (c) MTSQD (4,4) collected at 8 K with 0.25 nm spectral resolution and ~5 nW excitation power (power density ~0.4W/cm 2 , ~23% of P sat ). Black dots represent the raw data and the green line shows a fit with two individual Lorentzian peaks ( 1 and 2 ) shown as the red curves. 288 Figure 4.16 Excitation power dependent integrated PL intensity of the two peaks (a) 1 and (b) 2 from MTSQDs (3,5) collected with 640nm, 80MHz excitation. The black line is a fit of measured data showing (a) I 1 ~P 0.94 and (b) I 2 ~P 0.75 . Panel (c) shows the ratio of the integrated area intensity of 1 and 2 . 292 Figure 4.17 Measured g (2) (0) dependence on excitation power for MTSQD (3,5) under 640nm 80MHz excitation with detection window of 0.4nm. The red solid line shows the fitted results based on Eq. 4.99. 295 Figure 4.18 Energy levels and transitions for the model described in the text. 296 xxii Figure 5.1 Schematic of the envisioned paradigm for realizing photonic quantum optical circuits based on an array of spatially ordered mesa-top quantum dots (pyramids) on-chip integrated with lossless dielectric building block (DBB) based light manipulating unit (LMU; blue elements) that provide multiple functions (resonant cavity/nanoantenna/waveguide) and beam splitters to direct light to detectors (purple elements). 305 Figure 5.2 Schematic of the planarization of the MTSQD surface morphology through the growth of an overlayer on the MTSQD (pyramids) array (panel a) resulting in the buried ordered array of MTSQDs (panel b) around which light manipulating units can be lithographically created within the designed footprint (broken white lines). 319 Figure 5.3 Schematic of the simplest MTSQD-DBB integrated unit structure to be fabricated and studied towards the goal of optical circuits in Fig.5.1. 320 xxiii List of Tables Table 2.1 List of samples representing growth on <100> edge oriented square mesas 52 Table 3.1 Notation and zone center Bloch function of “most significant” bands. 105 xxiv Abbreviations AFM: Atomic Force Microscope APD: Avalanche Photodiode CB: Conduction Band CCD: Charge-Coupled Device CW: Continuous-Wave DBB: Dielectric Building Block DOS: Density Of States ECR: Electron Cyclotron Resonance EPS: Entangled Photon Source FIB: Focused Ion Beam FWHM: Full Width At Half Maximum HBT: Hanbury-Brown-Twiss HH: Heavy Hole IRF: Instrument Response Function LH: Light Hole LME: Light Manipulating Elements LMU: Light Manipulating Unit LO: Longitude Optical LPE: Liquid Phase Epitaxy MBE: Molecular Beam Epitaxy ML: Monolayer MOCVD: Metal Organic Chemical Vapor Deposition xxv MTSQD: Mesa Top Single Quantum Dot NIM: Nuclear Instrumentation Module PECVD: Plasma Enhanced Chemical Vapor Deposition PhC: Photonic Crystal PL: Photoluminescence PLE: Photoluminescence Excitation QD: Quantum Dot QIP: Quantum Information Processing RHEED: Reflection High Energy Electron Diffraction SAQD: Self-assembled Quantum Dot SEM: Scanning Electron Microscope SESRE: Substrate-encoded Size-reducing Epitaxy SO: Spin Orbit Split-Off SPS: Single Photon Source SQD: Single Quantum Dot SSMBE: Solid Source Molecular Beam Epitaxy STM: Scanning Tunneling Microscope TAC: Time-To-Amplitude TEM: Transmission electron microscope TRPL: Time Resolved Photoluminescence TTL: Transistor-Transistor Logic UHV: Ultra-High Vacuum UV: Ultra Violet xxvi VB: Valance Band VLS: Vapor Liquid Solid VPE: Vapor Phase Epitaxy xxvii Abstract This dissertation contributes to the subject of realizing on-chip integrated nanophotonic systems comprising light source, light manipulating elements (LMEs such as cavity, waveguide, etc.), and detectors operating down to a single photon level for quantum information processing. We envision a scalable architecture for building such information processing circuits with ordered single quantum dot based single photon and entangled photon source integrated with either conventional 2D photonic crystal or, as we propose, dielectric building block (DBB) based multifunctional light manipulation unit (LMU). The dissertation covers several different issues that underlie the challenge of realizing co-designed scalable, ordered, and spectrally uniform single photon source (SPS) and entangled photon source (EPS) arrays integrable with LMEs. Specifically, it addresses: (1) The spatially selective synthesis of ordered and spectrally uniform on-chip integrable single quantum dot (SQD) arrays with controlled size and shape; (2) The excitonic optical emissions and implications for the underlying electronic structure of the synthesized SQD array as probed through systematic photoluminescence (PL), PL excitation (PLE) and time-resolved PL studies to extract quantum confinement-related and carrier relaxation dynamics-related parameters to assess the application potential of SQD array as SPS and EPS; (3) The single photon emission properties and processes affecting the purity of single photon emission in the synthesized SQD arrays. xxviii We employed, for the first time, the substrate-encoded size-reducing (SESRE) approach that exploits growth on patterned nanoscale mesas designed to induce surface-curvature stress gradients that, during growth, preferentially direct atom migration from mesa sidewalls to the top to form shape and size controlled SQD selectively on mesa tops only. The evolution of the mesa top size reducing growth and the formation of the shape controlled mesa-top SQD (MTSQD) is established. Two types of MTSQDs can be realized utilizing SESRE on <100> edge oriented square nanomesas on GaAs(001) surface: MTSQD with {101} facets and MTSQD with {103} facets. Ordered 5 × 8 array of In 0.5 Ga 0.5 As MTSQDs bounded by {103} side wall with base length ~13nm and height ~3nm is synthesized using MBE technique whose optical property is studied in detail in this dissertation. To study the optical properties of MTSQDs and assess their potential use as SPS and EPS, a unique optical system was set up which enables time-integrated PL, PLE, TRPL and photon correlation measurements of each individual MTSQD in the array. The setup is applied to examine the excitonic complexes, underlying electronic structure, and the single photon emission properties of the MTSQD array. The synthesized MTSQD arrays are found to have high optical emission yield with near unity quantum efficiency. Highly efficient emission from single neutral exciton decay with spectral uniformity of ~8nm, significantly better than the typical randomly formed self-assembled island quantum dots in prevalent use, has been demonstrated. Strikingly, in the array there are several pairs of MTSQDs with emission wavelengths within the instrument resolution of 0.2nm. xxix From the systematic PL/PLE/TRPL studies, the MTSQDs are found to have comparable single carrier and exciton capture rates that result in the observed emission of the negatively charged trion as well as biexciton emission, in addition to the ground state neutral exciton transition. The ground state neutral exciton is found to be well confined (the first excited state is separated by ~40meV) and exhibiting decay lifetime of ~1ns, suitable for acting as SPSs. Single photon emission from the MTSQD’s neutral exciton decay is studied through photon intensity correlation measurements. Single photon emission purity ~90% at 8K and purity ~81% at 77K is obtained and found to be limited by, (1) the fluctuating charges experienced by the QD (2) the 0.2nm (~300μeV) instrument resolution that prevents the collection of only single excitonic state emission from the MTSQD. From polarization dependent PL studies, the MTSQD’s confinement potential is found to be likely of symmetry lower than C 2v and ground state emission involving hole states with mixed heavy hole and light hole character. The fine structure splitting between “bright” excitons is found to be likely <150 μeV, below the resolution of the setup. The emitting “dark” excitons are probably ~300 μeV and ~800 μeV away from the “bright” exciton. The collection of the emission from excitonic complexes within the instrument limited detection window limits the purity of the single photon emission from these MTSQDs. The findings noted above demonstrate a promising approach to the realization of scalable spatially ordered and spectrally uniform MTSQD based SPS xxx array with high single photon emission purity up to 77K. Through the growth of a planarizing overlayer these MTSQDs can be readily integrated with lithographically fabricated DBB based LMUs to create optical circuits envisioned for information processing. These findings make a compelling case for pursuing investigations of the full potential of this class of SQD arrays for building quantum optical circuits. 1 Chapter 1. Introduction §1.1 Motivation and Objective Realization of information processing systems using the laws of quantum mechanics, dubbed quantum information processing (QIP), to go beyond the continual incremental improvements in classical computing and optical communication technologies has been under considerable investigation in the last couple of decades led by the significant theoretical developments in quantum information science, specifically in quantum computation [1.1], quantum cryptography and communication [1.2, 1.3]. It has been established that photons can act as the information carrier in QIP systems utilizing different degrees of freedom such as photon number and polarization [1.4-1.6]. On-chip integrated nanophotonic optical information processing systems using many, few, and single photons are, therefore, a promising practical way of realizing QIP systems for conducting quantum computation [1.4], quantum cryptography [1.5] and quantum communication [1.6]. At the core of such systems is the on-demand single photon and entangled photon sources [1.4, 1.7], co-designed and integrated with light manipulating passive elements (LME) and detectors in on-chip scalable architectures [1.8-1.10]. To this end, realizing spatially ordered and spectrally uniform single photon sources (SPSs) and entangled photon sources (EPSs) that can be co-designed and integrated on-chip with LMEs providing functions such as emission rate enhancement, emitted photon guidance and propagation, and photon interference is a 2 critical need and an important milestone to be reached for constructing on-chip nanophotonic QIP systems. Semiconductor single quantum dots (SQDs) offer an attractive implementation scheme for realizing on-demand SPSs and EPSs. Semiconductor SQDs are nanometer-sized three-dimensional structures of material A of sizes smaller than the de Broglie wavelength of electrons and holes in the material, embedded in a higher bandgap material B. The quantum confinement effect arising from the small size and a confinement potential arising from discontinuity in the bulk Brillouin zone center energy band states between the two semiconductor materials results in discrete single particle (electron and hole) energy levels as shown in Fig. 1.1 and a nearly δ-function form of discrete density of states (DOS). Figure 1.1 Schematic of confinement potential arising from the alignment of the Brillouin zone center conduction and valence band edges of the QD (A) and barrier (B). The discrete electron and hole energy states and the corresponding wavefunctions due to quantum confinement are schematically shown along with the discrete δ form of density of states. 3 Populating electron states and depopulating hole states (optically or electrically) enables electron-hole recombination transitions that produce photons and thus make SQDs act as SPSs [1.9,1.10]. Different exciton complexes formed in optical pumping of the SQDs, i.e. the exciton (single electron-hole pair, denoted |X>) [1.9, 1.10], trions (single electron-hole pair with one additional electron or hole, denoted |X ± >) [1.9, 1.11] and biexciton (two electron–hole pairs, denoted |XX>) [1.11, 1.12] can be utilized to emit single photons. Their size-dependent tailorability of energy differences and thus the emitted photon energy, stability with respect to the environment, and integrability with LMEs and detector makes semiconductor SQDs attractive for implementing SPSs over other systems with discrete energy states, such as atoms [1.13], ions [1.14], molecules [1.15] and deep level defect color centers [1.16]. More importantly, semiconductor SQDs can be utilized to generate polarization-entangled photons through the sequential recombination occurring in their excitonic complexes, specifically in the biexciton-exciton cascade [1.12, 1.17]. After optical excitation of two electron–hole pairs (shown in Fig. 1.2(b)) in a QD, the biexcitons are created with energy 00 2 2 4 ( , ) ( , ) ( , ) xx e h coulomb coulomb coulomb E E E E e h E e e E h h      (Eq. 1.1) where 0 e E and 0 h E are the single particle energy of the lowest electron and hole state of the QD, respectively. ( , ) coulomb E e h , ( , ) coulomb E e e and ( , ) coulomb E h h are the Coulomb interaction between electron-hole, electron-electron and hole-hole. The biexciton decays via a two-photon cascade process through intermediate states (excitons shown in Fig.1.2(a) with energy E x ) emitting photons of two different wavelengths λ 1 4 and λ 2 owing to difference in Coulomb interaction energy in |XX> and |X> states as given by Eq. 1.2 and Eq. 1.3 below: 00 1 ( , ) x e h coulomb h E E E E e h      (Eq. 1.2) 2 xx x h E E  (Eq.1.3) The biexciton-exciton two-photon cascade process is schematically captured in Fig. 1.2 (c). For the III-V compound semiconductor based SQDs of interest here, the underlying atomic s and p orbitals of the constituent atoms results in the top of the bulk solid valence band at the Brillouin zone center to be of p-character and doubly degenerate with a heavy and a light hole band. Upon quantum confinement in the QD, the heavy and light hole bands split with the lowest confined state being the heavy hole derived with a total angular momentum J=L+S =3/2 and the next one being the J=1/2 light hole derived state. Accounting for total angular momentum projection, conventionally along the growth (vertical) direction taken as the z- direction, the lowest confined hole state in the QD is two-fold degenerate (J z = 3/2, - 3/2) with dominantly heavy hole character. The lowest confined electron state is derived from the s-orbital dominated bulk conduction band that is two-fold degenerate. For dipole driven transitions from such a confined electron state to the lowest hole states, angular moment selection rules dictate that only |1/2, -3/2> and |- 1/2, 3/2> combinations are allowed (here the first and second values represent the total angular momentum of the electron and the hole). When the splitting of these otherwise two degenerate single exciton states |X> arising from the electron – hole exchange interaction is smaller than the radiative linewidths, the two decay paths 5 will be indistinguishable and the two photons generated will be polarization- entangled. Figure 1.2 Panel (a) and (b) shows schematically the charge configuration of exciton and biexciton states of QDs, respectively. Panel (c) shows the schematic energy diagram of biexciton (|XX>), exciton (|X>) and ground state (|0>). The biexciton- exciton cascade emission of photons of wavelength λ 1 and λ 2 are polarization entangled. Fine structure splitting is excluded in this energy diagram. The compound semiconductor SQDs thus have the potential of generating on-demand single photons as well as entangled-photon pairs to meet the light source characteristics required for realizing on-chip integrated nanophotonic systems for QIP. The motivation and objective of this dissertation work is to synthesize and study the needed spatially ordered arrays of spectrally uniform SQDs that can act as SPSs as well as the EPSs for building scalable on-chip nanophotonic QIP systems. Although discussed in detail later in this dissertation, it is to be noted here that in spite of the nearly two decades of studies of SQDs as SPS, it is a surprising fact that the vast majority of studies have revolved around a SQD selected from a class of QDs that form at spatially random locations and are part of nearly random distribution of size, shape, and composition fluctuations, thus exhibiting significant 6 (~50meV) fluctuation in their spectral emission. These two features of this class of QDs, based upon spontaneous formation of defect-free nanoscale 3D islands driven by lattice mismatch induced strain energy minimization in strained heteroepitaxy and thus dubbed self-assembled quantum dots (SAQDs), render them unsuitable for constructing quantum optical circuits underlying QIP systems. Approaches to realizing spatially ordered placement of a few or a single QD have been pursued but largely in architectures that enable vertical (to the substrate) integration with LMEs such as cavity and waveguide but are essentially unsuited for creating interconnections (between these basic building units) needed for on-chip scalable quantum optical circuits. Hence the above stated motivation and objective of this dissertation. To achieve on-chip scalable architectures centered around SQDs in spatially ordered arrays, in this dissertation, we employ the substrate-encoded size-reducing (SESRE) approach [1.18] introduced by our group in 1990s and discussed in detail in Chapter 2. Briefly, it exploits growth on patterned nanoscale mesas designed to induce surface-curvature stress gradients that, during growth, preferentially direct atom migration from mesa sidewalls to the top to form shape and size controlled SQD selectively on mesa tops only. For the (001) surface oriented substrates of the tetrahedrally-bonded cubic semiconductors of groups IV, III-V, and II-VI, the <100> edge orientations of square mesas provide four-fold symmetry and thus potentially symmetric migration of adatoms from the all four sidewalls to the top to reduce, in- situ, the as-patterned mesa top size to the desired size utilizing homoepitaxy under 7 controlled growth kinetics [1.18-1.21], thus enabling the subsequent heteroexpitaxial growth of a SQD on top. The QD shape and size can be correspondingly fully controlled by controlling the evolution of the growth front on the mesa top [1.19, 1.21] thereby enabling highly spectrally uniform optical response from the synthesized QD arrays. We implemented the SESRE approach using molecular beam epitaxy (MBE) growth technique on nanolithographically patterned appropriately oriented mesa arrays and have demonstrated [1.22, 1.23]: (1) Realization of spatially ordered and spectrally highly uniform mesa-top SQD (MTSQD) arrays; (2) Highly efficient emission from excitonic complexes in such ordered MTSQDs with spectral uniformity 5 to 10 times better than the typical self- assembled quantum dots in prevalent use; (3) Single photon emission from such ordered and highly uniform SQD arrays up to 77K with ~90% purity at 8K. (4) And the potential use of such MTSQDs for entangled-photon pair generation. These findings are briefly summarized below. Figure 1.3 (a) shows a SEM (scanning electron microscope) image of part of the realized 5 × 8 array of the SESRE based ordered GaAs(001)/InGaAs mesa top SQDs (MTSQDs) synthesized using MBE. These ordered MTSQDs are highly spectrally uniform with σ λ (standard deviation) of 8nm, nearly an order of magnitude better than the standard self- 8 assembled QDs [1.24] or the solution-grown colloidal nanocrystal QDs [1.25]. Remarkably, as seen in Fig.1.3 (b) and Fig. 1.3 (c), several pairs (marked by like- color circles) of these MTSQDs emit at wavelengths within 0.2nm of each other, the measurement being limited by the instrument resolution. Such ordered and spectrally uniform MTSQDs act as efficient SPSs as exemplified by the measured photon count rate histogram of the second order intensity correlation function at zero time-delay (i.e. coincident), g (2) (0). A typical behavior, shown in Fig.1.3 (d), shows a g (2) (0)~0.15 at 8K [1.22], indicating nearly 90% purity of single photon emission from the MTSQD. Indeed, these MTSQDs act as SPSs up to 77K [1.22]. Besides single photon emission, the excitonic complexes i.e. neutral and charged exciton and biexciton in these MTSQDs, are examined through photoluminescence (PL), photoluminescence excitation (PLE) and time-resolved photoluminescence (TRPL) studies with 0.2nm instrument resolution for their intrinsic behavior which controls their potential use as sources for entangled photons and sources for creating photon interference from a single or two different MTSQDs, needed for building MTSQDs based nanophotonic QIP system. 9 Figure 1.3. Panel (a) shows a 60° tilted SEM image of a part of 5 × 8 array of single quantum dots on mesas tops. Inset is a magnified SEM image of a single MTSQD. Panel (b) shows the color map of spectral emission distribution from all 40 MTSQDs in the array. Pairs of like-color circles indicated the MTSQDs with emission within the instrument resolution limit of 0.2nm. Panel (c) shows PL from the pair of such MTSQDs marked with yellow circles. Panel (d) shows a typical coincident photon counting revealing a highly efficient single photon emission. 10 §1.2 MTSQD Arrays and On-chip integrated optical circuits To experimentally study entangled-photon emission from individual MTSQDs and interference of photons from a single or two different MTSQDs, the MTSQDs need to be integrated with LMEs such as cavities, waveguides and beam splitters. For realizing photon interference and photon entanglement, an important figure of merit is the wavepacket overlap (V) of the two photons emitted from the SQD with a time difference τ. The wavepacket overlap V is determined by the ratio of QD coherence time T 2 and decay time T 1 as given by [1.10], 2 00 2 1 00 ( ) ( ) 2 ( ) ( ) ( ) ( ) dt d a t a t T V T dt d a t a t a t a t                     (Eq. 1.4) The typical radiative lifetime of the MTSQDs, as determined from time- resolved photoluminescence (TRPL) is ~1ns [1.22,1.23]. Although the decoherence time of these MTSQDs has not yet been measured, it is expected to be comparable to the InGaAs self-assembled QDs with similar size and material composition, i.e. ~100ps to 200ps [1.26,1.27]. To obtain photon interference and photon entanglement, the wavepacket overlap of the two photons need to be maximized by bringing T 1 and T 2 to be comparable with each other. Therefore, MTSQDs need to be integrated with a cavity to enhance QD the emission rate by at least a factor of five through Purcell enhancement [1.10, 1.28]. Besides the physical requirement of cavities to shorten the MTSQD radiative lifetime, waveguide and beam splitters are needed to guide photon propagation on-chip towards desired location to create two-photon interference. All 11 these demand integrating MTSQDs with the LMEs that provide the above mentioned light manipulating functions. Unlike currently popular approaches to creating SQD arrays, the MTSQDs are naturally integrable with LMEs through an overgrowth of a morphologically planarizing overlayer following the QD formation as shown schematically in Fig. 1.4. Past the desired mesa pinch-off stage for the SQD formation, continued growth of an overlayer under suitably chosen growth kinetics optimized for balancing the competition between facet-dependent incorporation kinetics and interfacet migration kinetics (driven by the evolving facet area-dependent atom migration) can enable planarization of the SQD array surface morphology as depicted in Fig. 1.4 (a). The planarized ordered array of buried MTSQDs (Fig.1.4 (b)) lends itself naturally for integration with LMEs based upon the well-developed 2D photonic crystal (PhC) approach and fabricated using nanolithographic approaches (Fig.1.5, left panel). Figure 1.4. Schematic of the planarization of the MTSQD surface morphology through the growth of an overlayer on the MTSQD (pyramids) array (panel a) resulting in the buried ordered array of MTSQDs (panel b) around which light manipulating units can be lithographically created within the designed footprint (broken white lines). 12 The 2D photonic crystal approach exploits Bragg diffraction in perfect periodic bulk-like structures where the ability to derive desired functionality (such as high Q resonant cavity and waveguiding) is obtained by introducing departures from perfect crystallinity through controlled introduction of particular types of defects in the photonic lattice [1.10, 1.29]. The 2D photonic crystal approach, so far, has been implemented almost invariably around a single quantum dot picked from a random distribution of spontaneously formed 3D island self-assembled quantum dots of varying shapes, sizes, and composition [1.30-1.35] to create cavities, waveguides, etc.. But now with the buried MTSQD arrays it can be implemented on the area around the known positions of the MTSQD based SPSs marked with broken white line in Fig. 1.4(b) reproduced as the topmost image in Fig. 1.5. This opens for the first time the possibility of using the conventional 2D photonic crystal approach to create regular arrays of MTSQD-LME units and interconnect them to create the needed nanophotonic optical circuits for QIP. Figure 1.5 Schematic of the conventional 2D photonic crystal based (left panel) and the new paradigm of sub-wavelength sized dielectric building block (DBB, blue blocks in the right panel) based implementation of light manipulating functions such as SQD emission rate enhancement via a resonant cavity or nanoantenna- waveguiding structure, etc.. 13 Besides the use of departures from Bragg diffraction of light in photonic crystals, the buried MTSQD array also lends itself to integration with a new paradigm for implementing the needed light manipulating elements. This alternative approach exploits the physics of the collective Mie resonances of interacting sub- wavelength sized dielectric building blocks to generate the needed multiple light manipulating functions noted above using a single collective mode of the whole system—a true light manipulating unit [1.36]. The basic idea is captured in Fig. 1.5, lower right panel. The blocks shown in blue represent the dielectric building blocks (DBBs) and the pyramid the embedded SQD. Utilizing Mie theory, we have demonstrated [1.36] that the collective Mie resonance of dipole mode (primarily magnetic) of interacting high index such as GaAs DBBs, (Fig. 1.5 right panel) can provide simultaneous multiple functions such as QD emission rate enhancement by a factor of ~7 at the QD location and guiding and lossless on-chip propagation of the emitted photons along the DBB array, i.e a local nanoantenna-waveguide unit with a small footprint of <10 microns surrounding the MTSQD [1.36]. Indeed, recently these simulations have been extended to include splitting of the beam and combining two beams from two adjacent and parallel rows [1.37]. This suggests a pathway for implementing and studying photon interference in between different MTSQD-DBB LME units. Simultaneous implementation of such broad range of functions on micron scale footprint carries high payoff towards the development of integrated quantum nanophotonic systems. 14 The demonstration of ordered uniform MTSQDs based on-chip integrable and scalable SPS array opens a pathway towards building photonic integrated circuits that is worthy of exploration towards the goal of realizing on-chip integrated nanophotonic QIP systems. §1.3 Organization of the dissertation The dissertation is organized as follows. Chapter 2 presents studies of the growth of InGaAs QDs on mesa top using SESRE approach. We discuss the approaches taken in the literature on realizing ordered QDs and their limitation followed by the discussion of our SESRE approach and its place in the literature. The concept of SESRE and the growth physics including the effect of inter-facet atom migration controlled by the surface curvature induced stress gradient and strain relaxation is addressed. Detailed discussion on growth process and growth front profile evolution on mesa top for the control of single QD formation is presented. Understanding of inter-facet atom migration under our employed growth condition and its use to control and synthesis QD with desired size and shape is also discussed. Structural information on the synthesized size-shape controlled MTSQDs used for optical studies reported in Chapter 3 and Chapter 4 is captured. Chapter 3 presents the optical studies of MTSQDs aimed at understanding their electronic structure and multiple excitonic states to assess their potential use as SPSs and more importantly as EPSs. Our home-built micro-PL setup is used for studying PL, PLE and TRPL of individual MTSQDs. Systematic studies of PL as a 15 function of excitation power, temperature and polarization are presented. These are complemented with PLE and TRPL studies to enable as consistent inferences as possible within the instrument spectral resolution limits. A central theme here is narrowing down the range of potential optical responses obtainable from the MTSQDs including the impact of shape on its electronic structure and excitonic emission. Chapter 4 presents the single photon emission behavior of the MTSQDs. We first address the physics of single photon emission from QDs followed by the instrumentation for the Hanbury-Brown and Twiss (HBT) setup for measuring the QD emission statistics through the second-order intensity correlation (g (2) (τ)). The single photon emission purity and efficiency of the MTSQDs are presented. The physical processes of charging and state mixing that limit the purity of single photon emission and their connection to the QD electronic structure are also addressed in this chapter. Chapter 5 summarizes the results presented in chapters 2 to 4, and suggests some directions for future research. There are four Appendices at the end of this dissertation. Appendix A covers the basic concept of QDs, implementation pathways for the QDs and detailed discussion of SAQDs (reference QDs used in this dissertation work) formation. Appendix B covers the major equipment used in the work discussed in Chapter 2 for MTSQD synthesis and structural characterization comprising molecular beam epitaxy, scanning electron microscopy, and atomic force microscopy, including 16 system calibration. Appendix C covers a detailed discussion of the instrumentation of our home-built micro-PL setup that is capable of studying PL, PLE, TRPL and g (2) (τ) of individual MTSQD, underpinning the work presented in Chapters 3 and 4. Appendix D covers the theoretical background of second quantization of the electro- magnetic field that underpins the discussion of theoretical background of single photon and entangled photon emission from QD in Chapter 4. Chapter 1 References: [1.1] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer”, SIAM J. Comput. 26, 1484 (1997). [1.2] L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack”, Phys. Rev. Lett. 79, 325 (1997). [1.3] C. Bennett, F. Bessette, G. Brassard, L. Salvail and J. Smolin, “Experimental quantum cryptography”, J. Cryptol. 5, 3 (1992). [1.4] E. Knill, R. Laflamme and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001). [1.5] N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, “Quantum cryptography”, Rev. Mod. Phys. 74, 145 (2002). [1.6] N. Gisin and R. Thew, “Quantum communication”, Nat. Photonics 1, 165 (2007) [1.7] T. Jennewein, C. Simon, G. Weihs, H. Weinfurter and A. Zeilinger, “Quantum cryptography with entangled photons”, Phys. Rev. Lett. 84, 4729 (2000). [1.8] J. L. O’Brien, A. Furusawa and J. Vučković, “Photonic quantum technologies”, Nature Photon. 3, 687 (2009). [1.9] S. Buckley, K. Rivoire and J. Vučković, “Engineered quantum dot single photon sources”, Rep. Prog. Phys. 75, 126503 (2012). [1.10] P. Lodahl, S. Mahmoodian and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures”, Rev. Mod. Phys. 87, 347 (2015). 17 [1.11] R.M. Thompson, R.M. Stevenson, A.J. Shields, I. Farrer, C.J. Lobo, D.A. Ritchie, M.L. Leadbeater and M. Petter, “Single-photon emission from exciton complexes in individual quantum dots”, Phys. Rev. B 64, 201302(R) (2001). [1.12] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs”, Nature Photon. 8, 224 (2014). [1.13] A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single photon source for distributed quantum networking”, Phys. Rev. Lett., 89, 067901(2002). [1.14] M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther, “Continuous generation of single photons with controlled waveform in an ion-trap cavity system”, Nature, 431, 1075 (2004). [1.15] B. Lounis and W. E. Moerner, “Single photon on demand from s single molecule at room temperature”, Nature, 407, 491(2000). [1.16] I. Aharonovich, S. Castelletto, D. A. Simpson, C-H. Su, A. D. Greetree and S. Prawer, “Diamond-based single-photon emitters”, Rep. Prog. Phys. 74, 076501 (2011). [1.17] O. Benson, C. Santori, M. Pelton and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot”, Phys. Rev. Lett. 84, 2513 (2000). [1.18] A. Madhukar, “Growth of semiconductor heterostructures on patterned substrates: defect reduction and nanostructures”, Thin Solid Films. 231, 8 (1993). [1.19] K.C. Rajkumar, A. Madhukar, P. Chen, A. Konkar, L. Chen, K. Rammohan, and D.H. Rich, “Realization of three-dimensionally confined structures via one-step in situ molecular beam epitaxy on appropriately patterned GaAs(111)B and GaAs(001)” , J. Vac. Sci. Technol. B 12, 1071 (1994). [1.20] A. Konkar, A. Madhukar, and P. Chen, “Creating three-dimensionally confined nanoscale strained structures via substrate encoded size-reducing epitaxy and the enhancement of critical thickness for island formation,” MRS Symposium Proc. 380, 17 (1995). [1.21] A. Konkar, K. C. Rajkumar, Q. Xie, P. Chen, A. Madhukar, H. T. Lin and D. H. Rich, “In-situ fabrication of three dimensionally confined GaAs and InAs volumes via growth on non-planar patterned GaAs (001) substrates”, J. Cryst. Growth. 150, 311 (1995). [1.22] J. Zhang, S. Chattaraj, S. Lu and A. Madhukar, “Mesa-top quantum dot single photon emitter arrays: Growth, optical characteristics, and the simulated optical 18 response of integrated dielectric nanoantenna-waveguide systems”, J. Appl. Phys. 120, 243103 (2016). [1.23] J. Zhang, Z. Lingley, S. Lu, and A. Madhukar, “Nanotemplate-directed InGaAs/GaAs single quantum dots: Toward addressable single photon emitter arrays,” J. Vac. Sci. Technol. B 32, 02C106 (2014). [1.24] M. Grundmann, Nano-Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. [1.25] A.P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots”, Science, 271, 933 (1996). [1.26] C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot”, Phys. Rev. Lett. 86, 1502 (2001). [1.27] X. Liu, H. Kumano, H. Nakajima, S. Odashima, T. Asano, T. Kuroda and I. Suemune, “Two-photon interference and coherent control of single InAs quantum dot emission in an Ag-embedded structure”, J. Appl. Phys. 116, 043103 (2014). [1.28] E. M. Purcell, H. C. Torrey and R. V. Pound, “Resonance absorption by nuclear magnetic moments in a solid”, Phys. Rev. 69, 37 (1946). [1.29] P. Yao, V.S.C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots”, Laser Photonics Rev. 4, 499 (2010). [1.30] K. H. Madsen, S. Ates, J. Liu, A. Javadi, S. M. Albrecht, I. Yeo, S. Stobbe and P. Lodahl, “Efficient out-coupling of high purity single photons from a coherent quantum dot in a photonic-crystal cavity”, Phys. Rev. B. 90, 155303 (2014). [1.31] S. Laurent, S. Varoutsis, L. L. Gratiet, A. Lemaître, I. Sagnes, F. Raineri, A. Levenson, I. Robert-Philip and I. Abram, “Indistinguishable single photons from a single-quantum dot in a two-dimensional photonic crystal cavity”, Appl. Phys. Lett. 86, 163107 (2005) [1.32] A. Schwagmann, S. Kalliakos, I. Farrer, J.P. Griffiths, G. A. C. Jones, D. A. Ritchie and A. J. Shields, “On-chip single photon emission from an integrated semiconductor quantum dot into a photonic crystal waveguide”, Appl. Phys. Lett. 99, 261108 (2011). [1.33] A. Laucht, S. Pütz, T. Günthner, N. Hauke, R. Saive, S. Frédérick, M. Bichler, M.-C. Amann, A.W. Holleitner, M. Kaniber and J.J. Finley, “A waveguide-coupled on-chip single photon source”, Phys. Rev. X. 2, 011014 (2012). 19 [1.34] M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide”, Phys. Rev. Lett. 113, 093603 (2014). [1.35] A. Faraon, A. Majumdar, D. Englund, E. Kim, M. Bajcsy and J. Vučković, “Integrated quantum optical networks based on quantum dots and photonic crystals”, New J. Phys. 13, 055025 (2011). [1.36] S. Chattaraj and A. Madhukar, “Multifunctional all-dielectric nano-optical systems using collective multipole Mie resonances: toward on-chip integrated nanophotonics”, J. Opt. Soc. Am. B. 33, 2414 (2016). [1.37] S. Chattaraj, J. Zhang, S. Lu and A. Madhukar, “On-chip quantum optical networks comprising co-designed and spectrally uniform single photon source array and dielectric light manipulating elements”, presented at IEEE summer topicals meeting, San Juan, Puerto Rico, July 10-12, 2017, Abstract number TuF1.4. 20 Chapter 2. Mesa-top single quantum dot synthesis: Substrate-encoded size-reducing epitaxy Given our objective of realizing spatially ordered and spectrally uniform SQD array that can act as single photon sources (SPSs) as well as the entangled photon sources (EPSs) for building nanophotonic quantum information processing (QIP) systems, we employ the substrate-encoded size-reducing epitaxy (SESRE) approach [2.1] and exploit its potential use to create such needed SQD arrays. In this chapter, we focus on the physics of the SESRE growth, the implementation of molecular beam epitaxy [2.2, 2.3] and control of the growth front profile evolution on patterned mesas to realize the mesa-top single quantum dot (MTSQD) array using the SESRE approach. This chapter is organized as the following: Section §2.1 captures the current approaches in the literature on realizing spatially ordered QDs and their limitations. Our SESRE approach and its place in the literature is discussed in section §2.2. The growth physics involved in controlling the morphology of the layer grown on the mesa top that includes the effect of the inter-facet atom migration controlled by the surface curvature induced stress gradient and strain relaxation on the mesa top is addressed. Section §2.3 contains (1) a brief discussion of our solid source molecular beam epitaxy (SSMBE) system used to implement the SESRE approach, (2) the substrate preparation before sample growth, and (3) the observed evolution of growth front on mesa top. Structural information of the growth front and the facets evolved on the mesa top discussed in this section are obtained from scanning 21 electron microscope (SEM) and transmission electron microscope (TEM) imaging. Details of the instrumentation of SEM and TEM used in this thesis work are captured in Appendix B. Based on the known information on the evolution of growth front, the inter-facet atom migrations under the employed growth condition for the size- reducing growth on mesa top is addressed. The obtained information on inter-facet atom migration along with the growth front evolution is used to predict and control the synthesis of the MTSQDs with desired size and shape utilizing the size-reducing growth on the mesa top. The synthesized size- and shape-controlled MTSQDs with {101} and {103} side facets formed at two different pinch-off stages of the mesa top are discussed in detail in section §2.4. The synthesized InGaAs MTSQDs with {103} side facets with designed emission wavelength within the silicon avalanche photodiodes (APDs) detection range are used for optical studies, discussed in detail in Chapter 3 and 4, of the QD electronic structure and the dynamics and statistics of photon emission in addressing the issue of the potential use of MTSQDs as SPSs and EPSs. §2.1 Current status of ordered quantum dot synthesis As discussed in Chapter 1, semiconductor SQDs with discrete energy states can act as SPSs and EPSs [2.4, 2.5]. Amongst different types of QDs, lattice mismatch strain driven spontaneously formed 3D islands QDs, dubbed self- assembled QDs (SAQDs) [2.6, 2.7] have been the dominant class of QDs studied for realizing SPSs and EPSs [2.3, 2.5]. Ever since the initial demonstration of a single SAQD acting as SPS at 4K [2.8], extensive studies on SPS have been carried out in a 22 variety of compound semiconductor SAQDs. A lot of progress has occurred on achieving single photon emission from SAQDs at high temperature (liquid nitrogen temperature to room temperature) [2.9-2.13] by controlling synthesis process and using wide band gap material (group II, V and group III-N) for the QDs. InGaAs/ AlGaAs SAQD [2.9, 2.10] and InP/AlGaAs SAQDs [2.11] provide single photon emission up to 80K and InGaN/GaN [2.12] and CdSe/ZnSe [2.13] SAQDs provide single photon emission up to room temperature. The emission wavelength of the single photons coming from QDs has also been expanded to optical communication wavelength, i.e. 1.3 μm [2.14] by controlling QD size and shape realized through engineering QD growth process. In addition, SAQDs have also been incorporated in p-i-n structures to achieve electrically pumped single photon emission at low [2.15] and elevated temperatures [2.16,2.17] in wavelength region <1μm. The emission wavelength of electrically pumped SAQD based SPSs has also been extended to ~1.3 μm [2.18]. Along with the above mentioned important development on improving operating temperature and extending operating wavelength of SAQD SPS to advance QD SPS technology, progress has occurred on integrating SAQD based SPS with light manipulating structures such as cavity and waveguide in an integrated structure to study the physics of light matter interaction to control the propagation of QD emission on-chip and reduce the QD emission lifetime to make it comparable to the QD dephasing time, a necessary condition for photon interference needed for building QIP systems. To date, approaches taken to realize integrated SQD-light 23 manipulating unit structures have been based on either micropillar/ nanowire [2.4, 2.19-2.27] or 2D photonic crystal [2.5, 2.28-2.33] platforms, the former inherently well-suited to architectures exploiting vertical emission of photons while the latter to horizontal emission. Following the vertical architecture with micropillar/nanowire approach, single photon emission rate enhancement of ~2-7 [2.4, 2.19-2.22], single photon emission purity >99% (g (2) <0.001) [2.19, 2.22, 2.23, 2.25, 2.26], single photon indistinguishability > 98% [2.20-2.22, 2.26] and photon collection efficiency of 40- 70% [2.21-2.26] have been achieved. All important figures of merit of SPS have been improved in the QD-micropillar/nanowire integrated structure. Similar to micropillar/ nanowire approach, single photon emission rate enhancement of ~5-6, single photon emission purity >97% (g (2) <0.05), single photon indistinguishability of 70-85% have been achieved in SAQD-2D photonic cavity structures [2.5, 2.28-2.29]. Besides improving QD single photon emission properties with cavities, it has been demonstrated that SAQD can have coupling efficiency from 70% up to 98% [2.30- 2.32] to 2D photonic waveguide to control on-chip light propagation for circuitry function. Integrating of SAQD with a single cavity and waveguide integrated structure is still an on-going effort [2.33]. No result on the performance of such SAQD-cavity-waveguide structure is available in the literature so far. In both approaches, great progress has been made on improving important figures of merit of single photon sources as discussed above. Additionally, with the QD radiative lifetime being shortened to be comparable to QD decoherence time in the QD-cavity structure, single SAQD has been demonstrated to emit polarization 24 entangled photon pairs through biexciton-exciton cascade process [2.21, 2.34] and act as EPSs, a critical element for QIP system. However, invariably all the reported SPSs and EPSs studies are based on pre-selected single SAQDs – light manipulating units, not array that acts as the starting point for building optical circuits. Even in the single SAQD – light manipulating unit, the physical study is done by selecting a randomly located SAQD in the integrated structure picked out of the large spectral range of SAQDs (Fig. 2.1 (a)). The large spectral non-uniformity and random positioning of SAQDs affect the coupling between QD and light manipulating elements (LME) [2.28, 2.32]. This is illustrated by Figure 2.1 (b) and (c) that show, respectively, the variation of the obtained Purcell factor, coupling efficiency to waveguide and photon collection efficiency amongst different SAQDs with different spectral response and located at different position in the integrated structure. The large spectral non-uniformity of the QDs is beyond the tuning range of the well- established on-chip spectral tuning technique [2.33, 2.35] and makes it difficult to spectrally match such QD with the LMEs for circuit operation. Figure 2.1 (a) AFM image of InAs islands showing the inherent variation in size and random spatial positions along with the resulting broad photoluminescence and extremely sharp emission lines (yellow line) from individual islands when capped with a protective layer. (b) Measured Purcell factor, collection efficiency and β- factor (coupling efficiency) of SAQDs within the 2D phonic crystal cavity with 25 different spectral mismatch with respect to cavity mode, from Ref. [2.28] (c) Calculated SAQDs photon emission rate into the 2D photonic crystal waveguide mode as a function of SAQDs position in the waveguide, from Ref. [2.32]. SQD that can act as SPSs and EPSs with predetermined formation position and spectral uniformity that falls within the tuning range of on-chip scalable spectral tuning technologies [2.33, 2.35] is an essential element needed for building integrated QD-light manipulating unit and interconnected units for information processing optical circuits architecture.  Approaches to fabricating ordered quantum dot based single photon source To achieve spatial regularity of QDs, approaches taken in the literature can be categorized as growth/deposition on planar substrates [2.1] with oxide masks and on non-planar substrates with patterned structures [2.1]. For the class of SAQDs, a lot of effort has been put in to create ordered array of SAQDs by growing them on pattered substrate mesa tops [2.36, 2.37], with nanoholes [2.38-2.42] as well as on planar substrate with aluminum oxide [2.43] stain layer. However, reproducible growth of single SAQD per site and of uniform spectral response is still a challenge [2.38-2.43]. Given the limitation of SAQDs, approaches have been taken in the literature to synthesize ordered non-SAQDs whose formation is not lattice-mismatch strain driven. Dominate approaches taken to realize such ordered non-SAQDs are based on deposition on substrates with appropriate masks - typically oxides – with patterned etched holes, leaving the underlying semiconductor substrate planar [2.1]. The 26 growth is typically done using all gaseous molecular sources such as in gas source MBE or metal organic chemical vapor deposition (MOCVD) as the slow dissociative reaction kinetics and fast desorption kinetics of the relevant molecular species on the oxide enables selective growth in the holes. The lateral size and depth of the holes has been exploited for selective placement of quantum dots, including single QD [2.44-2.46]. However, such growth, being spontaneous, is sensitive to the fluctuations in the size and shape of the holes and does not allow the needed control on the shape and size of QDs formed. Most recently the holes-in-oxide-mask approach has been combined with the metal nanoparticle-seeded VLS (vapor liquid solid) growth [2.47, 2.48] by creating the seed metal nanoparticles selectively in the holes. The non-SAQD quantum dots in such nanowires have been demonstrated to be single photon sources [2.48, 2.49] with the nanowire bearing the QD to be designed to boost the collection efficiency of the emitted photons from the QD [2.49- 2.51]. They can also exhibit improved spectral uniformity, ~6meV, compared to SAQDs [2.52]. Nevertheless, certain challenges remain to be tackled, including the undesired incorporation of metal point defects that adversely impact the optical properties of the QDs [2.53], the fluctuation of nucleation events underlying VLS growth, and variation in the seed nanoparticle sizes [2.54] which together make predictable control on QD vertical location and emission wavelength difficult. In addition to the growth of non-SAQDs on planar substrate, non-SAQDs grown on patterned substrate with recesses/ holes to achieve ordered QD array have also been explored [2.55-2.59]. QD growth is done using MOCVD on patterned 27 GaAs (111) substrate with recesses and holes created through chemical etching. The curvature of the patterned structure leads to the net atom migration from the planar region into the recess/ holes which results in the expansion of the initial bottom area under conformal growth and the formation of QDs in the recesses/ holes [2.1, 2.55, 2.56]. Spectral uniformity of ~10meV has been achieved with this type of QDs [2.55, 2.56]. Progress has also occurred on the integration of these QDs with 2D photonic crystal based LMEs to the study of the optical property of the integrated structure [2.57-2.59]. However, this growth mechanism cannot incorporate high lattice mismatch strains without defects due to the expansion of the initial bottom area which in turn limits the material system of QDs that can be synthesized with and the corresponding emission wavelength of the synthesized QDs. One another limitation of this approach comes from the bad material quality coming from carbon impurities inherent to MOCVD growth [2.58]. §2.2 Substrate-encoded size-reducing epitaxy Substrate-encoded size-reducing epitaxy approach (SESRE) that exploits surface curvature induced surface stress gradients to preferentially drive adatoms, during growth, to migrate from mesa sidewalls to the top for appropriately patterned mesas [2.1] can be implemented using molecular beam epitaxy growth technique to realize spatial selective formation of size-shape controlled QDs in ordered arrays. For pattern designs that induce net migration from the side facets to the mesa top, preferential incorporation at the mesa-top leads to growth-controlled mesa size reduction during growth which enables the in situ formation of defect-free size and 28 shape controlled QDs with lattice matched and even highly lattice mismatched materials due to strain relaxation at the mesa top, overcoming the limitation in the approaches of forming ordered non-SAQD mentioned before [2.1]. In this section, we discussed the concept of SESRE, the effect of the surface curvature induced stress gradient on inter-facet atom migration in a classical continuous model and atomic model and the growth physics involved in controlling the grown layer morphology on mesa top including inter-facet atom migration and strain relaxation that play a critical role in realizing ordered and size, shape controlled InGaAs SQD arrays. §2.2.1 Continuum solid model: capillary effect All surfaces with curvature have built-in spatially-dependent surface stress fields and thus surface stress gradients. For the continuum model of a solid valid on the macroscopic level, Gibbs pointed out this as the basis for the capillary effect. A general expression for the average chemical potential [2.60, 2.61] just beneath the surface of a facet bounded by N other facets is 00 1 0 ( csc cot ) N i i i i A             (Eq.2.1) In this expression, 0  is the chemical potential in the bulk of the crystal,  is the atomic volume, 0  is the surface free energy of the facet, 0 A is its area, and i  is the acute dihedral angle between the ith bounding facet (of surface energy i  ) and the facet of interest. For the geometry of nonplanar morphology shown in Figure 2.2, 29 the spatial varying surface stress profile attendant to this morphology can be expressed based on Eq. 2.1 in the following form 0 2 TOP T l     (Eq.2.2) 0 2 BOTTOM B l     (Eq.2.3) csc cot SW       (Eq.2.4) where  and SW  represents the surface free energies of the top/bottom and side wall facets, T l and B l represents the width of the top and bottom facets, respectively, as indicated in Fig. 2.2. Figure 2.2. Schematic of the surface profile of a patterned mesa. SW represents “side wall”. The surface dependent chemical potential can drive atom to diffuse along the surface with velocity, V, s B D V k T s     (Eq.2.5) where s D is the surface diffusivity, B kT is the thermal energy, and the derivative with respect to s is taken along the surface. The opposite signs for the convex and concave regions of the surface profile shown in Eq. 2.2 and 2.3 are a magnification 30 of capillary forces that would drive atom motion, usually through surface diffusion, in a direction that moves from the top to the bottom region. On a length scale greater than some local length (such as local unit cell), as a first approximation, one may ignore the discrete atomistic nature of the surface and, with in continuum solid picture, represent the surface chemical potential using Eq. 2.2 and 2.3. The capillary forces drive the atom migration and will lead to changes in the evolution of the growth front file [2.62-2.64] given by 2 2 1/2 2 1/2 0 2 [1 ( ') ] [1 ( ') ] s B D h n h F h t k T s        (Eq.2.6) where h denotes the height of the grown overlayer, 0 n the number of atoms per unit area, F the incident flux, and ' h the first derivative with respect to position. Additionally, quite independent of the presence or absence of a deposition flux, the capillary forces will drive mass transfer and move the surface profile to a flat, planar, surface with uniform chemical potential everywhere [2.60]. §2.2.2 Atomistic model: Significance of Step Orientation and Density Growth on the nonplanar patterned structures shown in Fig.2.2, was first carried out by Tsang and Cho [2.65]. The mesas patterned along [110] and [1-10] directions shown in Figure 2.3 on GaAs (001) surface were used to grow GaAs on top. Tsang and Cho showed that the growth on mesas with these two different edge orientations is quite different. The growth on mesas of type shown in Fig. 2.3 (a) leads to mass transport from the side facet to the top and was thought to be useful for realizing laser structures. Wu et al [2.66] and Arakawa and Yariv [2.67] explored the 31 growth of laser structures on GaAs (001) patterned ridges using MBE while Kapon et al [2.68] using MOCVD. At the core of such growth lies the fact that mass migration between contiguous facets and variations in atom incorporation on different facets can be explored to create a difference in the thickness and composition of layers continuing from one facet to another. Further information on the presence of inter-facet migration during growth on patterned substrates was obtained by Smith et al [2.69], Mannoh et al [2.70, 2.71] and Guha et al with direct measurement of inter-facet migration [2.72, 2.73]. It has been found that mass migration is from mesa side wall to mesa top [2.65, 2.69-2.73] for ridge mesa along [1 -1 0] direction or vice versa for ridge mesa along [110] direction [2.69-2.73]. Figure 2.3 Schematic of the ridge mea with edges along (a) [1-10] and (b) [110] direction. The interaction between steps and the orientation of arsenic dangling orbitals with respect to the mesa edges determines the atom migration direction marked with red arrows. This observed orientation dependent atom migration between contiguous facets cannot be easily explained using the classical continuous solid model discussed in section §2.2.1. To understand the observed inter-facet atom migration behavior, the atomic nature of the structure should be taken into account. On an atomic scale, the spatial variation of the surface stress can be viewed in terms of the 32 variation in the local stress due to the particular displaced atomic arrangement with respect to the bulk position. The surface profile contains discrete steps with varying terrace widths between steps. In a pure geometric picture which ignores the specific chemical nature of the atomic orbitals and charge distribution attendant to the particular crystallographic edges and facets, expressions for surface step-step interactions and surface stress have been derived [2.74]. But these only account for mechanics without surface chemistry. What is needed is to take into account of surface chemistry in addition to the surface curvature related mechanics and to express surface potential as surface mechano-chemical potential to obtain a more realistic description to arrive at even a qualitatively correct atomistic understanding of the nature of the inter-facet migration. Madhukar et al take the note that [1-10] direction for GaAs substrate corresponds the direction parallel to the arsenic dangling orbitals while the [110] direction corresponds the direction perpendicular to the arsenic dangling orbitals shown in Fig. 2.3 [2.1, 2.75]. This dangling orbital orientation with respect to edges can give rise to distinct surface stress. Guha and Madhukar [2.75] showed the ledge-ledge interaction via stress field attendant to the displacements of atoms for surfaces with different orientations of the dangling orbitals and pointed out the central role of arsenic dangling orbital orientation in affecting the inter-facet migration. The directionality of the interfacet migration is determined by the relative balance of the surface chemical potential gradient arising from the differences in the vapor flux on the mesa tops and side walls and the ledge- ledge interaction energy on the side walls. From this atomic mode, it has been found 33 that atom migration direction is from the side facets, {113} /{114} facets, towards mesa top for mesas with edge along [1-10] direction where arsenic dangling orbitals are parallel to the edges and the migration direction is reversed for mesas with edge along [110] direction where arsenic dangling orbitals are perpendicular to the edges [2.75]. This atomic model accounting for the surface atom orientation explains the observed inter-facet atom migration behavior on patterned mesa strips. In the atomic scale model, the position dependent surface-mechano-chemical potential affected by the surface step density and its step edge orientation with respect to the surface atom dangling orbital provides the driving force to direct atom migration in between facets. The inter-facet migration become another important actor besides mesa size and elastic strain, i.e. lattice mismatch strains of the materials used, to control the morphology of the grown homoepitaxial or heteroepitaxial layer. Therefore, the actual growing layer morphology on the mesa is controlled fully by the competing length scale of mesa size (l M ), the inherent elastic strain field (l s ) set by the elastic properties of the material involved, of great importance for heteroepitaxial growth, and the adatom migration lengths (l m ) of the growth rate controlling species at the growth conditions employed. These length scales are schematically depicted in Figure 2.4. The interplay of these three fundamental length scales and the directionality of inter-facet atom migration controlled by the mesa edge orientation and surface curvature induced stress gradient determine the growth rate of continuous facet on the mesa and hence the morphology of the grown homoepitaxial or heteroepitaxial layer on mesa top. Illustration of the competition of 34 these length scales in affecting the grown layer morphology, especially heteroexpitaxial layer, on mesa top is captured in detail in the following section. Figure 2.4. Schematic picture representing the most critical characteristic length scales to control the growth on patterned substrate and the growing layer morphology on the patterned structure. L M is the size of mesa, l s is the elastic strain field propagation length, l m is the atom migration length. The superscripts T, B and SW represents “top”, “bottom” and “side wall”. §2.2.3 Substrate-encoded size-reducing epitaxy As mentioned in section §2.2.2, the directionality of the inter-facet migration is determined by the relative balance of the chimerical potential gradient arising from the differences in the vapor flux on the mesa tops and side walls and surface stress gradient arising from the position dependent surface mechano-chemical potential. The growth profile on patterned mesa (back line in Fig. 2.5) can have the following three different scenarios shown in Fig. 2.5: (1) mesa top size reducing (red line in Fig 2.5), (2) mesa top size remaining constant (black dashed line in Fig. 2.5) and (3) mesa top size expanding (blue line in Fig. 2.5) during growth by controlling the atom migration dynamics of the growth rate controlling species shown in Figure 2.4 represented in the control of growth rate of the mesa top surface and side walls facets. 35 Figure 2.5. Schematic of the growth profile on mesa top representing mesa top reduction (red line), mesa top constant (black dashed line) and mesa top expansion (blue line) with the corresponding required growth condition of growth rate on mesa top (τ T )and side wall (τ SW ). The control on inter-facet migration and the resulting growth layer morphology on patterned substrates can be utilized to create ordered nanostructures. Kapon et al explored the growth condition for surface expansion during growth in V- groove trenches on GaAs (111)B substrate to realize quantum wells and quantum wires [2.56, 2.68]. Smith et al [2.69], Mannoh et al [2.70, 2.71] and Madhukar et al [2.72, 2.73, 2.76] demonstrated the suitability and formation of quantum wells and quantum wires on strip mesas on GaAs (100) substrate with mesa edges along [1-10] direction. In these cases, the starting mesa strip width reduces under the chosen growth conditions due to preferential migration of group III atom from side wall to the mesa top [2.76]. Size-reducing growth on patterned substrate can be used to realize spatially selective formation of not only quantum wells and quantum wires but also 3D confined quantum dot through proper choice of the pattern design. Madhukar et al proposed the use of mesas with edge orientation along <100> direction where arsenic dangling orbitals are aligned 45° with respect to mesa edges 36 (Fig. 2.6 (a)). The four-fold symmetry in these mesas provide the needed surface stress gradients to drive growth rate controlling adatoms symmetrically from the all four sidewalls to the mesa top under appropriately chosen growth kinetics (Fig 2.6(b)) [2.1, 2.36, 2.37] to realize mesa top size-reducing from all four side facets. Such size-reducing growth on the square mesas can enable the in situ one step realization of quantum dot on mesa top. Figure 2.6 (c) shows the SEM image of the array of GaAs quantum dot on mesa top. The inset in panel (c) shows a cross-sectional TEM image revealing GaAs (dark) quantum dot surrounded by AlGaAs (light) barrier. Figure 2.6. Shows an SEM image (panel c) of an array of pyramidal structures resulting from directed adatom migration from sidewalls to the mesa top (panel b) during GaAs/AlGaAs heteroepitaxial growth on starting square mesas with edges oriented such that As dangling orbitals are at 45° degrees with respect to the edge (panel a), thus maintain four-fold symmetry. Inset in panel (c) shows the vertically- stacked GaAs quantum dots grown near the mesa top. (From Ref. 2.76). Symmetric size-reducing growth on mesa top can also be realized on GaAs (111) B substrate. The mesas patterned are of triangular shape with edges along <1- 10> directions. The arsenic dangling orbitals (Fig. 2.7 (a)) are symmetric along all three edges providing symmetric surface stress gradients to drive atom migration from all side walls to mesa top [2.37, 2.76]. A quantum dot can be formed on top through mesa top size-reducing growth with proper chosen growth conditions. Figure 37 2.7 (c) shows the SEM image of the array of GaAs quantum dot on mesa top. The inset in panel (c) shows a cross-sectional TEM image revealing GaAs (dark) quantum dot surrounded by AlGaAs (light) similar to the case of quantum dot grown on <100> orientated square mesa on GaAs (100) substrate. Figure 2.7. Shows an SEM image (panel b) of an array of triangular pyramidal structures resulting from directed adatom migration from sidewalls to the mesa top during GaAs/AlGaAs heteroepitaxial growth on starting triangular mesas with edges along <1-10> direction (panel a). Inset in panel (b) shows the vertically-stacked GaAs quantum dots grown near the mesa top. (From Ref. 2.37) The mesa top size-reduction during homoepitaxial growth can thus be “encoded” by the choice of the mesa edge orientation on the pre-chosen substrate using appropriate range of growth condition. Hence the name substrate-encoded size- reducing epitaxy (SESRE) [2.1]. The spatially varying and edge orientation dependent nature of surface stress profile and its contribution to the evolving surface mechano-chemical potential at the growth front is a basic underlying factor giving SESRE [2.1, 2.36, 2.37, 2.76, 2.77]. Therefore, for tetrahedrally-bonded semiconductors of groups IV, III-V, and II-VI, with pattern designs that induce net migration from the side facets to the mesa top, preferential incorporation at the mesa- 38 top leads to growth-controlled mesa size reduction, enabling in-situ preparation of contamination and defect-free nanomesa arrays from the as-patterned array via homoepitaxy. Subsequent heteroepitaxy then enables synthesis of quantum confined nanostructures such as quantum dots on pristine nanotemplates through surface stress engineering with atom migration directed in preferred manner using SESRE. For the lattice matched heteoepitaxy, such as GaAs/AlGaAs, the undulating surface stress on a structurally pre-patterned substrates can be exploited to direct formation of ordered arrays of truncated pyramidal shape quantum dots (Fig. 2.6 and 2.7) using SESRE. For lattice mismatched heteoexpitaxy, such as InGaAs/AlGaAs, the lattice mismatched strain and stress can be thus combined with and manipulated by the stress variations associated with structural templating of the starting substrates. The actual growing layer morphology on the mesa is determined by the relative length scales of the structurally patterned surface features, elastic strain set by the properties of the material used, and the relevant effective migration lengths of atoms during growth as captured in Fig. 2.4. Lattice mismatch strain can lead to the formation of island SAQDs [2.6, 2.7]. For the mesa strip with long lengths but narrow widths, strain relaxation will begin in the mesa strips of sufficient narrow width due to free surfaces but no strain relaxation is possible along the length of the stripe. There is still a sufficiently strong lattice misfit induced stress that can drive the formation of 3D islands QDs on top of the strip during the growth of InGaAs on GaAs strips. With reducing length of the mesa, sufficient stress relaxation at free surfaces of the small mesas at certain length scale will begin to set in and the 39 formation of the 3D islands will become susceptible to manipulation. Shown in Fig. 2.8 is the cross-section TEM image of 12ML InAs deposited on a square GaAs (001) mesa with base of 46nm after mesa top size-reduction homoexpital growth. At this nanoscale, substantial strain relaxation has occurred owing to the presence of mesa free sidewalls as demonstrated in multi-million atom molecular dynamics simulations [2.78, 2.79] and results in the flat morphology and the absence of defects for lattice mismatched quantum dot [2.36, 2.37]. It has been shown that ~120nm is the mesa size below which significant stress relaxation begins for the InAs/GaAs systems [2.80]. In addition, the thickness of InAs film grown on mesa top of length ~46nm shown in Fig. 2.8 shows self-limiting behavior with limited thickness of 12ML due to the reversal of indium migration direction coming from the build-up of residue strain in the film, a signature of capillary effect. Figure 2.8 (a) Cross-section TEM image of a 12ML thick InAs film on a 46nm square GaAs (001) mesa with edge orientation along <100> direction from Ref. 2.37. (b) Multimillion atom molecular dynamics simulation of hydrostatic stress distribution in a 12ML InAs film on a 40.7nm square GaAs (001) mesa from Ref. 2.79. It is important to note that growth on mesa top, unlike in recess, enables accommodation of large lattice mismatch strain without generation of defects owing 40 to strain relaxation in the mesa free surfaces during growth and is thus suitable for implementation using a wide variety of material combinations covering optical emission from the UV (such as in the III-Nitrides) to the midinfrared (such as in the small bandgap InSb). Therefore, ordered quantum dot structures of flat morphology with truncated pyramidal shape on mesa top can be realized using either lattice matched [2.1, 2.74, 2.77] and lattice mismatched [2.1, 2.36, 2.37, 2.80] material system through the control of the relative length scales of the structurally patterned surface features, lattice misfit generated stress fields, and the relevant effective migration lengths of atoms during growth with chosen pattern design. Moreover, depending upon the chosen as-etched sidewall crystallographic planes, the size- reducing growth can offer more than one pinch-off stages [2.36] dominated respectively by {103} and {101} crystallographic planes, thereby allowing control on not only size but also the shape of the QD formed subsequently via heteroepitaxy with {101} or {103} side walls. The QD shown in Fig. 2.6 (c) and Fig. 2.8 (a) are the ones bounded with {101} side wall planes. SESRE can thus also allow controlled vertical stacking and coupling of QDs in addition to the permit of control on shape and size of the formed QD on mesa top. The controlled formation of uniform SESRE QD array, however, requires atomistic level control on the growth kinetics and nanometer precision of the starting mesa lateral size which made its implementation technologically difficult prior to the availability of nanolithography. With the development of nanolithographic technologies that have enabled nanometer 41 precision control over large areas, the SESRE approach offers a path worthy of exploration as demonstrated below. §2.3 Mesa-top Single Quantum Dot: Growth As the maturing of nanolithography now enables precise nanometer scale control on the size of nanomesas structurally patterned on substrates, we revisited the SESRE approach and synthesized spatially-ordered arrays of GaAs(001)/InGaAs/GaAs mesa top single quantum dots (MTSQDs) using our solid source molecular beam epitaxy (SSMBE) and explored such MTSQDs’ potential use as SPSs and EPSs for information processing. In this section, we discuss briefly the SSMBE system used to implement SESRE approach, the preparation of patterned GaAs (001) substrate before sample growth, the mesa top size-reducing growth through homoepitaxy on patterned mesas and the observed the growth evolution of the growth front on mesa top. The structural information of growth front on mesa top was obtained via extensive SEM and TEM studies. All cross-sectional dark field TEM images on patterned mesas were obtained by Dr. Zachary Lingley, my fellow graduate student now graduated. Inter-facet atom migration under the employed growth conditions for the size-reducing growth on mesa top are addressed based on the known information on the evolution of the growth front and are used to control and predict the size-reducing growth on mesa top to create QD with desired size and shape. 42 §2.3.1 SESRE implementation using MBE SESRE approach is implemented using molecular beam epitaxy growth technique to grow structures on patterned GaAs (001) substrates containing square mesas with edges orientated along <100> direction. The experimental results shown in this dissertation were obtained from samples grown in our RIBER 3200P SSMBE machines. The RIBER MBE machine at USC is part of a six UHV (ultra high vacuum) chamber interconnected Growth-Processing-Characterization system as shown in Fig. 2.9 which contains (a) molecular beam epitaxical (MBE) growth chamber, (b) UHV chemical vapor deposition (UHV-CVD) chamber with electron cyclotron resonance (ECR) plasma source, (c) an RF-plasma enhanced chemical vapor deposition (PECVD) chamber, (d) a focused ion beam (FIB) assisted direct – write patterning chamber, (e) a metallization/H-cleaning chamber, and (f) a UHV STM/AFM system. Figure 2.9. Ultra high vacuum inter-connected semiconductor growth, processing, and in-situ characterization system in the Madhukar group at USC. FIB Chamber H-Cleaning/ e-beam metal deposition Chamber A B PECVD Chamber E C D F G H III-V MBE Growth Chamber Dry Glove Box ECR plasma-assisted UHV-CVD/etching Chamber 1 UHV-STM/AFM chamber UHV sample transfer modules Heating station FIB Chamber H-Cleaning/ e-beam metal deposition Chamber A B PECVD Chamber E C D F G H III-V MBE Growth Chamber III-V MBE Growth Chamber Dry Glove Box ECR plasma-assisted UHV-CVD/etching Chamber 1 UHV-STM/AFM chamber 1 UHV-STM/AFM chamber UHV sample transfer modules Heating station 43 In MBE growth, materials are delivered to substrates from sources as vapor under UHV (~10 -9 torr) condition with long mean free path (> 100cm). The impinging flux thus reaches the substrate ballistically without collision between the atoms and/or molecules unlike in MOCVD. Consequently, MBE offers several advantages as epitaxial techniques in comparison with others such as: (1) UHV environment with utilization of pure sources (typically 7N or better) minimizes impurity incorporation (typically, < 10 14 /cm 3 background impurity); (2) MBE allows integration with surface analyzer tools (such as RHEED, reflection high energy electron diffraction, requires UHV environment) to monitor in-situ the surface condition and its dynamic evolution (i.e. monolayer growth) during growth; (3) growth thickness can be calibrated using RHEED and controlled to obtain accuracies of one atomic layer or even down to sub-atomic layer. The sub-monolayer (ML) control on the deposition amounts originates from the Ballistic nature of the imping flux which allows sharp interruptions through the control of the K-cell shutter shut off time to obtain ~ 0.01 – 0.1 ML deposition accuracy; (4) as the substrate temperature can be independent of the temperature of source materials, growth can be performed at relatively low temperature compared to other traditional epitaxial growth techniques such as liquid phase epitaxy (LPE) or standard vapor phase epitaxy (VPE), which minimizes inter-diffusion of materials at interfaces; and thus, the abrupt junctions of heterostructures can be created. These features make MBE the most suitable technique to enable atomistic level control on the growth kinetics for the growth MTSQDs with high material quality and sub-ML precision in the 44 control of material deposition which in turn provides the needed degree of control on the QD size. During the sample growth, the growth condition, i.e. the growth rate of growth control species (Group III), growth temperature and arsenic flux, is calibrated and checked against physical conditions of the GaAs surface using the in-situ surface analyzer tool RHEED to control the growth structure with reproducibility from run to run. The high energy electron beam (~10-20keV, electron wavelength ~0.12A) from the RHEED incidents on the substrate with a glancing angle satisfying the off- Bragg diffraction condition of interference from the electron beam from diffracted from the adjacent layers of the substrate. Given the GaAs lattice monolayer thickness of 0.28A, the glazing angle is adjusted to 0.62° for characterizing the surface of GaAs during growth. The electrons are diffracted by the surface lattice following the general Bragg condition and the diffraction pattern from the surface that can be constructed from the Ewald sphere is observed as schematically shown in Fig. 2.10. Figure 2.10. Schematic picture of RHEED with geometry of sample, diffracted beams and RHEED screen. The incident beam is at glancing angle with respect to the substrate surface. The diffracted electron beam is projected on fluorescent screen. 45 The diffraction pattern represents the surface atom arrangement. GaAs (001) surface exhibits many reconstructions and the GaAs (001) surface reconstruction can be changed from As (2x4) to C(4x4) or (3x1) with the change of the chosen gallium partial pressure, arsenic pressure and substrate temperature. The presence of different surface reconstructions is inherent to the physical condition of the surface, i.e. the coverage percentage of arsenic atom on the surface [2.81]. Therefore, surface phase diagram represents the intrinsic material property of the substrate and is routinely used/ checked by us to maintain the reproducibility of experimental conditions such as arsenic flux and substrate temperature to compensate the possible errors caused in the machine reading of growth parameter from our SSMBE system. The growth rate of growth controlled species under the employed condition is calibrated through the oscillation behavior of the RHEED specular intensity [2.2, 2.82]. The specular spot intensity of the RHEED specular beam is qualitatively inversely related to the surface step density. Therefore, the specular spot intensity can be used to indicate the surface morphologic condition to grow structures with highest surface smoothness. During growth the specular intensity oscillates due to the monolayer (ML) high step density oscillation during the layer-by-layer (2D) growth of crystals. The oscillation is tied to ML coverage of the surface and its periodicity represents the growth rate of the growth controlled adatoms. Detailed information on our MBE system instrumentation, RHEED and system calibration and control are captured in Appendix B. 46 §2.3.2 Substrate preparation The GaAs (001) substrates are patterned ex-situ via electron-beam lithography after cleaning and spin coated with electron beam resist, i.e. HSQ used in our case. As illustrated in Fig.2.11, the substrates include a L-shaped unpatterned region that serves as a reference for calibrating and controlling gallium and indium fluxes (and thus growth rates, composition, and thickness) to atomic layer deposition precision during growth by enabling real-time RHEED pattern and intensity oscillation dynamics as described in section §2.3.1. Figure 2.11. Schematic drawing of the layout of the pattern design containing non- patterned and patterned regions with nanomesas array for the synthesis of mesa top quantum dot. In the patterned region there are two types of mesa patterns depicted in blue designed for structural characterization using cross-sectional TEM and red designed for optical (micro-PL) and (AFM and SEM) structural characterization. The 47 layout of nanomesa in each type of pattern is shown in the zoomed out images on the right side of the figure. The patterned region in Fig.2.11 contains total 16 patterned areas, 8 (the red blocks) designed specifically to enable optical access to individual mesas and 8 (the blue blocks) designed specifically to enable access to individual mesas for TEM studies. All nanomesas in the patterned regions have their four edges along the <100> directions to enable symmetric adatom migration from the four side walls to the mesa top for the controlled size reduction of mesa top during growth. As depicted, each of these two types of 8 patterned areas are arranged in a 2 × 4 array but each type contains a different designed layout of the mesas inside as indicated by the two blow-outs in Fig.2.11. The red pattern areas, designed to assist post growth optical and SEM studies, are marked “Opt” and the blue areas, designed for post growth structural studies using cross-sectional TEM, are marked “TEM”. Each Opt pattern contains 1 × 2 array of as-patterned mesas of sixteen different sizes labeled as S0 to S15 (see blow up of Opt pattern in Fig. 2.11). For each of the 16 mesa sizes, there is a 5 × 8 mesa array with inter-mesa separation ~ 5 μm chosen to facilitate optical characterization of each MTSQD in the array using our home built micro- photoluminescence setup discussed in Chapter 3 and Appendix C. Each “TEM” pattern contains four different mesa sizes chosen out of the sixteen sizes S0 to S15 in the “Opt” pattern. For each of these four mesa sizes, there are 30 × 8 mesas with inter-mesa separation of 10 μm (see the blow up of the “TEM” pattern in Fig. 2.11). Mesas in each row have an offset in the lateral direction to allow the electron beam to pass through a single mesa without being blocked by other mesas, thus enabling 48 imaging using cross-sectional TEM. Four out of the 2 × 4 arrays of the TEM patterns have same four mesa sizes S2, S3, S4, S5 picked out of sixteen sizes within each pattern. The other four “TEM” patterns have four mesa size S6, S8, S10 and S12. The sixteen different chosen as-patterned mesa sizes in the “Opt” pattern allow us to investigate evolution of the growth front on mesa top and different stages of pinch- off in the same growth run using SEM since at the same stage of deposition mesa pinch-off occurs later on mesas with larger starting lateral size. The eight sizes covered in the 8 “TEM” patterns for TEM studies provide a wide size range for studying the morphology of the grown layer on mesa top, evolution of the growth front and atom migration in between facets during the mesa top size-reducing growth with the assistance of marker layers grown (discussed in section §2.3.3) on mesa top. The patterned GaAs (001) substrate with layout shown in Fig. 2.11 has as- patterned size from 1390nm (S0) to 1810nm (S15) with size increment of ~30nm. Such patterned GaAs (001) substrate is developed using a solution of 4 wt% NaCl and 1 wt% NaOH for approximately ~ 40s to remove electron-beam resist (HSQ, used in our case) that has not been exposed to electron beam. The developed substrate is then etched with wet chemical etching using etchant composed of NH 4 OH:H 2 O 2 :H 2 O = 4:1:20 which gives etch rate of ~30 nm/sec to create nanomesas arrays with as-etched lateral size in the range of 50-500 nm and mesa vertical depth ~500 nm. NH 4 OH:H 2 O 2 :H 2 O etchant provides isotropic etch rate along [110] and [1-10] direction and enables the realization of square etched mesas with edge along <100> direction. The etched substrate is dipped in 50% HF solution for 49 30 sec to remove the resist on the etched mesas and etched again in the NH 4 OH:H 2 O 2 :H 2 O for 6 sec to clean up the surface. After rinsing in DI water, the substrate with nanomesas is mounted on Molyblock, loaded into heater station in Mod C (Fig. 2.9) for thermal cleaning and then loaded into MBE growth chamber (Fig. 2.9) for growth. Figure 2.12 below captures schematically the major preparation steps mentioned above for the preparation of patterned substrates. The patterned substrate fabricated has arrays of mesas with a nearly vertical sidewall. Figure 2.13 (a) shows an illustrative SEM image of a mesa array with 5 μm pitch , mesa lateral size 324 nm and mesa depth ~500 nm. Fig. 2.13 (b) shows a 45 ˚ tilted SEM image of an individual as-etched mesa picked out of the array. The as-etched mesa has near vertical side walls controlled by the fluid dynamics of etching process. Such etched mesas are used as the starting template on which MTSQDs are synthesized and studied for their optical properties in this dissertation work. 50 Figure 2.12. Schematics of the major preparation steps needed for the preparation of patterned substrate. 51 Fig.2.13 (a) Top view SEM image of part of an as-etched 5 × 8 GaAs(001) nanomesa array and (b) 45 ˚ tilted view of a typical individual as-etched nanomesa inside the array. §2.3.3 Growth evolution on square mesa with <100> edges To understand and control the size-reducing growth of as-etched mesa tops in arrays such as shown in Fig. 2.13 and to thus synthesize ordered size- and shape- controlled MTSQD arrays, several samples were grown and studied as listed in Table 2.1. For all samples, a thin AlGaAs layer was deposited right after oxide desorption at the start of the growth to demarcate the starting mesa profile just before growth. This was followed by size-reducing homoepitaxial growth of GaAs, interspersed with multiple thin AlGaAs depositions. The AlGaAs layers are used as marker layers, an approach introduced by Madhukar et al [2.35, 2.60], to study the evolution of growth front profile and inter-facet atom migration through post growth cross- sectional TEM studies. 52 Table 2.1 List of samples representing growth on <100> edge oriented square mesas Sample Mesa size (size/height) Growth condition Growth structure RG111105 780,1885,2850, 3842nm/470nm P As#1 =0.9E-6, P As#2 =1.6E-6, τ Ga =4sec/ML, τ Al =4sec/ML, T Pyro =607C, [40ML Al 0.5 Ga 0.5 As+60ML GaAs]×7 RG120315 80-470nm/ 470nm P As#2 =1.6E-6, τ Ga =4sec/ML, τ Al =4sec/ML, T Pyro =607C, rotation 15rmp [10ML AlAs+80ML GaAs]×3+40ML AlAs+20ML GaAs+40ML AlAs+100ML GaAs RG130321 30-550nm/ 530nm P As#1 =1.0E-6, P As#2 =1.0E-6, τ Ga =4sec/ML, τ Al =4sec/ML, τ In =12sec/ML T Pyro =595C, rotation 15rmp [24ML Al 0.33 Ga 0.67 As +71ML GaAs]×2+24MLAlGaAs+61ML GaAs +12ML In 0.25 Ga 0.75 As +130ML GaAs RG130625 60-480nm/ 400nm P As#1 =1.0E-6, P As#2 =1.0E-6, τ Ga =4sec/ML, τ Al =4sec/ML, τ In =4sec/ML T Pyro =595C, rotation 15rmp 18ML Al 0.33 Ga 0.67 As +42ML GaAs+18ML Al 0.33 Ga 0.67 As +51ML GaAs+15ML In 0.2 Ga 0.8 As(MEE)+30MLGaAs(MEE)+1 5ML In 0.2 Ga 0.8 As (MEE)+110ML GaAs(MEE)+12ML InAs(MEE)+ 100ML GaAs +18ML Al 0.33 Ga 0.67 As +100ML GaAs RG130828 100-550nm/ 550nm P As#1 =1.0E-6, P As#2 =2.0E-6, τ Ga =4sec/ML, τ Al =4sec/ML, τ In =16sec/ML T Pyro =595C, rotation 15rmp [18ML Al 0.33 Ga 0.67 As +75ML GaAs]×2 +12ML In 0.25 Ga 075 As +100ML GaAs+18ML Al 0.33 Ga 0.67 As +100ML GaAs RG130916 65-500nm/ 490nm P As#1 =0.6E-6 P As#2 =2.0E-6, τ Ga =4.58sec/ML, τ Al =9.14sec/ML, τ In =4.58sec/ML T Pyro =600C, rotation 15rmp [18ML Al 0.33 Ga 0.67 As +110ML GaAs]×3+4.25ML In 0.5 Ga 0.5 As +100ML GaAs+18ML Al 0.33 Ga 0.67 As +100ML GaAs 53 Given our main objective of fabricating MTSQDs on <100> oriented square mesas, it was instructive to first check whether size-reducing growth occurs on <100> oriented square mesas and how the growth profile of the growth front on the mesa top evolve with increasing deposition. Sample RG111105 was patterned with only 4 different mesa sizes ranging from 0.7 - 4 μm, as listed in Table 2.1, to verify the size- reduction growth on mesa top. The large mesas of micro-meter scale are chosen to maintain consistency to the patterned mesa growth done by A. Konkar on <100> orientated square mesas of 1-5μm as a built-in check. The growth structure contains seven periods of 40ML Al 0.5 Ga 0.5 As and 60ML GaAs, resulting in GaAs/AlGaAs multilayered structure on mesa top. The growth was carried out at 607°C. GaAs was grown at the growth rate of 4 sec/ML under As 4 pressure of 1.6 x 10 -6 torr while the Al 0.5 Ga 0.5 As marker layer was grown at the growth rate of 2 sec/ML with As 4 pressure of 2.5 x 10 -6 torr. To maintain high quality GaAs/Al 0.5 Ga 0.5 As interface with minimum step density and composition fluctuation, growth interruption [2.83, 2.84] of 60 sec was employed after the growth of GaAs to allow recovery of the GaAs surface for the growth of Al 0.5 Ga 0.5 As layers. Growth interruption was utilized in all the GaAs/ Al 0.5 Ga 0.5 As multilayers in the grown samples listed in Table 2.1. Figure 2.14 (a)-(d) shows the top-view SEM images of one mesa from each of the four different mesa sizes on the sample after the growth GaAs/ Al 0.5 Ga 0.5 As multilayered structure. On the mesa top, there are two additional type of facets, facets A and B labeled in Fig 2.14, evolving in addition to the (001) top surface with the continued deposition on mesa top. The growth of these two types of facets leads to the size- 54 Figure 2.14. Top view SEM image of the mesa profile after 700ML growth of mesa with as-etched sizes (a) 3842nm, (b) 2850nm, (c) 1885nm and (d) 780nm and 30° tilted SEM image of the mesa profile of mesas sizes (e) 1885nm and (f) 780nm from sample RG111105. reducing growth of mesa top (001) surface. During the size-reducing growth, (001) top surface evolves from a near square shape (Fig. 2.14 (c) and (e)) to a rhombus 55 shape (Fig. 2.14 (d) and (f)) with all edges bounded by facets B. These indicates that facets B grows faster than facets A. By comparing the top view and 30° tilted SEM image of mesa of size 1880nm and 780nm, the two type of facets A and B are found to be {101} and {103} facets respectively. With continued growth on mesa top, the rhombus shape (001) surface should reduce its sizes dominated by the growth of the surrounded {103} facet and can eventually lead to the pinch-off of mesa top. It is confirmed from this growth that mesa top size-reducing growth occurs on our etched <100> orientated square mesa with near vertical side walls. The evolution of mesa profile is similar to that reported by A. Konkar on <100> orientated square mesa with ~{101}, side walls [2.76, 2.80]. To explore the mesa top size-reducing growth on <100> orientated square mesas of desired widths from 50-500nm and examine the evolution of the growth front profile on such mesa tops, three additional samples, RG120315, RG130321 and RG130625, were grown. All these samples have the pattern design as shown in Fig. 2.11 with a total of sixteen different as-etched mesa sizes and were overgrown with different deposition amounts to fully explore the evolution of the profile of the growth front on such <100> orientated square mesas. To enable high degree of flux uniformity, the samples were grown with the substrate rotating at 15rpm. The grown structure on sample RG120315 contains only GaAs/AlAs multilayered structures similar to that of RG111105. The sample RG130321 contains a single layer of 12ML In 0.25 Ga 0.75 As grown at T Pyro ~520°C, deposited with a monolayer growth time of 3 sec/ML, in addition to the three GaAs/Al 0.33 Ga 0.67 As layers shown in Table 2.1. The 56 sample RG130625 was grown with the knowledge of mesa top size reducing evolution of the growth front obtained from previous three sample growths and designed to have two 15ML In 0.2 Ga 0.8 As layers separated by 30ML GaAs and one 12ML InAs layer with the objective of forming quantum confined QD structures on multiple mesas of different sizes in the same sample using the fact that mesa pinch- off occurs later on mesas with larger starting lateral size at the same stage of deposition. To minimize the intermixing of In and Ga, as well as maintaining high interface quality between InGaAs and GaAs, the two 15ML In 0.2 Ga 0.8 As, the 12ML InAs and the GaAs spacer grown in between these InGaAs structures were grown at 480°C with migration enhanced epitaxy (MEE). Group III and Group V flux supply are alternated. 1ML of group III material is deposited without group V flux to enhance group III atom migration length for better material quality at low temperatures. Then, group V flux was supplied for 6 sec to form 1ML of compound on the surface with the shutter opening time calibrated through arsenic induced RHEED intensity oscillations discussed in Appendix B. The In 0.2 Ga 0.8 As layers in the planar regions serve as a reference quantum well and enable the optical characterization of InAs QD formed on the mesa-top before the mesa pinch-off. 57 Figure. 2.15. Top view SEM image of the growth front profile of mesas with as- etched sizes of (a) 550nm, (b) 480nm, (c) 430nm mesa after 417ML of growth on top. Panel (d) shows a cross-section (87°) tilted SEM image of the growth front profile of a 310nm as-etched mesa projected along [1 -1 0] direction. All these results are from sample RG130321. The dashed white line marked on panel (d) represents the angle between side facet and (001) surface. Figure 2.15 (a)-(c) shows the top-view SEM image of the growth front profile on mesas with as-etched mesa sizes of 550nm, 480nm and 430nm after 417ML total amount of material deposition and the panel (d) shows the cross-section (~87°) tilted SEM image of the growth front profile of a 310nm as-etched mesa projected along [1 -1 0] direction. {101} and {103} type of new facets showed up and evolved on the mesa top with continued growth similar to that observed in sample RG111105. With continued growth on mesa top, (001) top surface changes its shape from square to 58 rhombus shape (Fig. 2.15 (a)) coming from the faster growth rate on {103} facet compared to that of {101} facet controlled by the surface curvature stress gradient driven inter-facet atom migration. The (001) rhombus shape top surface further reduces its sizes (Fig. 2.15 (b)) and eventually leads to the pinch-off of mesa top as seen in the top view SEM image in Fig. 2.15 (c) and in the cross-section (~87°) SEM in Fig 2.15 (d). This mesa top pinch off is controlled dominantly by the growth of the {103} facet surrounding the (001) top facet. The angle between side facet and (001) plane, marked as θ in Fig. 2.15(d), is measured to be ~19°, confirming that the side facet that dominates top surface size-reducing growth and leads to the mesa top pinch off is {103} facet present on the mesa. To find out the growth front evolution after the {103} plane dominated mesa top pinch-off, sample RG130629 was growth. Figure 2.16 below shows the structural information obtained from sample RG130629 on mesas with growth front evolution beyond the {103} plane controlled mesa top pinch off. Fig. 2.16 (a) shows the top view SEM image of the growth front on mesa of as-etched size of 530nm after 529ML total material deposition. The growth front on the mesa top only contains a (001) top facet bounded by four {101} side facets as shown in Fig. 2.16 (a). This indicates that continued growth on the mesa top after {103} plane controlled mesa top pinch-off results in the shrinkage of {103} plane by the growth of {101} planes and a reopening of square shape (001) top surface evolved from {103} facets. The mesa pinch-off with {103} facet modifies surface curvature stress and in turn affects the relative growth rate of {101} planes and {103} planes controlled by inter-facet 59 atom migration. The resulting faster growth rate on {101} facet leads to the shrinkage of {103} plane by the growth of {101} planes. At a certain point of continued growth, the {103} plane starts evolving into planes close to (001) resulting in a (001) top surface bounded by {101} sidewalls (Fig. 2.16 (a)). The continued growth of {101} facet and inter-facet atom migration from {101} to (001) surface results in the second mesa top pinch off bounded by {101} facet as seen in the top view SEM image of the growth profile of an as-etched 430nm mesa (Fig. 2.16 (b)). Fig. 2.16 (c) and (d) shows respectively the cross-section (87°) SEM image of the growth profile on mesas before and after {101} plane controlled mesa top pinch-off shown in Fig. 2.16 (a) and (b). The angle between side facet and (001) plane, marked as θ in Fig. 2.15(d), is measured to be ~42°, confirming that the facet lead to this mesa top pinch off is {101} facet present on the mesa. 60 Figure 2.16 Top view SEM image of the growth front profile of mesas with as- etched sizes of (a) 530nm, (b) 430nm after 529ML amount of growth on top. Panel (c) and (d) shows a cross-section (87°) tilted SEM image of the growth front profile of the mesa shown in panel (a) and (c) projected along [1 -1 0] direction respectively. All these results are from sample RG130629. The dashed white line marked on panel (d) represents the angle between side facet and (001) surface. From the SEM studies of growth front profile evolution shown in Fig. 2.15 and Fig. 2.16, the growth front evolution on square mesa with edges along <100> direction has two pinch-off stages: (1) the {103} plane dominated mesa top pinch-off and, (2) the {101} plane dominated pinch-off. The {101} plane dominated pinch-off happens after the {103} plane dominated pinch-off. 61  Cross-Sectional TEM studies To verify the pinch-off, cross-section TEM imaging was carried out on one of the mesa of size ~280nm present in the T pattern (pattern for TEM) from sample RG130629. The mesa imaged have mesa top already pinched with the {101} side facet. The dark field cross-section TEM imaging was carried out under the condition of g=(0,0,2) with imaging geometry schematically captured in Fig. 2.17 (a). The electron beams incidents along [110] direction, perpendicular to the (001) growth direction, in to mesa. Fig. 2.17 (b) shows a cross-section TEM image of the mesa. The large thickness of the mesa structure and strains in the structure prevents the clear observation of GaAs, AlGaAs and InGaAs layers grown on mesa top. But it can still be seen from the image in Fig. 2.17 (b) that there are two mesa top pinch-off states occurred on the mesa top with the first one being {103} facet dominated and the second {101} facet dominated. To shine light on the origin of {101} and {103} facet evolved on the mesa top during mesa top size-reducing growth, cross-section TEM imaging at the corner of the mesa has been carried out to identify the starting mesa profile. During the growth, the first growth layer of AlGaAs marks the initial starting profile of the mesa before growth. The cross-section TEM image at the corner of mesa shown in Fig. 2.17 (c) does not clearly reveal the starting first AlGaAs layer. The geometry of the mesa corner reveals that there are side facets present on the corner of the mesa after chemical etching and thermal deoxidation in the MBE chamber marked with white dashed line in Fig. 2.17 (c). No reliable information on the facet index information can be obtained from the image. 62 Therefore, we can only conclude that the small side facets present at the corner of the mesa present after chemical etching and thermal deoxidation and these facets act as the precursor of the evolution of {103} and {101} facet on the mesa top. Higher resolution TEM imaging on the mesa is needed to provide more information on the types of side facets at the corner of the mesa to understand the presence and evolution of {103} and {101} facet during growth on mesa top. Figure 2.17 Panel (a) shows the TEM imaging geometry. Panel (b) shows the cross- section g(002) dark field image of the mesa with as-etched size of 400nm with {101} plane dominated mesa-top pinch-off after 529ML material growth taken with experiment geometry shown in panel (a). The {101} plane dominated pinch-off occurs after the first {103} plane (marked with white dashed line) dominated pinch- off. The dark white fringes arise from strains in the mesa and thickness contrast. Panel (c) is the zoomed in image of the corner of the mesa, indicating the presence of side facets around corner. [Images courtesy of Zachary Lingley of our group] From the study of growth front profile evolution on mesa top as a function of deposition, it is found that there are two mesa top pinch-off stage: the first one is {103} plane dominated and the subsequent one is {101} plane dominated. Figure 63 2.18 below capture schematically the observed growth front evolution on square mesas with edge along <100> directions. After etching and thermal deoxidation, small side facets showed up around the corner of the mesa. Upon growth, these side Figure 2.18. Schematic of growth evolution on square mesa with edges along <100> direction under employed growth condition. facets evolve into the growth of {101} and {103} planes on the mesa top (Fig. 2.18 (a)). With continued growth, (001) mesa top size reduces and changes its shape from square like to a rhombus shape along with the growth of {101} and {103} planes (Fig. 2.18 (b)). The rhombus shape (001) surface has its elongated diagonal direction along [1-10] direction. The four edges have ~ 22°±4° offset with respect to [1-10] direction as measured from the SEM image in Fig. 2.14 (d) and Fig. 2.15 (b). The rhombus shape base is symmetric with respect to the (1-10) and (110) mirror image and has arsenic dangling orbital ~22°±4° with respect to the edges as shown in Fig. 2.19 (a). The symmetric arsenic dangling orbital arrangement results in the symmetric atom migration from side facets to the top. The inter-facet migration of atoms driven by surface curvature gradients in between {101} and (001) and {103} and (001) facet leads to size reduction growth of (001) surface. The mesa top size further reduces and finally reaches first mesa top (001) pinch-off controlled by {103} 64 side facets (Fig. 2.18 (c)) indicating the faster growth rate of {103} facets compared to {101} facet. After the (001) mesa top surface pinch-off by {103} planes, continued growth on mesa results in the shrinkage of {103} plane suggesting a faster growth rate of {101} planes (Fig. 2.18 (d)) compared to {103} planes. The mesa pinch-off with {103} facet modifies surface curvature stress and in turn affects the relative growth rate of {101} planes and {103} planes controlled by inter-facet atom migration. At a certain point, the {103} plane starts to evolve into planes close to (001) resulting in a square (001) top surface bounded by {101} sidewalls (Fig. 2.18(e)). The square (001) top surface has edges along <100> direction and arsenic dangling orbitals at 45° with respect to the edges (Fig. 2.19(b)). The four fold symmetry of the newly opened (001) top surface enables symmetric migration of atom from side facets to the top. Therefore, the continued growth of {101} facet and inter-facet migration from {101} facet to (001) facet would lead to the second (001) mesa top pinch-off (Fig. 2.18 (f)) bounded by {101} facets. Fig. 2.19 Schematic of (a) the rhombus (001) top surface before {103} pinch-off (b) the square (001) top surface before {101} pinch-off with the mirror plane and arsenic dangling orbital orientation along edges shown. 65 Given the presence of two mesa top pinch-off stages, quantum dot bounded by {103} sidewall facets (Fig. 2.18 (c)) or bounded by {101} sidewall facets (Fig. 2. 18 (f)) can be grown at the apex of mesas. By controlling the size-reduction of (001) top surface through the evolution of the growth front on mesa top, quantum dot of controlled size and shape can be hence formed deterministically on mesa top. §2.3.4 Inter-facet atom migration To gain control on the size-reduction growth of (001) top surface through the evolution of the growth front on mesa top towards the objective of synthesizing quantum dot with desired size and shape on mesa top, understanding of the inter- facet atom migration and its effect on controlling the grown material thickness on mesa top and the mesa top size reduction is a critical need. The first careful study of the nature of inter-facet migration on patterned structures in the literature is done in our group by employing systematic marker layer growth on patterned GaAs (100) substrate and subsequent examination via high resolution TEM studies [2.72, 2.73]. In this section, we present our latest studies on inter-facet migration during the growth on our as-etch square mesas (Fig. 2.13) of size 50-500nm with <100> orientation employing the same approach undertaken in our group before. Multiple AlGaAs marker layers have been grown on all the samples containing square mesas listed in Table 2.1 to mark the growth front profile at different growth stages on the mesa top to assist the study of inter-facet migration using high resolution TEM imaging with measurement geometry shown in Fig. 2.17 (a). Due to limited TEM result on our samples, the study of inter-facet atom migration on the growth front 66 evolution is limited to the first {103} facet dominated pinch-off and presented in this section. Figure 2.20 shows a cross-section dark field TEM image from a mesa of as- etched 264nm size after the growth of 417ML material from sample RG130321 taken under the diffraction condition g=(0,0,2). Under such condition, the contrast of different material is obtained from the ratio of the difference of the f value of the atoms, i.e. 22 | | / | | Al As Ga As f f f f  . Ga and As having atomic number off by just two have nearly identical f values where Al has a quite different f value. Thus the scattered intensity of electron beam from GaAs is much lower and high contrast can be obtained using g=(0,0,2) with GaAs appears darker and AlGaAs appears lighter in the collected TEM image. On sample RG130321, there 24ML Al 0.33 Ga 0.67 As marker layer was grown on mesa top with inter-marker layer separated by 71ML GaAs spacer. The bright layers, marked with white arrows, in between darker layers imaged and shown in Fig. 2.20 is the grown 24ML Al 0.33 Ga 0.67 As layers. The thickness of the grown Al 0.33 Ga 0.67 As marker layer and the GaAs layer in between marker layers on the mesa top that contains information on inter-facet atom migration can be directly measured from the cross-section TEM image of the grown mesas. 67 Figure 2.20. Cross-section dark field TEM image with g=(0,0,2) of a 264nm as- etched mesa from sample RG130321 after 417ML material growth. [Image courtesy of Zachary Lingley of our group] Fig. 2.21 (a) summarizes the measured thickness of material grown on mesa top after four different material deposition amount (95ML, 119ML, 190ML and 214ML deposition) marked by the marker layers in TEM images obtained on mesas with four different as-etched mesa sizes of 264nm, 300nm, 392nm and 780nm. The information on the first three mesas is from sample RG130321 while the last one is from sample RG111105. These two samples are grown with same marker layer structure under same growth condition, i.e. temperature, arsenic pressure and growth rate as seen in Table 2.1, on mesas of same as-etched curvature. Therefore, the inter- facet atom migration is expected to be same for these two growth runs. Data obtained from these two grown runs are combined and shown in Fig. 2.21 for the analysis and understanding of inter-facet atom migration during growth on mesa top. The actual amount grown on the mesa top is always larger than the deposited amount as seen in Fig. 2.21 (a) and more clearly in Fig. 2.21 (b) that shows the normalized thicknesses of the grown layers, ratio of the actual amount grown on mesa top and the deposited amount, as a function of mesa sizes for the four different deposition amounts. The 68 normalized value being larger than one indicates inter-facet atom migration from the side facets onto (001) top surface resulting in the larger thickness of the material grown on mesa top (001) surface. Figure 2.21. Data on (a) the actual and (b) normalized thickness, normalized with respect to deposition amounts, of materials with different deposition amount grown on mesa top from four different mesa sizes obtained from TEM studies on 69 nanomesas. The mesa of size 264nm, 300nm, and 392nm are from sample RG130321 while the mesa of size 780nm is from sample RG111105. If inter-facet atom migration rate from the sidewalls to the mesa top were independent of the mesa top size, then the growth rate of mesa top will be expected to increase with decreasing mesa top size. Though the data on 392nm and 780nm mesa followed this trend, for the mesa top size < 390nm, mesa top growth rate is not increasing with decreasing mesa top size, but rather decreasing with decreasing mesa top size as shown in Fig. 2.21. Such a variation of growth rate with reducing mesa top size is related to the change of net lateral adatom migration and possible changes in effective growth condition on such fairly small mesa top sizes induced by the lateral atom migration itself. However, such a growth rate variation is helpful in reducing the size fluctuations in the confined structure arising from variations in the starting mesa size in the same array and thus plays an important role in the fabrication of size-shape controlled quantum dots with uniform optical response. The varying inter-facet atom migration plays an important role in controlling the mesa top size-reduction growth. The variation in the growth rate of the (001) surface and the observed non-linear nature of this variation indicates a varying inter-facet atom migration rate, hence a varying inter-facet atom migration length with growth. Given this varying nature of atom migration, knowing the average inter-facet atom migration under employed growth condition on square as-etched mesas with <100> edge orientation is, therefore, of great importance help to predict and guide the size- 70 reducing growth of the (001) top surface utilizing homoepitaxial growth and in turn control the size and shape of the heteroexpitaxially grown SQD on mesa top.  Analytical Modelling To understand the observed inter-facet migration, we employed an analytical model to calculate the average atom migration rate from the side facets, {103} and {101} facets, to the top (001) facet based on the known information on thickness of material grown on mesa top after different material deposition amount. The presented analytical model is a simple phenomenological model based on the known facet geometry on the mesa top obtained from SEM studies of growth front evolution discussed in previous section. In the presented analysis below, we assume that atom migrates from {103} facets to the top (001) facet and from {101} facets to {103} facets. Direct inter-facet atom migration from {101} facets to (001) surface has been ignored and atom from {101} facets migrate to (001) top surface through {103} facets, resulting in faster growth rate of {103} facet compared to {101} facets which is consistent with SEM findings shown in Fig. 2.14 and Fig. 2.15. Symbols used in the model are listed as following: a: the length of the square mesa edge x: the deposition amount on the mesa h: the thickness of the amount of material grown on mesa top η 1 : the atom migration rate from {101} facet to {103} facet, percentage of atoms impinged on {101} facet migrating to {103} facet η 2 : the atom migration rate from {103} facet to (001) facet, percentage of 71 atoms impinged on {103} facet migrating to (001) facet S top : the surface area of (001) top surface S {101} : the surface area of {101} facets on the mesa projected into (001) plane S {103} : the surface area of {103} facets on the mesa projected into (001) plane Following the conservation law of mass, the material deposited on the (001) surface combined with the material migrated from {103} and {101} facets to the (001) surface should be equal to the actual amount grown on (001) facet. We, therefore, have: {103} 2 {101} 1 2 top top S dh S dx S dx S dx       (Eq. 2.7) The surface area of different facets are taken to be the area of the surface projected to the (001) plane to take into account the variation in the incoming flux at the growth direction of each different facets. Guided by the SEM results on mesa top and 3D facet geometry before {103} plane dominated mesa top pinch-off, the projected facets surface area are {101} 2 (2 / tan ), / (1 1/ tan ) 2 , / (1 1/ tan ) h a h h h a S ah h a            (Eq. 2.8) 2 {103} 22 12 cos , / (1 1/ tan ) 2 ( 2 2 2 ) tan , / (1 1/ tan ) h h a S a ah a h h a               (Eq. 2.9) 22 2 2 (2 / tan ) 12 cos , / (1 1/ tan ) ( 2 2 2 ) tan , / (1 1/ tan ) top a h a h h h h a S a h h a                    (Eq. 2.10) representing the two phases of (001) top surface reduction with (001) top surface of near square shape and rhombus shape (Fig. 2.22) before the {103} plane pinch-off. 72 Angle θ and α used in Eq. 2.8-2.10 representing the geometry of {101} side facet and the top (001) facet and are marked and shown in Fig. 2.22. Figure 2.22 Schematic of the two phases of (001) top surface reduction with (001) top surface of (a) near square shape and (b) rhombus shape. Angle θ and α used in defining the projected area of {101} and top (001) facets are labeled in red where θ=15°±2° and α~22°±4°. The material thickness grown on (001) surface, h, as a function of deposition amount, x, can be calculated by inserting Eq. 2.8-2.10 into Eq. 2.7 with inter-facet atom migration rate η 1 and η 2 as input parameter. Therefore, the average inter-facet atom migration rate η 1 and η 2 can be found by fitting the calculated h-x relation with measured date of h and x from mesas. Figure 2.23 (a) and (b) show two representative examples of the fitted result on the measured h and x from mesa of as- etched size of 300nm and 780nm using the analytical model captured in Eq. 2.7. For the 300nm mesa, the atom migration rate from {101} to {103} facets is found to be 40% and the atom migration rate from {103} facets to (001) top surface is 10%. Compared with 300nm mesa, the atom migration rates are higher for the 780nm mesas. The atom migration rate from {101} to {103} facets is found to be 85% and the atom migration rate from {103} facets to (001) top surface is 70% for the 780nm mesa. Fig. 2.23 (c) shows the obtained result on atom migration rate η 1 and η 2 from 73 the above mentioned four mesas of size 264nm, 300nm, 392nm and 780nm. The atom migration rate in between {101} and {103} facets increases from 30% to 85% while the migration rate in between {103} and (001) facets increase from 8% to 70% as the as-etched mesa sizes increases. The migration rate, the percentage of impinging atom on the facet migrated to a different facet, is determined by the atom migration length on that facet under the employed growth condition. Given that at the same stage of deposition mesa pinch-off occurs later on mesas with larger starting lateral size, the increasing migration rate with mesa sizes suggests that the atom inter-facet migration length is decreasing as growth continues on mesa top. Guided by the limited data of growth on mesas of different sizes, we found that the Gallium atom migration length along {101} surface indirectly derived from data in Fig. 2. 23(b) is in the range of ~80-660nm while the Ga atom migration length along {103} surface is in the range of ~20-180nm during the growth on mesa top. 74 Figure 2.23 Fitted results (dark line) of the measured thickness of material (dark dot) grown on (a) 300nm and (b) 780nm mesa top as a function of deposition using analytical model captured in Eq.2.7. (c) atom migration rate from {101} facet to {103} facet (black dot) and from {103} facet to (001) facet (red dot) obtained from the fitted results on four different mesas of size 264nm, 300nm, 392nm and 780nm. The analytical model with known facets arrangement on the mesa top provides a phenomenological understanding of the average inter-facet atom migration during growth. The results obtained from such model on inter-facet atom migration rate is used, therefore, phenomenologically to predict and control the size- reduction growth of (001) top surface for the controlled formation of MTSQDs with desired sizes before the {103} facets dominated mesa top pinch-off discussed in section §2.4. 75 The analytical model used here provides only guiding information on average inter-facet migration on pre-known existing facets on the mesa without the ability to identify why the particular migration take place, how the inter-facet migration affects the shapes of evolving mesa profiles and hence to understand the growth on mesas. To understand the growth on mesas, one needs to learn more about the competing dynamical processes of atom migration (hop to adjacent sites), atom evaporation into vapor phase, surface chemical reactions, and atom incorporation into crystal on different evolving facets, and take these into account in simulations based on an atomistic kinetic model. Only a macroscopic and phenomenological framework capturing the atom migration through a phenomenological atom diffusion coefficient D and the atom desorption and incorporation through lifetimes t v and t c respectively during MBE growth, introduced by Ohtsuka and coworkers [2.85], exsists. The quantities D and t v are assumed to be independent of facet orientation while t c is dependent on facet orientation. Dynamics of growth evolution are captured in differential equations containing the above mentioned quantities as input parameters. This approach provides after-the-fact help in assessing the effect of inter-facet migration on the shapes of evolving mesa profiles during growth without the ability to identify why the particular migration take place and hence to understand the growth on mesas. On atomic level, considering a Ga adatom on any general surface, there is a finite concentration of adatoms on the surface even in the absence of any Ga flux. 76 Assuming local equilibrium between the kink sites and the surface adatoms, the adatom concentration is expected to be an activated process given by [2.80, 2.85]: 0 exp( / ) eq a B C C E k T  (Eq. 2.11) where E a is the activation energy required to transform an atom present at a kink site into the adatom phase. C 0 is the pre-exponential factor which contains the entropy contributions and is related to the equilibrium density of kink sites. During growth the surface concentration of adatoms is modified due to the incoming flux. Therefore, the adatom concentration in the steady state during growth can be written as: gr eq C C F   (Eq. 2.12) where  is the total adatom lifetime on the surface given by  = (  c  e /(  c +  e )) and F is the flux impinging on the surface. The values of the above parameters that determine the adatom concentration will depend on the nature of the planes being considered. Due to the different nature of the various planes, the concentration of adatoms on the various planes during growth is different. The values of C gr of different facets determine the inter-facet adatom migration and hence growth front profile. As seen in Eq. 2.12, term C eq plays an important role in affecting the steady state adatom surface concentration C gr during growth but it unfortunately has been ignored in the macroscopic and phenomenological models [2.86]. The representation of growth that faithfully captures its nature should be based on the steady state adatom surface concentration C gr and the net dynamical change of the adatom concentration. 77 The net adatom concentration on both the mesa top and the sidewall planes is determined by the balance between the vapor flux and the rate at which the adatoms incorporate or re-evaporate on these planes. Depending on the relative value of t v and t c for the mesa top and the sidewall planes, the net adatom concentration gradient could be from the sidewall to the mesa top even though the flux impinging on the sidewall is less than that impinging on the mesa top that could result in inter-facet migration from the sidewall to the mesa top. The rates for adatom incorporation and evaporation are given by the inverse of their lifetimes and these rates themselves are activated processes with activation energies for incorporation (E i ) and evaporation (E e ), respectively. It is the atomic understanding based on activated processes of atom concentration, incorporation and evaporation that determines the inter-facet migration and controls the growth evolution of different facets on mesa top. From our observed atom migration from {101} and {103} facets to (001) facets discussed in section §2.3.3, we can conclude that C eq (001) < C eq {101} /{103} and C gr (001) < C gr {101} /{103} . The observed rhombus shape (001) facets with four nearest facets being {103} facets suggests C gr {101} < C gr {103} . Deeper understanding of activation energies controlling atom concentration, incorporation and evaporation can be obtained in the systemic work on exploring the effect of temperature on atom migration and simulations based on atomic nature of the involved dynamics process. §2.4 Mesa-top single quantum dot arrays Based on the evolution of growth front on mesa top, MTSQDs with {103} side wall and {101} side wall can be formed on mesa top before {103} and {101} 78 facets dominated mesa top pinch-offs by controlling the amount of homoepitaxial growth to reduce mesa to (001) surface to desired pinch-off stage with desired (001) surface sizes. Two primary MTSQDs samples are of central importance to our SESRE- based approach to realizing spatially regular and spectrally uniform arrays of single quantum dots that act as single photon sources. These are: (1) sample RG130625 with 12ML InAs MTSQDs bounded by {101} side facets and, (2) sample RG130916 with 4.25ML In 0.5 Ga 0.5 As MTSQDs bounded by {103} side facets. Guided by the existing 3D island based SAQDs, as well as our groups past work on SESRE based InAs/GaAs mesa-top quantum dots [2.35, 2.36], the binary InAs based quantum dots emit at >1000nm wavelengths. By contrast, the 50% InGaAs/GaAs based quantum dots are expected to emit in the 920 to 950nm range. Though discussed later, it is appropriate to note here that, given our focus on single photon emitters, this dissertation work has focused upon the latter wavelength regime as Si APDs provide reasonable efficiency single photon detectors in this wavelength regime and no efficient and affordable detectors were available in the longer wavelength regime. As listed in Table 2.1, homoepitaxial growth of GaAs buffer layer with a few monolayer thick AlGaAs marker layers interspersed on sample RG130625 is carried out at T pryo =595˚C, P As4 =2.0E-6 Torr, and Ga delivery time of τ Ga =4 sec/ML to (1) recover from any residual damage remaining after chemical etching and deoxidation 79 and (2) control the mesa top size reduction to bring it to the desired size < 30 nm for the growth of flat single QD on mesa top. The InAs QDs are grown at τ In =4 sec/ML with T pyro =480 ˚C. The L-shaped unpatterned region on the sample is used for RHEED to monitor the surface and calibrate growth conditions. The QDs are capped by 200 ML GaAs to create three dimensional confinements and to protect the QDs from impurities and defects on the GaAs surface. Figure 2.24 (a) shows schematically the grown layer structure on mesas with mesa size ~125nm with expected InAs MTSQD of base length of ~15 nm and height ~5 nm (Fig. 2.24 (b)) formed on mesa top after 299ML GaAs buffer growth interspersed with AlGaAs and InGaAs marker layers. The size of MTSQDs formed before {101} facet controlled mesa top pinch off is estimated based on growth evolution of the growth front on mesa top and 3D geometry of mesa. Figure 2.24 Schematic of the grown structure (panel a) on mesa of size 125nm with targeted InAs QD bounded with {101} side walls (panel b) on the mesa top of sample RG130625. 80 Photoluminescence (PL) from individual mesa of as-etched size ~125nm has been collected and studied using our home-built micro-PL setup. Details of the instrumentation of the setup is captured in Appendix C. To identify the emission from MTSQDs on mesa top, PL data from the planar region were collected as a built-in reference for the sample. Due to 7% lattice mismatch in between InAs and GaAs, InAs grown on the planar GaAs forms 3D islands self-assembled quantum dots (SAQDs) after a critical deposition of 1.5ML of InAs [2.6, 2.7]. The accumulated InAs deposition amounts on GaAs can lead to the formation of defects in the QDs to relax the accumulated lattice mismatch strain in the grown InAs structure [2.6, 2.7] when the deposition amount has gone beyond the amount needed for forming well developed coherent QDs. The low PL signal in the QD emission wavelength ~1050nm from the 12ML InAs in the planar region of sample RG130625 shown in Fig. 2 25 (a) indicates that the 12ML InAs SAQDs formed on the planar region are defected QDs with low quantum efficiency and optical yield. The peak ~880nm in Fig. 2.25 (a) is the PL response from the grown 15ML In 0.2 Ga 0.8 As quantum wells (QWs) in the planar region. The 12ML InAs QD formed on mesa top of size ~15nm should be defect free due to the sufficient stress relaxation at the free surfaces of the mesa [2.37, 2.79]. Figure 2.25(b) shows the PL spectra collected one mesa of as-etched size ~125nm that contains a 12ML InAs MTSQD. The clear peak emission peak at wavelength ~1122nm, which is absent in the planar region, showed up when collecting PL from mesa. Power dependent studies (Fig. 2.25 (c)) of this peak show that the PL intensity saturates at very low power of 4 μW (~130W/cm 2 ) 81 indicating the 3-dimensionally confined nature of the electron states involved. Therefore, the observed emission peak at 1122nm is identified as the emission from 12ML InAs MTSQD on the mesa top. The low PL emission from 1000-1100nm seen in Fig. 2.25 (b) is from 12ML InAs SAQDs in the valley region in between mesas collected under the 1.25μm diameter optical beam. The observed high emission intensity and the longer wavelength of the 12ML InAs QD emission peak indicates the strain relaxation on the mesa top which results in the formation of defect free QD on mesa top. The strain relaxation lowers the bandgap of InAs and in turn modifies of QD confinement potential which results in the modification and elongation of the QD emission wavelength compared to that of strained SAQDs. The strain relaxation of the mesa with top size <30nm after the homoepitaxial size-reducing growth helps the formation of QDs with highly lattice mismatched materials, enriching the material combinations that can be utilized to form QDs with large range of spectral response, i.e. 800-1200nm using InGaAs/AlGaAs material. 82 Figure 2.25 PL collected at 77K using micro-PL setup from (a) the planar region and (b) a single mesa of size of 125nm from sample RG130625 containing, respectively, InAs SAQDs and mesa-top single quantum dot. Excitation is CW at 780nm with power 375nW (power density ~30W/cm 2 ) and spectral resolution of 1nm. Panel (c) shows the power dependence of the observed 1122 nm PL peak intensity at 77.4 K.  InGaAs Alloy Quantum Dots Guided by the objective of assessing the potential of SESRE grown MTSQDs as ordered spectrally uniform SPSs and EPSs and the practical issues of the limited detection range, less than 950nm, of the standard Si avalanche photodiodes (APDs), we focused on creating InGaAs MTSQDs with {103} side walls that has smaller volume compared to {101} facet bounded MTSQDs with same base size and hence 83 shorter emission wavelength aimed at creating QDs with emission wavelength less than 950nm. InGaAs deposition in the planar region can lead to the formation of SAQDs. Therefore, the chosen InGaAs grown on patterned sample should satisfy two conditions: (1) QD emission wavelength is within the APD detection range and (2) the corresponding InGaAs SAQDs formed during the growth in the planer region of the sample should have very low QD density, i.e. ~1-2/μm 2 . For InGaAs SAQDs, the well-developed In 0.5 Ga 0.5 As SAQDs, i.e. 6ML In 0.5 Ga 0.5 As, have QDs base ~50nm and height ~10nm and high QD density > 1000/ μm 2 with PL emission centered around 970nm. Given the targeted MTSQD size of < 30nm, In 0.5 Ga 0.5 As material combination can provide the desired MTSQD emission wavelength within the APD detection region for optical studies of the dynamics and statistics of emitted photons from the QD. To minimize the optical signal from the SAQDs formed in the planer region during the In 0.5 Ga 0.5 As MTSQD growth, the In 0.5 Ga 0.5 As deposition amount needs to be controlled to limit SAQDs density to be ~1-2 /μm 2 . A series of In 0.5 Ga 0.5 As SAQDs sample were grown with different deposition amount to find the optimal condition for the growth of low density SAQDs. Three different delivery amounts, 4.25ML, 4.5ML and 6ML, of In 0.5 Ga 0.5 As were chosen and used to grow In 0.5 Ga 0.5 As SAQDs for structural and optical characterization. These deposition amounts were chosen based on the understanding of the growth evolution of 3D islands QD formation obtained by Ramachandran et al [2.87, 2.88] through systematic structural and optical study of InAs grown on GaAs as a function of InAs deposition. It is found that 3D islands evolve from 2D clusters (i.e. 1ML high) and 84 the density of 3D islands increases rapidly with increasing InAs delivery amount after the onset of 3D islands formation. To realize formation of low density SAQDs, the deposition amount should be very close to the deposition amount required for the onset of 3D islands. Correspondingly, the delivery amount for In 0.5 Ga 0.5 As SAQDs growth is chosen to be 4.25ML and 4.5ML which are the amount just enough for the onset of SAQDs (~4.2ML) formation. The In 0.5 Ga 0.5 As SAQDs with above mentioned deposition amounts have been grown with and without the 170ML GaAs capping layer for optical and structural characterization respectively. Figure 2.26 (a) and (b) shows the atomic force microscope (AFM) image obtained from the uncapped In 0.5 Ga 0.5 As SAQDs sample with 4.25ML and 4.5ML deposition amount. The 4.25ML In 0.5 Ga 0.5 As SAQDs has QD density ~2/μm 2 and the 4.5ML In 0.5 Ga 0.5 As SAQDs has QD density ~400-500/μm 2 . To evaluate the QD density optically, SAQDs samples are excited with excitation energy below GaAs band gap excitation (863nm excitation) and collected over 300 μm diameter region to sample the ensemble PL behavior of the SAQDs. The PL intensity measured under such condition directly reflects the density of QDs in the sample. The emission intensity from the large-area PL spectra of the 4.25ML In 0.5 Ga 0.5 As SAQDs is ~100 times lower than that from the 4.5ML In 0.5 Ga 0.5 As SAQDs as shown in Fig. 2.26 (c), indicating the low QDs density of less than 5/μm 2 in the 4.25ML In 0.5 Ga 0.5 As SAQDs sample consistent with AFM results. Compared to the well-developed In 0.5 Ga 0.5 As SAQDs with 6ML deposition amount, the SAQDs formed with deposition amount just above the critical amount needed for the formation of SAQDs 85 are of smaller sizes as can be seen in the shorter emission wavelength (~910nm) of 4.25ML and 4.5ML In 0.5 Ga 0.5 As SAQDs with respect to the 6.0ML In 0.5 Ga 0.5 As SAQDs. Guided by these above mentioned results regarding QD density and emission wavelength, 4.25ML In 0.5 Ga 0.5 As is chosen to be the MTSQDs structure with QD emission response lying within the APD detection region and with minimum density of SAQDs in the planar region to act as build in reference in the sample without interfering the signal from MTSQD. Figure 2.26 Panels (a) and (b) show AFM images of the 4.25ML (panel (a)) and 4.5ML (panel (b)) In 0.5 Ga 0.5 As SAQDs. A low density of <5/μm 2 in the 4.25ML In 0.5 Ga 0.5 As SAQDs sample is seen. Panel (c) shows the large area PL spectra collected at 77K with 863nm excitation (excitation power density ~17W/cm 2 ) from 86 three In 0.5 Ga 0.5 As SAQD samples with 4.25ML (RG130307-3-F2), 4.5ML (RG130215-2-F2) and 6.0ML (RG121115-2-F2) deposition. The PL intensity ratio of emission from 4.25ML and 4.5ML In 0.5 Ga 0.5 As SAQDs. Sample RG130916 was grown with 4.25ML In 0.5 Ga 0.5 As layer on mesa top surface size-reduced after 384ML GaAs buffer growth interspersed with AlGaAs markers as shown in Figure 2.27 (a). The GaAs buffer layer with a few monolayer thick AlGaAs marker layers interspersed is carried out at T pryo =600˚C, P As4 =2.5E-6 Torr, and Ga and Al delivery time of τ Ga =4.58 sec/ML and τ Al =9.14 sec/ML on sample RG130625. The thickness of the overall buffer layer grown before the deposition of InGaAs is guided by the growth front evolution observed on mesa and the inter-facet atom migration rate obtained from the analytical model discussed in the previous section to reduce the mesa top (001) surface of size 430nm to (001) top surface of rhombus shape with edge of ~13nm before the {103} facet pinch-off. The 4.25ML In 0.5 Ga 0.5 As was grown at τ In =4.58 sec/ML with T pyro =520 ˚C after the growth of GaAs buffer to form QD bounded by {103} side facets. Subsequent 200ML GaAs interspersed with an 18ML AlGaAs layer was grown under the same condition for GaAs buffer growth to cap the InGaAs QD and protect it from the surface defects. The formed QD is of rhombus base with estimated base edge length ~13nm. Based on the side facet orientation and the shape of the rhombus base shown in the SEM image in Fig.2.15(b) and the schematics in Fig. 2.22(b), the base edges are found to be along ~[1 -3 0] and ~[3 -1 0] direction. The 4.25ML deposition amount on the mesa top results in the formation of MTSQDs with height of 3 nm estimated based on the known growth evolution of the growth front on mesa top and 87 the dependence of thickness of material grown on mesa top shown in Fig. 2.23. Schematic of the formed MTSQD with {103} side facet along with the edge and facet orientation is shown in Fig. 2.27 (b). Figure 2.27 (a) Schematic of the grown structure (panel a) on mesa of size 430nm with the targeted 4.25ML In 0.5 Ga 0.5 As QD bounded with {103} side walls on the mesa top of sample RG130916. (b) Schematic of the formed QD size and shape bounded with {103} side walls (upper one). The QD base shape and edge orientation is shown in the lower panel. The grown 5×8 array of mesas with size ~430nm containing MTSQD is shown in Figure 2.28 below. The top view and the 60° tilted SEM image of part of the 5 × 8 array are shown in Fig. 2.28 (a) and (b), respectively. The inset in these two panels shows the magnified image of a single mesa in the array. The MTSQDs in the array contain a single QD on top of each mesa. The position of the SQD on mesa top is captured in the schematic drawing of the cross-section of the grown mesa along the (1 -1 0) facet shown in Fig. 2.28 (c). 88 Figure 2.28 A (a) top view and (b) a 60° tilted SEM image of the SESRE-grown 4.25ML In 0.5 Ga 0.5 As MTSQD array with the MTSQD residing on the top of each nanomesa. The inset is a magnified image of a MTSQD-bearing single nanomesa (scale bar of 300 nm). Panel (c) shows the schematic of the cross-section of the mesa along (1-10) facet indicating the geometry and position of QD (in red) on mesa top. The dot dashed black line represents the growth front of the capping layer on top. Optical properties of this synthesized array of 4.25ML In 0.5 Ga 0.5 As MTSQD with {103} side walls are studied systematically using our home-built micro-PL setup to assess their potential use as ordered spectrally-uniform SPSs and EPSs as discussed in Chapters 3 and 4. The low density In 0.5 Ga 0.5 As SAQDs in the planar region act as a built in reference for the optical studies of the MTSQDs. Chapter 2 References: [2.1] A. Madhukar, “Growth of semiconductor heterostructures on patterned substrates: defect reduction and nanostructures”, Thin Solid Films. 231, 8 (1993). 89 [2.2] A. Madhukar and S. V. Ghaisas, “The nature of molecular beam epitaxial growth examined via computer simulations,” CRC Critical Rev. Solid State Mater. Sci., 14, 1 (1988). [2.3] A. Madhukar, “A unified atomistic and kinetic framework for growth front morphology evolution and defect initiation in strained epitaxy”, J. Cryst. Growth. 163, 149 (1996). [2.4] S. Buckley, K. Rivoire and J. Vučković, “Engineered quantum dot single photon sources”, Rep. Prog. Phys. 75, 126503 (2012). [2.5] P. Lodahl, S. Mahmoodian and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures”, Rev. Mod. Phys. 87, 347 (2015). [2.6] A. Madhukar, “Stress engineered quantum dots: Nature’s Way”, Chapter 2 in Nano-Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. [2.7] V. A. Shchukin, N. N. Ledentsov and D. Bimberg, Epitaxy of Nanostructures , Springer Verlag: Berlin, 2004. [2.8] C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot”, Phys. Rev. Lett. 86, 1502 (2001). [2.9] X. M. Dou, X. Y. Chang, B. Q. Sun, Y. H. Xiong, Z. C. Niu, S. S. Huang, H. Q. Ni, Y. Du, and J. B. Xia, “Single-photon-emitting diode at liquid nitrogen temperature,” Appl. Phys. Lett. 93,101107 (2008). [2.10] L. Dusanowki, M. Syperek, A. Maryński, L. H. Li, J. Misiewicz, S. Höfling, M. Kamp, A. Fiore, and G. Sęk, “Single photon emission up to liquid nitrogen temperature from charged excitons confined in GaAs-based epitaxial nanostructures,” Appl. Phys. Lett. 106, 233107 (2015). [2.11] W. -M. Schulz, R. Roβbach, M. Reischle, G.J. Beirne, M. Bommer, M. Jetter, and P. Michler, “Optical and structural properties of InP quantum dots embedded in (AlxGa1-x) 0.51 In 0.49 P,” Phys. Rev. B. 79, 035329 (2009). [2.12] S. Kako, C. Santori, K. Hoshino, S. Gtzinger, Y. Yamamoto and Y. Arakawa, “A gallium nitride single-photon source operating 200 K”, Nat. Mater. 5, 887 (2006). [2.13] O. Fedorych, C. Kruse, A. Ruban, D. Hommel, G. Bacher, and T. Kümmell, “Room temperature single photon emission from an epitaxially grown quantum dot,” Appl. Phys. Lett. 100, 061114 (2012). [2.14] M. B. Ward, O. Z. Karimov, D. C. Unitt, Z. L. Yuan, P. See, D. G. Gevaux, A. J. Shields, P. Atkinson, and D. A. Ritchie, “On-demand single-photon source for 1.3 μm telecom fiber”,Appl. Phys. Lett. 86, 201111 (2005). 90 [2.15] A. J. Shields, “Semiconductor quantum light sources”, Nat. Photonics, 1, 215 (2007). [2.16] M. Reischle, G. J. Beirne, W. M. Schulz, M. Eichfelder, R. Rosbach, M. Jetter, and P. Michler, “Electrically pumped single-photon emission in the visible spectral range up to 80K”, Opt. Express, 16, 12771 (2008). [2.17] S. Deshpande, T. Frost, A. Hazari, and P. Bhattacharya, “Electrically pumped single-photon emission at room temperature from a single InGaN/GaN quantum dot,” Appl. Phys. Lett. 105, 141109 (2014). [2.18] M. B.Ward, T. Farrow, P. See, Z. L. Yuan, O. Z. Karimov, P. Atkinson, K. Cooper, and D. A. Ritchie, “Electrically driven telecommunication wavelength single-photon source”, Appl. Phys. Lett. 90, 063512 (2007). [2.19] D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel and Y. Yamamoto, “Photon antibunching from a Single Quantum-Dot- Microcavity System in the Strong Coupling Regime”,Phys. Rev. Lett. 98, 117402 (2007). [2.20] V. Giesz, O. Gazzano, A. K. Nowak, S. L. Portalupi, A. Lemaȋtre, I. Sagnes, L. Lanco and P. Senellart, “Influence of the Purcell effect on the purity of bright single photon sources”, Appl. Phys. Lett. 103, 033113 (2013). [2.21] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs”, Nature Photon. 8, 224 (2014). [2.22] X. Ding, Y. He, Z. –C. Duan, Niels Gregersen, M.-C. Chen, S. Unsleber, S. Maier, C. Schneider, M. Kamp, S. Höfling, C. Lu and J. Pan, “On-demand single photons with high extraction efficiency and near-unity indistinguishability from a resonantly driven quantum dot in a micropillar”, Phys. Rev. Lett.116, 020401 (2016). [2.23] N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gónez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G.White, L. Lanco and P. Senellart, “Near-optimal single-photon sources in the solid state”, Nature Photon. 10, 340 (2016) [2.24] J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lelanne and J. M. Gerard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire”, Nature Photon. 4, 174 (2010). 91 [2.25] M. E. Reimer, G. Bulgarini, N. Akopian, M. Hovevar, M. B. Bavinck, M. A. Verheijen, E. P.A.M. Bakkers, L. P. Kouwenhoven and V. Zwiller, “Bright single- photon sources in bottom-up tailored nanowires”, Nature Commun. 3, 737 (2012). [2.26] G. Bulgarini, M. E. Reimer, M. B. Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers and V. Zwiller, “Nanowire waveguide launching single photons in a Gaussian mode for ideal fiber couping”, Nano Lett. 14, 4102 (2014). [2.27] M. E. Reimer, G. Bulgarini, A. Fognini, R. W. Heeres, B. J. Witek, M. A. M. Versteegh, A. Rubino, T. Braun, M. Kamp, S. Höfling, D. Dalacu, J. Lapointe, P. J. Poole and V. Zwiller, “Overcoming power broadening of the quantum dot emission in a pure wurtzite nanowire”, Phys. Rev. B. 93, 195316 (2016). [2.28] K. H. Madsen, S. Ates, J. Liu, A. Javadi, S. M. Albrecht, I. Yeo, S. Stobbe and P. Lodahl, “Efficient out-coupling of high purity single photons from a coherent quantum dot in a photonic-crystal cavity”, Phys. Rev. B. 90, 155303 (2014). [2.29] S. Laurent, S. Varoutsis, L. L. Gratiet, A. Lemaître, I. Sagnes, F. Raineri, A. Levenson, I. Robert-Philip and I. Abram, “Indistinguishable single photons from a single-quantum dot in a two-dimensional photonic crystal cavity”, Appl. Phys. Lett. 86, 163107 (2005) [2.30] A. Schwagmann, S. Kalliakos, I. Farrer, J.P. Griffiths, G. A. C. Jones, D. A. Ritchie and A. J. Shields, “On-chip single photon emission from an integrated semiconductor quantum dot into a photonic crystal waveguide”, Appl. Phys. Lett. 99, 261108 (2011). [2.31] A. Laucht, S. Pütz, T. Günthner, N. Hauke, R. Saive, S. Frédérick, M. Bichler, M.-C. Amann, A.W. Holleitner, M. Kaniber and J.J. Finley, “A waveguide-coupled on-chip single photon source”, Phys. Rev. X. 2, 011014 (2012). [2.32] M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide”, Phys. Rev. Lett. 113, 093603 (2014). [2.33] A. Faraon, A. Majumdar, D. Englund, E. Kim, M. Bajcsy and J. Vučković, “Integrated quantum optical networks based on quantum dots and photonic crystals”, New J. Phys. 13, 055025 (2011). [2.34] O. Benson, C. Santori, M. Pelton and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot”, Phys. Rev. Lett. 84, 2513 (2000). 92 [2.35] R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. Ritchie and A. J. Shields, “Two-photon interference of the emission from electrically tunable remote quantum dots”, Nature Photon. 4, 632 (2010). [2.36] A. Konkar, K. C. Rajkumar, Q. Xie, P. Chen, A. Madhukar, H. T. Lin and D. H. Ritch, “In-situ fabrication of three-dimensionally confined GaAs and InAs volumes via growth on non-planar patterend GaAs (001) substrates”, J. Cryst. Growth 150, 311 (1995). [2.37] A. Konkar, A. Madhukar, and P. Chen, “Creating three-dimensionally confined nanoscale strained structures via substrate encoded size-reducing epitaxy and the enhancement of critical thickness for island formation”, MRS Symposium Proc. 380, 17 (1995). [2.38] P. Atkinson, S. Kiravittaya, M. Benyoucef, A. Rastelli, and O.G. Schmidt, “Site-controlled growth and luminescence of InAs quantum dots using in situ Ga- assisted deoxidation of patterned substrates”, Appl. Phys. Lett. 93, 101908 (2008). [2.39] J. Skiba-Szymanska, A. Jamil, I. Farrer, M. Ward, C. Nicoll, D. Ellis, J. Griffiths, D. Anderson, G. Jones, D. Ritchie, and A. Shields, “Narrow emission linewidths of positioned InAs quantum dots grown on pre-patterned GaAs(100) substrates”, Nanotechnology. 22, 065302 (2011). [2.40] T. Hsieh, J. Chyi, H. Chang, W. Chen, T. Hsu, and W. Chang, “Single photon emission from an InGaAs quantum dot precisely positioned on a nanoplane”, Appl. Phys. Lett. 90, 073105 (2007). [2.41] K. D. Jöns, P. Atkinson, M. Müller, M. Heldmaier, M. Ulrich, O.G. Schmidt, and P. Michler, “Triggered indistinguishable single photons with narrow line widths from site-controlled quantum dots”, Nano Lett. 13, 126 (2013). [2.42] L. Yang, S. G. Carter, A. S. Bracker, M. K. Yakes, M. Kim, C. S. Kim, P. M. Vora and D. Gammon, “Optical spectroscopy of site-controlled quantum dot in a Schottky diode”, Appl. Phys. Lett. 108, 233102 (2016). [2.43] A. Strittmatter, A. Holzbecher, A. Schliwa, J. Schulze, D. Quandt, T. D. Germann, A. Dreismann, O. Hitzemann, E. Stock, I. A. Ostapenko, S. Rodt, W. Unrau, U. W. Pohl. A. Hoffmann, D. Bimberg and V. Haisler, “Site-controlled quantum dot growth on buried oxide stressor layers”, Phys. Status Solidi A. 209, 2411 (2011). [2.44] P. J. Poole, D. Dalacu, J. Lefebvre and R. L. Williams, “Selective epitaxy of semiconductor nanopyramids for nanophotonics”, Nanotechnology. 21, 295302 (2010). 93 [2.45] K. Choi, M. Arita, S. Kako and Y. Arakawa, “Site-controlled growth of single GaN quantum dots in nanowires by MOCVD”, J. Cryst. Growth. 370, 328 (2013). [2.46] M.J. Holmes, K. Choi, S. Kako, M. Arita and Y. Arakawa, “Room- temperature triggered single photon emission from a III-nitride site-controlled nanowire quantum dot”, Nano Lett. 14, 982 (2014). [2.47] G. Bulgarini, M.E. Reimer, T. Zehender, M. Hocevar, E.P.A.M. Bakkers, L.P. Kouwenhoven, and V. Zwiller, “Spontaneous emission control of single quantum dots in bottom-up nanowire waveguides,” Appl. Phys. Lett. 100, 121106 (2012). [2.48] D. Dalacu, K. Mnaymneh, J. Lapointe, X. Wu, P.J. Poole, G. Bulgarini, V. Zwiller, and M.E. Reimer, “Ultraclean emission from InAsP quantum dots in defect- free wurtzite InP nanowires,” Nano Lett. 12, 5919 (2012). [2.49] M. A. Versteegh, M.E. Reimer, K.D. Jöns, D. Dalacu, P.J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nature Commun. 5, 5298 (2014). [2.50] M. E. Reimer, G. Bulgarini, N. Akopian, M. Hovevar, M. B. Bavinck, M. A. Verheijen, E. P.A.M. Bakkers, L. P. Kouwenhoven and V. Zwiller, “Bright single- photon sources in bottom-up tailored nanowires”, Nature Commun. 3, 737 (2012). [2.51] G. Bulgarini, M. E. Reimer, M. B. Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers and V. Zwiller, “Nanowire waveguide launching single photons in a Gaussian mode for ideal fiber couping”, Nano Lett. 14, 4102 (2014). [2.52] Y. Chen, I. E. Zedeh, K. D. Jöns, A. Fognini, M. E. Reimer, J. Zhang, D. Dalacu, P. J. Poole, F. Ding, V. Zwiller, and O. G. Schmidt, “ Controlling the exciton energy of a nanowire quantum dot by strain field,” Appl. Phys. Lett. 108, 182103 (2016). [2.53] J.E. Allen, E.R. Hemesath, D.E. Perea, J.L. Lensch-falk, Z.Y. Li, F. Yin, M.H. Gass, P. Wang, A.L. Bleloch, R.E. Palmer, and L.J. Lauhon, “High-resolution detection of Au catalyst atoms in Si nanowires,” Nature Nanotech. 3, 168 (2008). [2.54] J. C. Harmand, F. Jabeen, L. Liu, G. Patriarche, K. Gauthron, P. Senellart, D. Elvira, and A. Beveratos, “InP1-xAsx quantum dots in InP nanowires: A route for single photon emitters,” J. Cryst. Growth. 378, 519 (2013). [2.55] M. Felici, P. Gallo, A. Mohan, B. Dwir, A. Rudra, and E. Kapon, “Site- controlled InGaAs quantum dots with tunable emission energy,” Small. 5, 938 (2009). [2.56] E. Pelucchi, V. Dimastrodonato, A. Rudra, K. Leifer, E. Kapon, L. Bethke, P. A. Zestanakis, and D. Vvedensky, “Decomposition, diffusion, and growth rate 94 anisotropies in self-limited profiles during metalorganic vapor-phase epitaxy of seeded nanostructures,” Phys. Rev. B. 83, 205409 (2011). [2.57] B. Rigal, C. Jarlov, P. Gallo, B. Dwir, A. Rudra, M. Calic, and E. Kapon, “Site-controlled quantum dots coupled to a photonic crystal molecule,” Appl. Phys. Lett. 107, 141103 (2015). [2.58] A. Lyasota, S. Borghaardt, C. Jarlov, B. Dwir, P. Gallo, A. Rudra and E. Kapon, “Integration of multiple site-controlled pyramidal quantum dot systems with photonic-crystal membrane cavities”, J. Cryst. Growth. 414, 192 (2015). [2.59] C. Jarlov, A. Lyasota, L. Ferrier, P. Gallo, B. Dwir, A. Rudra and E. Kapon, “Exciton dynamics in a site-controlled quantum dot coupled to a photonic crystal cavity”, Appl. Phys. Lett. 107, 191101 (2015). [2.60] J. W. Gibbs, The collected works of J. W. Gibbs, Yale University, New Haven, (1948) [2.61] C. Herring, “Diffusion viscosity of a polycrystalline solid”, J. Appl. Phys. 21, 437 (1950). [2.62] Q. Xie, P. Chen and A. Madhukar, “InAs island-induced-strain driven adatom migration during GaAs overlayer growth”, Appl. Phys. Lett. 65, 2051 (1994). [2.63] D. J. Srolovitz, “On the stability of surfaces of stressed solids”, Acta metal. 37, 621 (1989). [2.64] D. E. Jesson, S. J. Pennycook, J. –M. Baribeau and D. C. Houghton, “Direct imaging of surface cusp evolution during strained-layer epitaxy and implicaitons for strain relaxation”, Phys. Rev. Lett. 71, 1744 (1993). [2.65] W. T. Tsang and A. Y. Cho, “Growth of GaAs-Ga 1-x Al x As over preferentially etched channels by molecular beam epitaxy: A technique for two-dimensional thin- film definition”, Appl. Phys. Lett. 30, 293 (1977). [2.66] Y. H. Wu, M. Werner, K. L. Chen and S. Wang, “Single-longitudinal-mode GaAs/GaAlAs channeled-substrate lasers grown by molecular beam epitaxy”, Appl. Phys. Lett. 44, 834 (1984). [2.67] Y. Arakawa and A. Yariv, “Quantum well lasers—Gain, Spectra, dynamics”, IEEE J. Quantum Electron 22,1887 (1987). [2.68] E. Kapon, J. P. Harbison, C. P. Yun and L. T. Florez, “Patterned quantum well semiconductor laser arrays”, Appl. Phys. Lett. 54, 304 (1989). 95 [2.69] J. S. Smith, P. L. Derry, S. Margalit and A. Yariv, “High quality molecular beam epitaxial growth on patterned GaAs substrates”, Appl. Phys. Lett. 47, 712 (1985). [2.70] M. Mannoh, T. Yuasa, S. Naritsuka, K. Shinozaki and M. Ishii, “Pair-groove- substrate GaAs/AlGaAs multiquantum well lasers by molecular beam epitaxy”, Appl. Phys. Lett. 47, 728 (1985). [2.71] T. Yuasa, M. Mannoh, T. Yamada, S. Naritsuka, K. Shinozaki and M. Ishii, “Characteristics of molecular-beam epitaxially grown pair-groove-substrate GaAs/AlGaAs multiquantum-well lasers”, J. Appl. Phys. 62, 764 (1987). [2.72] S. Guha, A. Mahukar, K. Kaviani, L. Chen, R. Kuchibhotla, R. Kapre, M. Hyugachi and Z. Xie, “Molecular beam epitaxical growth of Al x Ga 1-x As on non- planar patterned GaAs (100)”, Mat. Res. Soc. Symp. Proc. 145, 29, 1989. [2.73] S. Guha, “MBE Growth of InGaAs and AlGaAs on patterned and nonpatterned GaAs(100): A study of inter-facet migration, morphology and defect formation”, Ph.D. dissertation, Materials Science, University of Southern California, 1991. [2.74] V. A. Shchukin and D. Bimberg, “Spontaneous ordering of nanostructures on crystal surfaces”, Rev. Mod. Phys. 71, 1125 (1999). [2.75] S. Guha and A. Madhukar, “An explanation for the directionality of interfacet migration during molecular beam epitaxial growth on patterned substrates”, J. Appl. Phys. 73, 8662 (1993). [2.76] K. C. Rajkumar, A. Madhukar, P. Chen, A. Konkar, L. Chen, K. Rammohan and D. H. Rich, “Realization of three-dimensionally confined structures via one-step in situ molecular beam epitaxy on appropriately patterned GaAs (111)B and GaAs (001)”, J. Vac. Sci. Tech. B 12, 1071 (1994). [2.77] K. C. Rajkumar, “Molecular beam epitaxical growth and characterization of III-V compounds on planar and patterned GaAs (111)B substrates towards the realization of quantum boxes,” Ph.D. dissertation, Physics, University of Southern California, 1994. [2.78] X. Su, R. Kalia, A. Nakano, P. Vashishta and A. Madhukar, “Million-atom molecular dynamics simulation of flat InAs overlayers with self-limiting thickness on GaAs square nanomesas”, Appl. Phys. Lett. 78, 3717 (2001). [2.79] M. A. Kakeev, R. K. Kalia, A. Nakano, P. Vashishta and A. Madhukar, “Effect of geometry on stress relaxation in InAs/GaAs rectangular nanomesas: 96 multimillion-atom molecular dynamics simulations”, J. Appl. Phys. 98, 114313 (2005). [2.80] A. Konkar, “Unstrained and strained semiconductor nanostructure fabrication via molecular beam epitaxical growth on non-planar patterned GaAs (001) substrates,” Ph.D. dissertation, Physics, University of Southern California, 1999. [2.81] S. B. Ogale, M. Thomsen and A. Madhukar, “Surface kinetic processes and the morphology of equilibrium GaAs(100) surface: A Monte Carlo study,” Appl. Phys. Lett. 52, 723 (1988). [2.82] A. Madhukar, P. Chen, F. Voillot, M. Thomsen, J. Y. Kim, W. C. Tang and S. V. Ghaisas, “A combined computer simulation, RHEED intensity dynamics and photoluminescence study of the surface kinetics controlled interface formation in MBE grown GaAs/Al x Ga 1-x As (100) quantum well structures”, J. Cryst. Growth. 81, 26 (1987). [2.83] A. Madhukar, P. Chen, F. Voillot, M. Thomsen, J. Y. Kim, W. C. Tang and S. V. Ghaisas, “A combined computer simulation, RHEED intensity dynamics and photoluminescence study of the surface kinetics controlled interface formation in MBE grown GaAs/Al x Ga 1-x As (100) quantum well structures”, J. Cryst. Growth. 81, 26 (1987). [2.84] F. Voillot, A. Madhukar, J. Y. Kim, P. Chen, N. M. Cho, W. C. Tang and P. G. Newman, “Observation of kinietcally controlled monolayer step height distribution at normal and inverted interfaces with ultrathin GaAs/Al x Ga 1-x As quantum wells”, App. Phys. Lett. 48, 1009 (1986). [2.85] W. K. Burton, N. Cabrera and F. C. Frank, “The growth of crystal and equilibrium structure of their surfaces”, Trans. Royal Soc. London 243, 299 (1951). [2.86] M. Ohtsuka and A. Suzuki, “Simulation of epitaxial growth over patterned substrates”, J. Cryst. Growth. 95, 55 (1989). [2.87] T. R. Ramachandran, R. Heitz, N. P. Kobayashi, A. Kalburge, W. Yu, P. Chen and A. Madhukar, “Re-entrant behavior of 2D to 3D morphology change and 3D island lateral size equalization via mass exchange in Stranski--Krastanow growth: InAs on GaAs(001)”, J. Cryst. Growth 175, 216 (1997). [2.88] R. Heitz, T. R. Ramachandran, A. Kalburge, Q. Xie, I. Mukhametzhanov, P. Chen and A. Madhukar, “Observation of Reentrant 2D to 3D Morphology Transition in Highly Strained Epitaxy: InAs on GaAs”, Phys. Rev. Lett. 78, 4071 (1997). 97 Chapter 3. Optical properties of MTSQDs Single quantum dot (QD) with discrete electron and hole energy levels, as discussed briefly in Chapter 1, can act as single photon sources (SPSs) and, under certain conditions also as entangled photon sources (EPSs). Our mesa-top single quantum dots (MTSQDs) with SESRE-controlled size and shape in a regular array, as discussed in Chapter 2, have the potential of providing the highly spectrally uniform QD arrays needed for use as SPSs and EPSs around which on-chip integrated photonic quantum information processing (QIP) systems can be built. This chapter thus reports on optical studies devoted to examining the nature of light emission from SESRE based MTSQDs. It is perhaps worth noting that no prior study of this class of SQDs exists in the literature—this dissertation work is the first exploration and thus the results presented in the following essentially constitute the equivalent of the behaviors reported over the past two decades for the self-assembled 3D island quantum dots. Specifically, in subsections 3.1 and 3.2 we provide, respectively, the relevant theoretical underpinnings of the tetrahedrally-bonded semiconductor quantum dot electronic structure and optical transitions, noting the significance of the confinement potential symmetry and accompanying notions of bright and dark excitons in photoluminescence (PL). Subsection 3.3 provides a description of the instrumentation employed in the PL studies of reference samples and micro-PL studies of individual MTSQDs in the array. This is followed in subsection 3.4 by a discussion of the systematic study of photoluminescence of individual InGaAs 98 MTSQDs as a function of incident power and temperature. The different excitonic transition types (neutral and charged single excitons and biexcitons) indicated are further examined through the polarization dependence of the emitted photons reported in subsection 3.5. The time-resolved photoluminescence behaviors are discussed in subsection 3.6. Finally, photoluminescence excitation (PLE) studies that shed light on excited states and involvement of phonons are discussed in subsection 3.7. Owing to the prohibitive cost of LHe, only limited, judiciously chosen, studies could be performed below liquid nitrogen temperature (~77K). These studies are aimed at understanding the MTSQD electronic structure with emphasis on single exciton and biexciton states to assess their potential use as SPSs and more importantly as EPSs. The PL studies on MTSQDs as a function of excitation power, temperature and polarization are complemented with PLE and TRPL studies to enable as consistent inferences as possible within the instrument spectral resolution limit of 0.2nm. With anticipated improved resolution to less than 0.02nm in future investigations we will narrow down the range of potential optical responses obtainable from the MTSQDs, including the impact of its shape. §3.1 Origin of Single Particle Electronic States in Bulk and QD Structures As discussed in Chapter 1, single photon emission can be obtained from QD exciton [3.1-3.3], biexciton [3.3, 3.4] and singly charged exciton decay [3.1, 3.4, 3.5]. The exciton and biexciton states are formed, respectively, by single and two electron-hole pairs. Singly charged exciton is formed, however, by a single electron- hole pair and one additional electron or hole inside the QD. The energy and 99 polarization of the emitted photons are controlled by the nature of electron and hole single particle states in the QDs and the selection rules for the dipole transitions between them. Understanding the origin of the nomenclature of the electronic states in a QD is thus central to the understanding of the properties of photon emitted from different multiple excitonic states of a QD. In order to develop an understanding of the electronic states in a quantum dot, it is helpful to recall briefly the description of the electron and hole states in tetrahedrally-bonded bulk semiconductors, the class of semiconductors underlying the quantum dots of interest in this work. The confinement effects characteristic of QDs have been found to be well-described using the terminology and nomenclature established for the description of bulk semiconductors. A good description of the bulk semiconductor and quantum well (confinement in only one direction) can be found in Dr. Li Chen’s dissertation [3.6] in our group. We capture here only the important information needed for the understanding of the three-dimensionally confined electronic state energies and charge distributions and the optical transitions between these states in a QD. §3.1.1 Bulk Crystalline Solids and Semiconductors In a crystalline bulk solid, within the commonly employed one-electron-at-a- time picture, the motion of electrons follows the Schrödinger equation, 2 [ ( )] ( ) ( ) 2 p V r r E r m   (Eq.3.1) where      i p ˆ is the momentum operator and () Vr is the periodic crystal potential satisfying the property, 100 ( ) ( ) V r T V r  (Eq.3.2) where T represents a lattice translation vector. Thus, according to Bloch’s theorem, the electronic wavefunctions take the form, ,, 1 ( ) exp( ) ( ) n k n k r ik r u r    (Eq.3.3) where  is the volume of the crystal, n is the index of the band, k  a good quantum number that is the wave vector which resides in the first Brillouin zone and identifies all independent wavefunctions, and ) ( , r u k n   is the Bloch function that has the same translational invariance as the crystal potential () Vr . Taking into account the spin of the particle, the electronic wavefunction is expressed as the product of the spatial wavefunction captured in Eq.3.3 and the spin of the particle: ,, ( ) ( ) spin n k n k rr   (Eq.3.4) From Eq. 3.3 and Eq. 3.4, one can obtain that the charge distribution at position r , () r  , within the bulk unit cell with typical linear dimensions ~ 0.5nm, can be given in terms of the complete set of { , () nk ur } as, 22 ,, ( ) | ( ) | | ( ) | n k n k nn kk r r u r       (Eq.3.5) The electron eigenvalue, , n k E are generally not a simple function of k  . Although the wavefunction k n  ,  in Eq.3.3 is the exact form of the solution of the Schrödinger equation for an electron or hole in a periodic potential, the exact form of Bloch function , () nk ur is typically unknown. The calculation of band structure is thus done 101 by approximating the Bloch function with functions that can realistically represent the charge distribution in the unit cell (Eq. 3.5). The above recalled standard approach to calculation of the band structure, when applied to crystalline semiconductors of interest to this dissertation, needs to account for the presence of spin-orbit coupling energy in the single particle Hamiltonian [3.7, 3.8] and the Schrödinger equation becomes [3.7, 3.9]: ) ( ) ( ) ˆ ( 4 ) ( 2 ˆ ) ( ˆ , , , 2 2 0 0 2 , r E r p V c m r V m p r H k n k n k n k n                             (Eq.3.6) where the        ) ˆ ( 4 2 2 0 p V c m term is the electron’s spin-orbit coupling energy in which   is the Pauli spin matrix given by, 0 1 0 , 1 0 0 xy i i                , 10 01 z       (Eq.3.7) Substituting Eq. 3.3 into Eq. 3.6 and considering that the electron crystal momentum k   is small compared to the momentum ) ( ˆ , r u p k n   in the atomic interior, the following equation for the Bloch function , () nk ur is obtained [3.9]: 01 2 2 2 22 , , , 0 0 0 0 ˆ ˆˆ ( ) ( ) ( ) ( ) 2 2 4 n k n k n k HH pk V r k p V p u r E u r m m m m c                             (Eq.3.8) Moreover, as in semiconductors the phenomena of typical interest such as electron transport and luminescence (central to this work) are dominated by the properties of 102 states near the fundamental band edges, only the wavefunction k n  ,  and energy k n E  , of states with k  near 0 are typically of the greatest importance. At 0  k  Eq. 3.8 can be reduced to, ) ( ) ( ˆ 0 , 0 , 0 , 0 r u E r u H n n n    (Eq.3.9) where 0 , n E and ) ( 0 , r u n  are the eigenvalues and eigenfunctions of 0 ˆ H in Eq. 3.8. Near 0  k  , 1 ˆ H is small compared to 0 ˆ H and hence can be considered as a perturbation to 0 ˆ H . k n E  , and ) ( , r u k n   in Eq.3.8 can be solved using perturbation theory. The piece ) ( , r u k n   can be expanded using zone center Bloch functions )} ( { 0 , r u n  from all bands as the zeroth order basis given that )} ( { 0 , r u n  is a complete set of functions capturing the symmetry of the crystal [3.6]: ) ( ) ( ) ( 0 , , , r u k C r u j j j n k n       (Eq.3.10) Correspondingly, we have, ) ( ) exp( ) ( ) ( 0 , , , r u r k i k C r j j j n k n           (Eq.3.11) where ) ( , k C j n  are the expansion coefficients. In an actual calculation for a specific semiconductor crystal, however, the zeroth order perturbation basis does not necessarily have to cover all bands, but only those m bands that contribute “most significantly”. In bulk semiconductors of group IV and III-V, the valance electrons of the atoms are in s and p atomic orbitals that dominantly form the local chemical bonds. 103 The charge distribution (Eq. 3.5) within their bulk unit cell has been approximated well by the limited set )} ( { 0 , r u n  taken as the linear combination of atomic orbitals for calculating and understanding their band structures [3.9, 3.10]. Given the in-plane rotation symmetry of the semiconductors, the orbital angular momentum, L, is a good quantum number. The Bloch function )} ( { 0 , r u n  can also be expressed in the form of spherical harmonics, | , ( , ) m ll l m Y   , that are eigenfunctions of orbital angular momentum operator L. Such a set can be used as )} ( { 0 , r u n  reflecting the symmetry of the crystal to solve for the crystal band structure using Eq. 3.8. The typical band structure of a direct bandgap III-V zinc-blende semiconductor near zone-center is shown in Figure 3.1 (a). Each E-k curve in figure 3.1(a) corresponds to two degenerate bands with respect to the z angular momentum of the electron (see table 3.1). The two lowest conduction bands C 6  arise from the group III s-orbital. The six highest valence bands V 8  (light hole and heavy hole, four fold degenerate at k=0) and V 7  (spin-orbit split-off hole doubly degenerate) arise from group V p-orbital. Around the extrema of the bands the energy dispersion is parabolic and can be approximated with the energy of a free electron with effective mass m. The effective mass is obtained from the curvature of the energy dispersion. This is how the two upper valence bands V 8  take their names: the heavy (HH) and light hole (LH). The HH and LH sub-bands are degenerate at the Γ point (k = 0) and split into two branches for nonzero wave vector k . The third sub-band in the valence band is the spin orbit split-off (SO) band. The above 8 bands are usually the “most 104 significant” bands and thus the 8 atomic orbital states are used to form a 8 × 8 Hamitonian matrix using the Hamiltonian in Eq. 3.8. The zone center Bloch functions and other properties of the eight “most significant” bands are summarized in Table 3.1. The states are represented by the total angular momentum ( J ) and its projection along z-axis ( z J ) in Table 3.1 as |, z JJ  . S, X, Y, Z denote Bloch functions with even parity, odd parity along x ((100) direction), y ((010) direction) and z ((001) direction) respectively. The arrows  and  represents the spin up and down state. Figure 3.1 Schematic of the band structure near k=0 in a direct bandgap semiconductor of zinc-blende structure: (a) without strain. The blue line depicts the lowest conduction band while the red and magenta lines depict the lowest HH and LH valance band, respectively. The purple line represents the spin orbit split off band. Panel (b) shows the conduction, HH, and LH valance bands of the semiconductor under biaxial compressive strain. 105 Table 3.1 Notation and zone center Bloch function of “most significant” bands. Band Z J J, V C Jz J u , , Group Notation   0  k E  Conduction Band 2 1 , 2 1  S 6  E g 2 1 , 2 1   S Heavy Hole (HH) 2 3 , 2 3 1 () 2 X iY  8  0 2 3 , 2 3    ) ( 2 1 iY X Light Hole (LH) 2 1 , 2 3     Z iY X 3 2 ) ( 6 1 2 1 , 2 3       Z iY X 3 2 ) ( 6 1 Spin Split- off Hole (so) 2 1 , 2 1     Z iY X 3 1 ) ( 3 1 7    2 1 , 2 1       Z iY X 3 1 ) ( 3 1 These zone center Bloch functions listed in Table 3.1 capture the symmetry of the conduction and valance bands and can be used as the basis to expand the Hamiltonian in Eq.3.8 as an m m  matrix to calculate the band structure at 0 k  [3.6-3.9]. As discussed later in the section §3.1, the zone center Bloch functions play an important role in understanding the optical transitions in the crystal.  Strain Effects Strain in the crystal induces shifts and potentially mixing in the energy levels of the unstrained conduction and valence bands. While hydrostatic strain merely 106 shifts the energy levels of a band, uniaxial and biaxial strain remove the band degeneracy. The effect of strain can be incorporated in semiconductor band structure calculations by incorporating a strain dependent term into the Hamiltonian shown in Eq. 3.8 called the Pikus-Bir Hamiltonian [3.6, 3.11] and is given by, ( , ) ( , ) ( ) 2 2 () ( ) 3 [( 3 ) . ] 6 [{ } . ] 3 c v c v v xx yy zz x xx v x y xy H a b L L c p d L L c p               (Eq.3.12) where ij  represents the component of the strain tensor, L the angular momentum operator, c.p. denotes cyclic permutations with respect to the rectangular coordinates, x, y and z. The {} indicates the symmetrized product of { } ( ) / 2 x y x y y x L L L L L L  . The parameter a is the hydrostatic pressure deformation potential. The parameter b and d are the uniaxial deformation potentials for tetragonal and rhombohedral symmetry, respectively, and are zero for the s-like conduction band. In the case of a biaxial compressive strain, the energy difference between conduction and valance band is enlarged and the degeneracy of the HH and LH band is lifted at Γ point (k = 0) with the LH band pushed to lower energy compared to HH (magenta line in Fig. 3.1 (b)). In the presence of biaxial tensile strain, the relative energy difference between HH and LH band is reversed compared to biaxial compressive strain. §3.1.2 Semiconductor Quantum Dot As noted in Chapter 1, semiconductor QDs are formed by two different semiconductors, one, with linear sizes smaller than the de Broglie wavelength of the electron, surrounded by the other that has a bandgap larger than that of the enclosed 107 material. Typically, the smaller bandgap of the enclosed material is contained entirely within the larger bandgap of the latter. Indeed, the relative energy alignment of the two bandgaps is a fundamental characteristic defining the properties of heterostructures and gives rise to different classes of heterostructures often referred to as type 1 (smaller bandgap entirely inside the larger gap), type 2 (bandgaps staggered), and type 3 (one bandgap entirely below the other). In spite of the critical significance of the band alignment in defining the confinement potential as further discussed later, in practice it is usually not well-known. The synthesized MTSQDs studied in this chapter are formed with deposited In 0.5 Ga 0.5 As (material A) surrounded by GaAs (material B) as schematically shown in Fig. 3.2 (a). The horizontal axis marked z in the figure is the growth direction of the mesa-top quantum dots grown and studied in this work. In such a situation a change in the crystal potential occurs across the materials as symbolically shown in Fig.3.2(b) assuming energy alignment with respect to a common flat vacuum level (E vac =0) as the zero of energy. Such an assumption implies also the alignment of the bulk energy bands of semiconductors A and B (depicted as a function of wave vector by the red and blue curves) with respect to the same common vacuum level, thus fixing the energies of the Brillouin zone center band edges of the two materials with respect to each other, as depicted in Fig.3.2 (b). The relative positions of the conduction and valance bands at the zone center define the concept of “band-offset” that introduces the notion of “confinement potential” at the core of heterojunctions and the resulting electronic states of the system, as discussed below. 108 Figure 3.2. (a) Schematic of a truncated pyramidal MTSQD structure. The horizontal axis marked as z in the figure is the growth direction of the mesa-top quantum dots grown and studied in this work. (b) Schematic of the change in the crystal potential across materials, aligned energetically assuming a common (flat) vacuum level (E vac ) as the zero of energy. Under such an assumption, shown also is the alignment of the bulk conduction (E CB (k)) and valance (E VB (k)) bands of semiconductors A and B. The bulk band states of the QD region (red) in the energy range shown find no matching states in the band gap of the barrier region (blue) and become quantized for dimensions smaller than the bulk de Broglie wavelength. What at the atomic level is a change in the crystal potential across material interfaces can, in a simplified continuum solid approximation, be modelled by the discontinuity arising from the alignment of the Brillouin zone center conduction (E CB (Γ)) and valence (E VB (Γ)) band edges of the QD (A) and barrier (B). The bulk band states of the QD region (red) in the energy range shown find no matching states in the band gap of the barrier region (blue) and become quantized for dimensions smaller than the bulk de Broglie wavelength. What at the atomic 109 level is a change in the crystal potential across material interfaces can, in a simplified continuum solid approximation such as this, be modelled by the discontinuity arising from the alignment of the conduction and valence band edges of the QD (A) and barrier (B). Establishing this confinement potential in 3D is a fundamental challenge and typically it is not well-known for any system owing to the inherent kinetics and thermodynamics of the material deposition processes involved in the synthesis of QDs as discussed in the preceding chapter. No clear structural and chemical boundary between the two materials as suggested by the vertical black dotted lines in Fig.3.2 (b) can be experimentally realized owing to the very existence of steps at the growth front and unknown (and often uncontrolled) inter-mixing of the chemical species distinguishing material A from B. This is inherent to heterojunctions [3.12]. Nevertheless, certain approximations, partly recalled below, have enabled a simple operational picture that replaces the actual crystal potentials by certain bulk band structure energies - - actually certain band edges - - of the two materials to define an operational “confinement potential” for electrons and holes, at least near the original bulk band edges. Within such a model picture, elaborated upon below, Fig. 3.2 (b) shows a schematic drawing of the confinement “potential” for electrons and holes along z direction (growth direction) along a line passing through the center of the QD. The small MTSQD lateral size (<15nm) and height (~3nm) compared to the exciton Bohr diameter for bulk In 0.5 Ga 0.5 As (~23nm) result in strong quantum confinement in the x-y plane as well as in the vertical direction. Consequently the 3D quantum 110 confined electronic states acquire discrete energies with separations that turn out to be significantly larger than the mutual Coulomb interactions in single particle states, as well as the particle exchange energies. Thus, electrons and holes in the QD ground state can be treated as non-interacting single particles.  Confinement Potential: The Band-Offset Model The electron and hole states of a QD can be obtained by solving the Schrödinger equation given in Eq.3.1 for single particle Hamiltonian provided the crystal potential () Vr could be specified for all space. The crystal potential () Vr is inherently different in the two spatial regimes defined by materials A and B and changes across their interface. Compared with bulk crystals, the loss of translational invariance of the crystal potential inherent to quantum dots ensures that the wave vector k  is no longer a good quantum number. Solution of the Schrodinger equation for the semiconductor quantum dots of interest here, Eq. 3.6, with appropriate () Vr that has lost translational invariance needs to be carried out in real space. Taking an atomistic view of the quantum dots thus necessarily demands calculating their electronic states using approaches such as the atomic pseudopotentials [3.13-3.15]. As a typical QD involves in excess of a million atoms, such calculations are implemented on powerful computational platforms (typically involving parallel computing). These provide the most detailed and quantitative results for the electron energy levels and charge distributions. However, such calculations are impractical for the exploration of the field of quantum dots. 111 Fortunately, for qualitative and semi-quantitative guidance purposes a continuum-solid view based calculation approaches have been developed that fruitfully exploit methodologies developed for calculation of the electronic structures of bulk semiconductors, including strain and piezoelectric effects [3.9, 3.11, 3.16- 3.19]. The loss of atomistic information is compensated by the gain in practical guidance by parametrically incorporating the dependence of the electronic states on material parameters. Below, we capture the essentials of one such approach that has been most extensively employed in the field of semiconductor heterojunctions in general and quantum dots in particular. The change of the crystal potential () Vr from material A to B has been found, in the continuum solid viewpoint, to be modelled well by the change in the bulk solid conduction and valance band edges that define the fundamental bandgaps of the two materials across the (assumed abrupt) interface [3.10, 3.20]. The corresponding “confinement potential” in Eq.3.1 can thus be approximated by the discontinuity of the bulk band edges: the lowest energy of the conduction bands and the highest energy of the valance bands. As captured in Fig. 3.2 (b), the potential () Vr that the electron of the QD sees can be represented as the bandgap offset [3.10,3.20,3.21] of the two materials A and B as below: ( ), () ( ), A B V r r QD Vr V r r QD       , (Eq. 3.13a) or, 112 for electron; QD for electron; QD () for hole; QD for hole; QD A CB B CB A VB B VB Er Er Vr Er Er               (Eq. 3.13b) where E CB and E VB refer, respectively, to the zone center values of conduction and valance bands of materials A and B as indicated by the superscript and by red and blue dotted lines in Fig.3.2(b) above. Thus, resetting the zero of energy to the conduction and valance band edge of the quantum dot material, the confinement potential for electrons and holes along z becomes, 0, QD () ( ), QD e BA CB CB r Vr E E r       (Eq. 3.13c) 0, QD () ( ), QD h BA VB VB r Vr E E r       (Eq. 3.13d) Figure 3.3 Schematic of confinement potential arising from the alignment of the measured zone center conduction and valence band edges of the QD (superscript A) and barrier (superscript B) bulk materials. The confinement potentials for electron and hole are marked with black double arrows. With the QD size smaller than the 113 bulk de Broglie wavelength, the electron and hole state energies become discrete (black lines). Figure 3.3 above captures the confinement potential for electrons (V e ) and holes (V h ) along z direction coming from the band-offset of the conduction and valance bands of QD (material A, red line) and barrier (material B, blue line). The solution of the Schrodinger equation in Eq. 3.1 needs to be carried out with () Vr as given by Eq.3.13. One widely used approach for such calculation is the envelop wave function approach. An arbitrary state function () r  in the solid can be expanded in terms of the Bloch form eigenfunctions )} ( { , r k n    of the bulk periodic solid given by Eq.3.11: , ,0 , ,, ,0 ( ) ( ) ( ) ( ) ( )exp( ) ( ) ( ) ( ) n n n j j nk j n k n k jj j r b k r b k C k ik r u r f r u r           (Eq.3.14) where ) (k b n  and , () nj ck are the expansion coefficients and     k n j n n j r k i k C k b r f      , , ) exp( ) ( ) ( ) ( is the envelop wavefunction. Thus () r  is represented in terms of the zone center Bloch functions from all bands (index j). In actual calculations, the expansion typically only employs a limited number of “most significant” bands ensuring the symmetry of the crystal structure. For the tetrahedrally-bonded III-V semiconductor QDs involved in our studies the relevant basis states remain the same as listed in Table 3.1 for bulk band structure calculations. The complete wave function of electrons in QDs can thus be expressed as a product of a spatial part that is itself the product of a slowly varying envelop 114 function (representing the localization of the particle within the confined potential region) and a “Bloch” part representing the spatial variation within the unit cell of the bulk solid, and a spin part. Following the work of Bastard et al [3.20, 3.21], the wavefunctions of electrons and holes in the QD (material A) and the barrier (material B) can be written as, , , , ,0 ( ) ( ) ( ). A B A B A B n n spin n r f r u r    (Eq.3.15) where n is the bulk band index. Given that most of the host materials for QDs (material A and B) display similar band structures and the periodic parts of the Bloch functions of the relevant band edges do not differ very much from one host material to the other, the zone center Bloch function in A and B are assumed to be the same i.e. ) ( ) ( 0 , 0 , r u r u B n A n    for the calculation of QD electronic states [3.20]. The one-to-one correspondence between Bloch functions of different materials involved requires that the different materials should have the same crystal structure and chemical bond. This assumption is reasonable in epitaxially grown semiconductor QDs, i.e. InGaAs/AlGaAs SAQDs and MTSQDs. The validity of the envelop function approach requires that the envelop wavefunction varies on a spatial scale much larger than the dimension of the unit cell. In addition, the envelop wavefunction approach assumes certain similarity between the materials involved in defining the QDs i.e the heterostructure. The one-to-one correspondence between Bloch functions of different materials involved indicates that the different materials should have the same crystal structure and chemical bond. These assumptions taken for the envelop function approach based calculation of QD 115 electronic structure are reasonable for the epitaxially grown semiconductor QDs, i.e. InGaAs/AlGaAs SAQDs and MTSQDs that are formed with materials of same crystal structure and are of size ~10-20nm, much larger than the unit cell diameter (~0.5nm). The QD electron wavefunctions and energy levels can be calculated by writing the Hamiltonian captured in Eq. 3.8 in real space as a m m  matrix of operators by expanding the wavefunctions using m “most significant” bands. The widely used Bloch function basis - that also captures the symmetry of the bulk solid - for studying QD electronic structures are the 8 bands listed in Table 3.1. Since the spin-orbit (SO) splittings for InAs and GaAs are large (380 and 340 meV, respectively), we exclude the SO states from our further consideration. Therefore the zone center basis for calculation contains the conduction, HH and LH bands ( 11 , 22 c  , 33 , 22 HH  and 31 , 22 LH  ). The calculations for a QD in real space can be carried out in a similar manner compared to the k space calculation for bulk semiconductors, except: (1) z y x k , ,  is replaced by momentum operator z y x z y x i p , , , , ˆ 1      with anti- symmetrization consideration and, (2) all parameters (such as Δ, L 1  ...) in the Hamiltonian can become space dependent in the presence of chemical composition or strain variation. 116 In general the Hamiltonian accounting for confinement potential, chemical composition and strain field for the heterostructure written formally in spatial representation [3.16-3.19] gives the Schrödinger equation, ( , , , , , , , , , , , ) ( , , ) ( , , ) x y z xx xy xz yy yz zz H x y z x y z E x y z            (Eq.3.16) For calculational purposes the Hamiltonian resulting from the approximations discussed above gives a Schrödinger equation in which ( , , ) x y z  is typically expanded using the conduction, HH and LH bands noted above. The solution of the differential equation resulting from Eq. 3.16 must satisfy the following boundary conditions: (1) the wavefunction is continuous across the interface and (2) the current density (related to the first order derivative of the wavefunction weighted by the effective mass of electron and hole) should also be continuous across the interface. Detailed description of band structure calculation of heterostructure and QDs can be found in Dr. Li Chen’s [3.6] and Dr. Siyuan Lu’s [3.22] dissertations in our group. Given the s-like nature of conduction band and p-like nature of valance band Bloch functions, the electron wavefunction of the electron ground state (e 0 ) of QD has 11 , 22  dominating character while the hole wavefunction of the hole ground state (h o ) of QD has the p-like HH and LH character as shown in Fig. 3.2 (b). The specific amount of HH and LH hole character of the confined hole states is defined in terms of the symmetry of the Bloch functions when projected in to axial angular 117 momentum components. The electron and hole wave functions can be expanded in the basis of s-like conduction band state 11 , 22  and p-like HH states 33 , 22  and LH states 31 , 22  in terms of the angular momentum basis , l lm . The spin dependent HH states in angular momentum basis , l lm are written as 33 , 1,1 22  and 33 , 1, 1 22     .Similarly, LH states can be written as 3 1 1 , 1,1 2 1,0 22 3     and 3 1 1 , 1, 1 2 1,0 22 3       . The envelop wave function are decomposed into angular momentum functions ( exp( ) l im  ) in a Fourier transform form, , 1 ( ) exp( ) 2 l l n n m l m f r f im     (Eq.3.17) The proportion of HH/LH character can be numerically determined through the norm of the Fourier coefficient in Eq. 3.17. The relative contribution of HH and LH character in QD hole state (h 0 ) affects the polarization of the electron-hole transition and correspondingly the polarization of light emitted from such transition as discussed in detail in section §3.2. §3.2 Optical Transitions in Bulk and QD structures Photons can be absorbed in or emitted from the QDs through creation of excitons (single electron-hole pairs) or the recombination / decay of excitons inside QDs. Given our objective of examining the QD as SPSs and EPSs, photon emission 118 process form QD is our primary interest. In order to develop understanding of the photon emission process and the corresponding properties of emitted photons, i.e. their energy and polarization, we recall briefly the description of the interaction of electron with radiation field in bulk structures and then discuss in detail the photon emission from QDs. For an electron under radiation field, the Hamiltonian can be written in the following form under semiclassical treatment with dipole approximation: 2 1 ( ) ( ) 2 H p eA V r m    (Eq.3.18) where A is the vector potential of the electromagnetic field. Under Coulomb gauge, 0 A  , Hamiltonian in Eq. 3.18 can be written as, 0 1 2 22 2 () 22 H H H p e e H V r p A A m m m     (Eq.3.19) which consists the Hamiltonian without radiation field ( 0 H ), and the parts representing the coupling of electron with the radiation field ( 12 , HH ). Considering only one photon related optical transition, 1 H is, therefore, the only term that need to be taken into account to represent the coupling between the motion of electron and radiation fields. The transition rate between a filled initial electronic state, | i  , and an empty final state | f  is determined by Fermi’s Golden rule, 2 2 2 , 2 | | | | ( ) f i f i ij e w e p E E m           (Eq.3.20) 119 where i E and j E are the energy of initial and final state. Eq. 3.20 represents the energy conservation during optical transitions. The transition matrix element can be calculated by integrating the wave functions over one Wigner-Seitz cell [3.6], | | | | exp[ ( ) ] f i f i WSC i j l l e p e p i k k R              | | ( ) f i WSC i j e p k k         (Eq.3.21) Crystal momentum conserves during optical transitions. Since  of interest is usually a factor of 10 3 smaller than the Brillouin zone dimensions, we obtain the transition rule, ij kk  . As seen in Eq. 3.21, the optical transition selection rule are related to the symmetry of the wave functions of the initial and final states and can be determined from the transition matrix element || f i WSC ep     . For atomic transitions, we have selection rules: 0, 1 J    , 0, 1 z J    and 1 L    . The optical transition 1 z J    are circularly polarized (σ ± polarized), as they involve net transfer of angular momentum with photons. The transition 0 z J  transitions are linearly polarized (π polarized). For bulk structures at 0 k  , the conduction and valance band wave functions exhibit s-like and p-like behavior. Thus the selection rules between the 11 , 22  conduction band, 33 , 22 HH  and 31 , 22 LH  valance bands imitates the selection rules of the atomic transition as shown in Fig. 3.4. The offset between HH and LH states is set for clarity of the diagram and does not impose any specific band ordering. 120 Figure 3.4 Schematic of the optical transition between electron (e) and hole (HH and LH) states with the transition polarization and transition strength labeled. The electron and the hole state is labeled by the projected value of total angular momentum, M j . All allowed optical transitions between different electron and hole states are marked with arrows. The arrows marked with same color have the same emitted light polarization property (i.e. circular polarized (σ ± ) or linear polarized (π z )). The number listed besides the arrow indicates the relative transition strength. To obtain the optical transition strength, we introduce the electric field vector of light as ( , , ) X Y Z e e e e  with 11 ˆ ˆ ˆ ˆ ˆ ( ), ( ), 22 X Y Z e x y e x y e z      along [110], [1-10] and [001] crystal direction chosen to represent the mirror planes of tetrahedral crystal structure of interest in this dissertation work. Following the defined electric field vector, the optical transition dipole moments between the conduction 11 , 22  and HH valance band 33 , 22  for light emission are: 3 3 1 1 , | | , ( ) 2 2 2 2 2 XY e p e ie      3 3 1 1 , | | , ( ) 2 2 2 2 2 XY e p e ie        121 3 3 1 1 3 3 1 1 , | | , , | | , 0 2 2 2 2 2 2 2 2 e p e p          (Eq.3.22) Similarly, the optical transition dipole moments between the conduction 11 , 22  and LH valance band 31 , 22  are 3 1 1 1 , | | , ( ) 2 2 2 2 6 XY e p e ie       3 1 1 1 , | | , ( ) 2 2 2 2 6 XY e p e ie        3 1 1 1 2 , | | , 2 2 2 2 6 Z e p e        3 1 1 1 2 , | | , 2 2 2 2 6 Z e p e      (Eq.3.23) where | | | | | | X Y Z X p S Y p S Z p S           following the notation in Ref. [3.23] . The HH transition with ΔJ=1 (first two equations in Eq. 3.22) and the two LH transition with ΔJ=1 (first two equations, Eq. 3.23) are all circularly polarized with transition strength differ by a factor of 1/3. The HH transition with ΔJ=2 (the last equation in Eq. 3.22) are not allowed transitions due to selection rule. The last two allowed LH transitions with ΔJ=0 (last two equations in Eq. 3.23) are linearly polarized along z direction with transition strength being 2/3 of the HH transition. The numbers listed in Fig. 3.4 represents the relative transition strength amongst all the allowed transitions with J=0, ±1.  Optical transitions in QDs: Bright and Dark Excitons 122 Same as bulk solids, the relevant quantity for QD photon absorption and spontaneous emission is also the dipole momentum matrix element || fi d e p      with i  and f  being the initial and final state wavefunction of the QD. Given our interest in the photon emission process, the dipole moment that governs the photon emission process is || he d e p      with e  and h  being the electron and hole wavefunction of QD following Eq. 3.15, ,0 ,0 ( ) ( ) | | ( ) ( ) | h h e e i i j j h e ij d f r u r e p f r u r        (Eq.3.24) The last term in Eq. 3.24 represents the spin of electron and hole. Eq. 3.24 leads to three selection rules for QD interband electron-hole optical transition: (1) the envelop function must have the same parity (2) the Bloch function must have opposite parity, and (3) the electron spin must remain unchanged. Figure 3.5 (a) captures schematically the allowed interband transitions in the QDs. For QD ground state transition between e 0 and h 0 state, the envelop function of electron and hole have same symmetry. A large matrix elements arise mainly from s-type Bloch function of the initial electron state and p-type Bloch function of the initial hole state. Besides interband transitions, strong intraband transitions can also occur when the electron and hole Bloch functions have same parity but envelop functions are of different parity. Experimentally transitions obeying the above mentioned selection rules have been observed to dominate [3.24]. Optical transitions in QD take place through the electron-hole pair (excitons) formed with electron and hole from QD electron and hole lowest state e 0 and h 0 , 123 respectively as discussed in Chapter 1. The optical transition energy or the total energy of an electron-hole exciton is: 00 exciton e h coulomb E E E E    (Eq.3.25) where 0 e E and 0 h E are the single particle energy of the lowest electron and hole state of the QD, respectively. coulomb E is the coulomb interaction between the electron and holes which has been ignored in the single particle calculation of QD electron and hole states. In a strong confinement regime (QD size < exciton Bohr radius, the case for SAQDs and for MTSQDs discussed before), the coulomb energy is smaller than the energy separation of the QD discrete states coming from quantum confinement and thus can be added as a first order energy correction: 22 2 ( ) ( ') ' eh cv coulomb r r e E drdr rr       (Eq.3.26) where  is the dielectric constant of the nanocrystal. Therefore, single particle states’ energy and the mutual coulomb interaction between electron and hole controls the energy of the photon coming out of the spontaneous decay of the exciton. In this section we focus on the transition dipole moments and the polarization properties of the exciton transitions in the QD. Detailed discussion of QD spontaneous emission process is presented in section §4.2 of Chapter 4. The optical transition polarization, the other important property for optical transitions, of the QDs can be written as the following form given that the electron and hole envelop functions have the same parity for the allowed optical transitions 124 ,0 ,0 ( )| ( ) ( ) | | ( ) | h e h e i j i j h e ji d f r f r u r e p u r          (Eq.3.27) Therefore, the polarization of light emitted from the allowed optical transition is controlled by the nature of the Bloch functions of the QD electron and hole states involved in the formation of excitons. As mentioned in section §3.1, the typical Bloch function used to expand the QD wavefunction and to understand QD optical transitions is the 11 , 22 c  , 33 , 22 HH  , and 31 , 22 LH  bands. The Bloch function of the QD confined lowest electron state (e 0 ) can be represent as 11 , 22  . For holes, the quantum confinement in the QDs breaks the HH and LH band degeneracy. The typical HH-LH splitting found in the QDs optical experiments, mostly SAQDs experiments, are on the order of ( ) 10 HH LH meV    . For QDs with in-plane symmetric confinement potential, the QD confined lowest hole states (h 0 ) are primarily heavy hole (HH) in origin and the next states (h 1 ) as primarily light hole (LH). If the QD confinement potential has in-plane asymmetry, total angular momentum is no longer a good quantum number. The HH and LH can thus mix with each other. Therefore, the contribution of LH admixture of the QD lowest hole state (h 0 ) can be enhanced [3.25-3.28] compared to QD with in-plane symmetric confinement potential. For QDs with HH dominating nature, the QD ground state exciton formed are four-fold degenerate owing to the four different possible spin orientations of the electron and hole. Accounting for the interaction of electron and hole, specifically the exchange interaction that couples the spin of the electron and 125 hole, the QD ground state excitons can be non-degenerate and result in the fine structures of excitons. To understand the QD ground state excitonic transitions as well as optical polarization and strength of each transitions, we consider three major different cases (1) QD has the D 2d in-plane potential symmetry (i.e. in-plane rotational symmetry about the vertical z direction), (2) QDs with lower C 2v symmetry (resulting from the inequivalence of the [110] and [1-10] directions in the confining potential owing to internal piezoelectric fields). In the (1) and (2) cases, the HH and LH are assumed to be uncoupled with large energy separation. In the last case, we consider HH-LH mixing in QDs with in-plane asymmetric confinement potential. To understand the exciton fine structure and transition polarization, we start with the exchange energy of electron and hole. In its general form, the exchange energy is proportional to the integral, 3 3 * 1 2 1 2 2 1 12 1 ( , ) ( , ) || exchange X e h X e h E d rd r r r r r r r r r rr           (Eq.3.28) where X  is the exciton wave function. e r and h r is the electron and hole coordinates. The exchange interaction consists of two parts: short ranged part and long ranged part. The short ranged part is given by the probability of finding an electron and hole in the same Wigner-Seitz unit cell while the long ranged part is the contribution when they are in different unit cells. The exchanges interaction can lifts the degeneracy of exciton transition with opposite electron and hole spins. 126 Considering the short range exchange interaction only, the Hamiltonian for the electron-hole interaction can be written as [3.23], 3 ,, ex i i i i i i i x y z H a S J b S J        (Eq.3.29) where S i and J i are the matrix of spin 1/2 electron and spin 3/2 the hole respectively. The a i and b i are the spin-spin coupling coefficient. Consider the QD has in-plane symmetry where the HH-LH states of QD are energetically apart and uncoupled, we have the well-known solution for the eigenvectors for pure HH related exciton states: 1 1 | (|1, 1 |1, 1 ) 2 hh        2 1 | (|1, 1 |1, 1 ) 2 hh        3 1 | (| 2, 2 | 2, 2 ) 2 hh        4 1 | (| 2, 2 | 2, 2 ) 2 hh        (Eq.3.30) with 3 3 1 1 | 2, 2 | , | , 2 2 2 2         3 3 1 1 | 2, 2 | , | , 2 2 2 2         3 3 1 1 |1, 1 | , | , 2 2 2 2         3 3 1 1 |1, 1 | , | , 2 2 2 2         (Eq.3.31) 127 where we have followed the notation | , | , | , z z z J J j j s s      used in Ref [3.23]. J and J z are the total and z-projected total angular momentum of the exciton states while j, j z (s, s z ) the corresponding quantities for the HH (electron). Considering the transition dipole of exciton states, we have, 1 || X hh e p e e     2 || Y hh e p e i e      3 | | 0 hh e p e    4 | | 0 hh e p e    (Eq.3.32) following Eq. 3. 22. Two exciton transitions are dark while the other two are bright. The two bright excitons produce linearly polarized light along [110] and [1-10] crystal direction. The four fold degeneracy of the excitonic states has been broken with electron-hole exchange interaction. The dark and bright excitons are split in energy by 3 27 2 , | | 4 16 zz K K a b  . The two bright states are split by 3 2 , | ( ) | 8 xy bb  and the two dark states by 00 3 2 , | ( ) | 8 xy bb     obtained from the solution of Eq. 3.29 [3.23]. For QDs with D 2d symmetry, we have xy bb  and thus 0   . Therefore, the two bright excitons 1 | hh  and 2 | hh  have the same energy resulting in a two-fold denigrate bright excitons states in the QDs. Given 1 | hh  and 2 | hh  states have linearly polarized dipole moment with same magnitude but along [110] and [1-10] direction, the light emitted from this degenerate bright exciton states is thus circularly polarized. Two dark excitons states 128 are non-degenerate with energy separation of 2 0  and are of energy 2K lower than the two degenerate bright excitons. While for QDs with C 2v symmetry ( xy bb  ), the degeneracy of the two bright excitons in the D 2d case is broken resulting in two non- generate bright exciton state with energy separation 0   . The two non-degenerate bright excitons have linear polarization property and can result in light emission with X  and Y  linear polarization. The schematic drawing of exciton states and the emitted light polarization behavior are shown in Fig. 3.5 (b) and (c) for QDs with D 2d and C 2v symmetry respectively. 129 Figure 3.5 (a) Schematic of QD envelop function and Bloch function symmetry for allowed optical transition. Panel (b) and (c) shows the schematic energy diagram of the HH and LH related excitons in QDs with D 2d (panel (b)) and C 2v (panel (c)) symmetry of the confinement potential. The energy separation in between different exciton transitions is only shown as schematic representation without imposed ordering and energy separation amount. The exciton states and the polarization of the transition (in parenthesis) are labeled in the schematic drawing. Similarly the well-known solutions for the eigenvectors for pure LH related exciton states are: 1 1 | (|1, 1 |1, 1 ) 2 lh        130 2 1 | (|1, 1 |1, 1 ) 2 lh        3 1 | (|1, 0 |1, 0 ) 2 lh        4 1 | (|1, 0 |1, 0 ) 2 lh        (Eq.3.33) with 3 1 1 1 |1, 1 | , | , 2 2 2 2         3 1 1 1 |1, 1 | , | , 2 2 2 2         3 1 1 1 |1, 0 | , | , 2 2 2 2         3 1 1 1 |1, 0 | , | , 2 2 2 2         (Eq.3.34) Considering the transition dipole of the LH related exciton states, we have 1 || 3 Y i lh e p e e      2 1 || 3 Y lh e p e e     3 2 || 3 Z lh e p e e      4 | | 0 lh e p e    (Eq.3.35) Out of the four exciton transitions, three are bright excitons and one dark exciton. Similar to the electron-HH exciton case, the four fold degeneracy of the excitonic 131 states has been broken with electron-hole exchange interaction with the following listed energy separation amongst the four exciton states obtained from the solutions of Eq. 3.29: 12 5 (| | ) ( ) 2 y x x y E lh lh a a b b        34 5 (| | ) ( ) 2 y x x y E lh lh a a b b        34 12 || || 1 ( (4 ) 2 2 8 zz lh lh lh lh E a b          (Eq.3.36) For QDs with D 2d symmetry, we have xy aa  and xy bb  . Therefore the bright excitons 1 | lh  and 2 | lh  are degenerate. Since 1 | lh  and 2 | lh  states have linearly polarized dipole with same magnitude but orthogonal polarization, the light emitted from such degenerate bright excitons is circularly polarized. The third bright exciton 3 | lh  produces linearly polarized light along z direction with energy separated from the degenerate bright exciton as shown in Fig. 3.5 (b). For QDs with C 2v symmetry, the three bright excitons are all non-degenerate with linearly polarized light emitting along [110], [1-10] and [001] directions as shown in Fig. 3.5(c). In all the above discussion of the QD ground state related exciton structures, the HH and LH are assumed to be energetically remote and uncoupled. In this case, the QD confined lowest hole states (h 0 ) are primarily heavy hole (HH) in origin and the next states (h 1 ) as primarily light hole (LH). As seen in Fig. 3.5 (b) and (c), the excitons from the lowest hole states (h 0 ) with HH nature and the next states (h 1 ) with LH nature has finely separated excitons states. The energy separation and transition 132 polarization properties are tied to the QD confinement potential symmetry. For QDs with in-plane anisotropy, the angular momentum is, however, no longer a good quantum number and the QD hole ground state, h 0 , thus contains both HH and LH nature. The QD in-plane shape anisotropy, strain, and chemical composition variation can be sources of in-plane anisotropy of the confinement potential. Under such situations, the QD hole states cannot be expressed as HH and LH separately but in a form with linear combination of HH and LH. The QD lowest hole state (h 0 ) wave function can be written in a perturbation approach as below: 3 3 3 1 | | , | , 2 2 2 2 h E           (Eq.3.37) where E  is the energy difference between HH and LH. The optical transition dipole can be expressed as 1 1 3 3 1 1 3 1 1 1 | | , , | | , , | | , 2 2 2 2 2 2 2 2 2 2 h e p e p e p E                  ( ) ( ) 26 X Y X Y e ie e ie E        1 1 3 3 1 1 3 1 1 1 | | , , | | , , | | , 2 2 2 2 2 2 2 2 2 2 h e p e p e p E                  ( ) ( ) 26 X Y X Y e ie e ie E        (Eq.3.38) Emission from these two states contains two circularly polarized light with orthogonal direction (left handed and right handed) with intensity ratio of 2 () 3 E   . 133 The emitted light is, therefore, elliptically polarized. The HH-LH coupling affects the polarization properties of the QD transitions. Additionally, the HH-LH mixing makes the otherwise “dark” exciton with total angular momentum J=2 in the QDs with pure HH nature become “bright” now and can have non-zero transition dipole moment: 1 1 2 | | , 22 6 hZ e p e E          (Eq.3.39) resulting in the emission of linearly polarized light. From the above discussion, it is clear that the symmetry of the QD confinement potential which is affected by the QD shape symmetry, strain, composition fluctuation and piezoelectric field, determines the QD ground state excitonic degeneracy, transition polarization, and transition strength. Both HH and LH character can show up in the QD ground hole state (h 0 ). Therefore, the study of ground state excitonic structure and transition polarization properties is essential to gain insight on the symmetry of the QD confinement potential.  Excitonic Complexes Quantum dots can contain multiple electrons and/or holes leading to additional transitions with different optical properties than the single neutral excitions discussed so far in this section. The simplest examples of multiple excitonic states are trions and biexcitons that are useful in quantum-optics systems. Optical emission properties of trions and biexcitons can be explained employing the notations used in discussing single neutral excitons. In its lowest energy state, trions shown in Fig. 3.6 (a), either negative (X - ) or positive (X + ), are two fold degenerate. 134 X - contains two electrons of opposite spins and a hole with two possible spin orientations. The complex can be considered as a hole interacting with a spin-singlet electron pair. Therefore, the electron-hole exchange interaction vanishes due to Kramer’s theorem for systems with odd number of fermions governed by a Hamiltonian that is symmetric under time reversal [3.28, 3.29]. Similarly, X + is a single electron interacting with a spin-singlet hole pair formed by two holes of opposite spin and thus does not have electron-hole exchange interacting as well resulting in two-fold degenerate states. The polarization of the trion transition dipole moment and the subsequent light emitted is controlled by the HH/LH character of the QD ground hole state (h 0 ). For QDs with D 2d or C 2v symmetry, QD lowest hole state (h 0 ) is HH dominated. The emission of X + (X - ) with holes of HH Bloch nature is controlled by the transition 13 || 22        ( 31 || 22         ), following the notation of |J z > with J z representing the projection of total angular momentum along z direction). Therefore, the two fold degenerate trion states produce transitions with 1 J  . The light emitted from the trions is circularly polarized as shown in Fig. 3.6 (a). For QDs with in-plane asymmetric confinement potential, the QD lowest hole state (h 0 ) is a mixture of HH and LH and the transition dipole is governed by 11 | | , 22 h ep      and 11 | | , 22 h ep      shown in Eq. 3.38, resulting in elliptical polarized emission from the trions. 135 Figure 3.6 Schematic of electron and hole configuration and corresponding emission polarization properties of (a) trions and (b) biexcitons in QDs with D 2d and C 2v potential symmetry. The biexcitons shown in Fig. 3.6 (b) have pseudospin configuration || XX        with  and  represents hole and electron spin and does not reveal a fine structure itself. As discussed briefly in Chapter 1, the biexcitons can decay to bright excitons through the biexciton-exciton cascade process. The polarization of light from biexciton is controlled by that of neutral single exciton. For QDs with D 2d symmetry, the biexciton decays in to one of the bright excitons and produces a circular polarized light as shown in Fig. 3.6 (b) (recaptured from Fig. 1.2 (c) in Chapter 1). For QDs with C 2v symmetry, the light emitted from the biexciton to exciton decay is linearly polarized depending on which finely separated bright 136 exciton it decays to. For QDs with the in-plane asymmetric confinement potential, the light emitted from the biexciton to exciton decay is then elliptically polarized. The exciton, trion and biexciton decay can be used to generate single photon emission. Since biexcitons contain two excitons, they are observed at higher excitation densities compared to excitons and trions with optical decay approximately twice as fast as excitons and trions due to the existence of twice the number of radiative decay channels [3.4]. Additionally, the biexciton-exciton cascade process can be utilized to produce polarization entangled photons. The basis states for studying photon emission process from exciton complexes are multiparticle states, containing two entities, excitons and emitted photons. Detailed discussion of QD spontaneous decay dynamics and the states of photons emitted from QDs as well as photon entanglement is captured in Chapter 4. §3.3 Photoluminescence (PL) and Micro-PL Instrumentation The quantum dot electronic structure and various excitonic transitions discussed in the preceding subsections, as well as QD emission uniformity and exciton lifetime etc., are characterized by studying the photoluminescence coming out of the MTSQDs. For this purpose we established a home-built PL measurement setup that includes the ability to examine a single MTSQD at a time, referred to here as the micro-PL setup. In this section, we first briefly discuss the photoluminescence process in QD structures, different photoluminescence spectroscopy methods including time-integrated photoluminescence (PL), photoluminescence excitation (PLE) spectroscopy, and time-resolved photoluminescence (TRPL). The micro-PL 137 system built under this dissertation work is thus capable of measuring time- integrated PL, PLE and TRPL spectra from individual MTSQDs. Details of the micro-PL setup instrumentation are provided in Appendix C. We capture in this section only the important features of this home built micro-PL setup §3.3.1 Photoluminescence Process in Bulk and QDs Photoluminescence in bulk semiconductors can be approximated by a three step process: photogeneration, relaxation to ground state, and recombination of carriers as schematically captured in Fig.3.7. Figure 3.7 Schematic showing the 3-step model of the PL process: (1) photon generation, (2) carrier relaxation, (3) radiative recombination. Electrons and holes are generated in the material through the absorption of photons above the material’s fundamental bandgap. Owing to the rapid electron- electron interaction (femtosecond), such optically created electrons and holes have a “distribution temperature” much higher than the lattice temperature and hence are referred to as “hot” carriers; These hot electrons and holes can quickly (sub- 138 picosecond) relax to the states near the bottom of the conduction band and the top of the valence band by interaction with the lattice (mainly through the Fröhlich coupling with the LO-phonons) due to strong electron-phonon coupling in bulk semiconductors as well as in confined structures such as quantum wells that have continuous density of states (DOS). Then, these electron and hole pairs (excitons) recombine radiatively or non-radiatively at a much slower rate (on the order of 1ns) compared to carrier relaxation. Radiative recombination generates photoluminescence that reflects the distribution of the carriers in the near bandedge states of the bulk solids or the ground states of low dimensional structures. The PL process in quantum dots approximates to the same three general steps as discussed above for bulk except for the added process of electron, hole, and exciton capture by the quantum dot for excitation above the band gap of the barrier material, as is typically the experimental situation discussed in section §3.6.2 for the MTSQDs. However, owing primarily to the discrete nature of the electron density of states arising from the 3D confinement, the details involved in the relaxation process are quite unique. A detailed discussion of photoluminescence process in QDs can be found in Ref. [3.30,3.31]. Briefly, due to the δ-function like DOS in QDs, the phonon mediated relaxation of carriers is expected to be slowed down dramatically (1 to 100ps) given the less probable situation of direct matching of the phonon energy to the energy difference between the discrete states. The electron energy relaxation can only occur via multiple phonon process, which is exponentially slower 139 as a function of the number of phonons involved. This phenomenon is well-known as the “phonon bottleneck” [3.32, 3.33]. Additionally, the δ-function DOS in quantum dots combined with the typically small number of electrons present precludes the notion of carrier distribution from being applicable to a single QD. The width of the transition line of the photon emission from the QD ground state is thus expected to be of Lorenzian shape [3.22, 3.31]: 2 2 0 2 0 ) ( ) (         A A (Eq.3.40) Here 0  is the resonant frequency of the QD ground state transition, 0 A is an arbitrary constant, and  is the width of the Lorentian PL line. The linewidth  is inversely proportional to the dephasing time of the ground state exciton. Common dephasing mechanisms include radiative/non-radiative recombination of exciton and the exciton-phonon coupling. If recombination is the only dephasing mechanism, then 1 2 1 T   , where 1 T is the recombination lifetime of the exciton (as measured in time-resolved PL experiment). Indeed narrow PL linewidths < 100μeV are observed in liquid helium temperature single QD PL experiments [3.34-3.36].  Photoluminescence Spectroscopy Methods Photoluminescence coming from the QDs through radiative recombination of the electron-hole pair in the QD ground states (i.e. radiative decay of the ground state exciton) can be characterized using different photoluminescence spectroscopy 140 methods to provide different information about the QD electronic structure. Photoluminescence spectroscopy can be generally categorized as time-integrated photoluminescence (PL) and time-resolved photoluminescence (TRPL). In the usual time-integrated PL measurements, excitation energy is fixed and the detection energy is scanned. For PL measurements of QDs, there are typically three excitation schemes dubbed: (1) “Above gap excitation” (excitation energy above the band gap of the confinement barrier of the QD and thus non-resonant with QD electronic transitions); (2) “Near-resonant excitation” (excitation energy nearly matching the states of QD); (3) “Resonant excitation” (excitation energy matching closely the QD transitions, typically the ground state. Above gap excitation is widely used in studying the single particle and multi-particle states of the QDs due to the efficient feeding of QDs with electron- hole pairs through its surrounding barrier material and the easy spectral discrimination of the excitation light from the QD emission. The other two are widely used to minimize (hopefully avoid) charging around the QDs as above gap excitation also creates high probability of individual electron and hole getting trapped at defect states in the material around the QD [3.37, 3.38]. Utilizing above gap excitation, time-integrated PL measurements at 77K were carried out for all of the 40 MTSQDs in the 5 × 8 array to determine emission spectral uniformity (§3.4 below). To understand the nature of the electronic states underlying the PL, a few MTSQDs were studied selectively as a function of temperature and excitation power 141 and one examined comprehensively (§3.4 and §3.5). Additionally, the emission polarization was examined for the dominant excitonic peaks (§3.5). Besides the time-integrated ground-state PL measurements, time-resolved photoluminescence (TRPL) measurements that probe the dynamics of the ground state PL emission as a function of time between the excitation and emitted photon detection is another important and useful tool that allows establishing the spontaneous decay rate of the QDs. The typical TRPL instrumentation has a resolution ~10-100 ps which is adequate for the typical excitonic state decay time of ~1ns found in the III-V compound semiconductor class of quantum dots. Thus the recombination times of different excitons in the MTSQD’s is studied here using TRPL spectroscopy briefly described in section §3.6. While PL probed in ways noted above provides critical information on the behavior of the lowest energy transitions there remains the need to probe the excited states of the quantum dots that control important properties such as the temperature dependence of the ground state behavior. In bulk solids absorption measurements provide information on the higher lying energy states through the impact of their density of states. However for quantum dots, particularly single quantum dots the absorption volume being miniscule direct absorption measurements are impossible. Fortunately a variation of PL does provide some valuable information on the excited states. It involves PL measurements at certain fixed detection energies (typically within the linewidth of ground state PL peak) and scanning the excitation energy through higher lying excited states. Such a spectrum of PL intensity as a function of 142 excitation energy is called the PL excitation (PLE) spectrum and it typically exhibits peaks where the excitation energy matches higher excited state transitions owing to their higher density of states. The PLE is thus similar to absorption except that it also depends on the choice of the detection energy, the relaxation process and the rate of the carriers capture, and thus the competition between radiative recombination and non-radiative recombination processes as well. This makes it different from the absorption spectrum. The PLE is nevertheless an important tool for the study of QD electronic structure. Details of this type of study on our MTSQDs is captured in section §3.7. §3.3.2 Micro-PL Instrumentation Large area PL/PLE and TRPL setup has been built in the group by former group members (Appendix B). The excitation beam diameter on the sample is ~200- 300 μm that limits the suitability of such setup to the study of ensemble averaged optical behavior of QDs not individual single QD, i.e. MTSQDs. The synthesized MTSQDs arrays discussed in Chapter 2 have QD-QD separation ~5 μm. To study the optical properties from individual MTSQD, a micro-PL system with excitation beam diameter of ~1-2 μm that is capable of measuring time-integrated PL, PLE and TRPL spectra from individual MTSQD has been built in the group under this dissertation work. This system is further expanded with Hanbury-Brown Twiss setup (discussed in Chapter 4) to study the photon correlation functions of the emitted photons from individual MTSQDs. We capture here in this section the important features of the 143 home built micro-PL setup used to explore the optical properties of MTSQDs through PL/PLE/TRPL studies discussed later in this chapter. Figure 3.8 below shows the schematic diagram of our home-built micro-PL setup used to study optical properties of the synthesized individual MTSQDs (Chapter 2) in the array. The setup contains three major parts: (1) laser system for optical excitation, (2) optical detection system for collecting and subsequent detection of photons emitted from MTSQDs and (3) signal processing system to record the electronic signal from spectrometer and detectors. Excitation light from either one of the lasers in the laser system including 640nm diode laser (PicoQuant model LDH-P-C-640B), Ti-Sa continuous-wave (CW) laser (spectraphysics 3900) and Ti-Sa ps mode-locked pulsed laser (Coherent Mira 900D) can be coupled into multimode optical fiber with core 50 μm and NA 0.2, filtered by a 900nm short pass and a 900nm dichromatic filters and focused down to ~ 1.25μm diameter through a 40× NA 0.65 objective lens on to sample mounted inside a continuous flow cryostat (Janis ST-500, 4K to 360K) for optical excitation. The small excitation beam diameter of 1.25 μm enables the optical excitation of individual MTSQDs. The emitted photons from an individual MTSQD is collected in the same vertical geometry using the same objective lens, filtered by a 900nm dichromatic and a 900nm long pass filter, coupled to a FC-PC adjustable collimator (Thorlab, CFC-2X- B) and focused into multimode optical fiber with core 25 μm and NA 0.1. Photoluminescence collected through the optical fiber is focused by a pair of NIR- coated achromatic 1” f=25mm lens into the entrance slit of an f=0.3m single stage 144 imaging spectrograph (Acton SP300i). The spectrometer used has diffraction limited spectral resolution of 0.2nm (~200 μeV at 1μm). The spectrally dispersed light with spectral resolution set by the spectrometer can be detected by the LN 2 cooled InGaAs CCD (charge-coupled device) detector (Princeton InGaAs detector) or Si APDs (Excelitas model SPCM-NIR-14-FC and PicoQuant model τ-SPAD). The excitation and subsequent detection of emission from a MTSQD under the beam enables the study of photoluminescence from each individual MTSQD in the synthesized array. Figure 3.8 Schematic drawing of micro-PL setup. The red line with arrows represents the MTSQD emission detection light path. The Blue line with arrows and the blue excitation fibers represents the excitation light path used for PL/TRPL measurement. The purple line with arrows and the purple dash curved line (fiber) 145 represent the light path for PLE measurement. The black line and black dash line represent the electrical control and processing signal path for the setup. For time-integrated PL measurements, the 640nm solid state diode laser with repetition rate of 80MHz (PicoQuant model LDH-P-C-640B) or an 532nm solid state laser (Coherent Verdi G8) pumped Ti:S laser (Mira 900D, tunable wavelength from 700-1000nm) with 76MHz repetition rate are used as the excitation source. The light emitted from MTSQDs and collected through the microscope is spectrally filtered by scanning the spectrometer wavelength and subsequently detected by either Si APD (for InGaAs MTSQDs, λ < 950nm) or InGaAs CCD (for InAs MTSQDs, λ > 950nm) based on QD emission wavelength to map out the emission energy spectrum of the MTSQD. For PLE measurements the frequency-doubled 532nm solid state laser (Coherent Verdi G8) pumped Ti:Sa CW laser (Spectra physics 3900, tunable wavelength from 700-1000nm) is used as the excitation light source. The wavelength of the Ti:Sa CW excitation laser is scanned by adjusting the position of the prism inside the cavity controlled by a step motor attached to the prism and thus tuning the wavelength. The motor position is computer controlled and can provide wavelength adjustment accuracy of 0.2A. The light emitted from MTSQDs and collected through the microscope is spectrally filtered with spectrometer at the MTSQD emission wavelength. The number of photons detected by Si APD as a function of Ti-Sa CW laser excitation wavelength (energy) is recorded. 146 For TRPL measurements, the 640nm solid state diode laser with repetition rate of 80MHz or Ti:Sa pulsed laser (Mira 900D) with 76MHz repetition rate are used as the excitation source. The electronic pulse (NIM (nuclear instrumentation module) pulse) representing the optical excitation pulse from the laser itself is used as the signal to represent timing of optical excitation event. In the TRPL measurement, Si APD is used as the detector and generates the TTL (transistor- transistor logic) electrical pulse signal recoding the PL detection event from MTSQD. The timing of the TTL pulses from the Si APD detector and the NIM pulsed signal from laser are registered using constant fraction differential discriminators (Ortec, model 9307) and fed to a time-to-amplitude (TAC) convertor (Ortec, model 457) as the start and stop signal respectively. Details of signal processing for TRPL measurement is discussed in Appendix C. The stop signal is delayed before getting into TAC. The TAC outputs are read by a multi-channel analyzer (Ortec Trump-PCI) to generate TRPL spectrum. The TRPL setup has a finite timing resolution limited by the APD time jitter and the jitter of the processing electronics. The time resolution of TRPL measurement is generally characterized by the instrument response function (IRF). Operationally the IRF is measured by focusing the excitation laser on the copper sample holder and measuring the time distribution of the scattered laser light collected and detected by the Si APD using exactly the same procedure and instrument configuration in which the TRPL measurements of MTSQDs will be performed. Figure 3.9 is an example of a measured IRF. The FWHM (full width at half maximum) is 200ps. The rise time (10%-90%) of IRF is ~150ps and the decay 147 time (10%-90%) is ~190ps. The FWHM of IRF represents the time-resolution of the TRPL setup. Figure 3.9. Measured instrument response function (IRF) of the TRPL setup. The FWHM, rise time and decay time of the IRF are, respectively, 200ps, 150ps and 190ps. §3.4 Photoluminescence study of MTSQDs: Emission Uniformity and Efficiency Photoluminescence based optical study of the MTSQDs is carried out employing PL/PLE/TRPL spectroscopy using our home-built micro-PL setup discussed in the previous section. Due to the prohibitive expense of liquid helium the PL/PLE/TRPL studies could largely be carried out at liquid nitrogen temperature with only limited results at liquid helium temperature. In this section, we discuss the time-integrated PL studies of the uniformity and emission efficiency of the MTSQD array. §3.4.1 MTSQD Emission: Identification of Single Neutral Exciton 148 The sample studied extensively in this dissertation is sample RG130916 that contains a 5 × 8 array of 4.25ML In 0.5 Ga 0.5 As MTSQDs with {103} side wall (details in Chapter 2, section 2.3.2, 2.3.3 and 2.4). The SQD formed on mesa tops with {103} side walls before the {103} planes pinch-off have a rhombus base with estimated base edge length ~13nm and height ~3nm. The identification of the MTSQD emission is carried out through time-integrated PL studies with the excitation beam of diameter ~1.25 μm focused on the nanomesa (bearing a single quantum dot). The valley area surrounding the nanomesa has, as discussed in Ch.2 (Sec. 2.4), a low density of lattice mismatch strain-driven 3D island self-assembled QDs (~2-5/μm 2 ) that provide a built-in reference. Indeed, the optical emission from these SAQDs in the planar (i.e. unpatterned; see chapter 2) region of the above referenced MTSQD sample, as well as SAQDs grown on non-patterned substrates under identical conditions (sample RG130307) were studied first to establish their expected distinct spectral identity. These SAQDs have an average QD base length ~51.5nm and height ~4nm as shown in Fig. 3.10(a) obtained from atomic force microscope (AFM) images from an uncapped 4.25ML In 0.5 Ga 0.5 As SAQD sample (RG130307) grown under the same conditions as the MTSQDs sample. The size distribution of the SAQDs in the planar regions shows biomodal characteristics with a narrow size distribution around base~ 45nm and height ~3nm and a broad size distribution around base ~60nm and height ~5nm. Given the SAQD density and optical excitation beam size of 1.25μm, the PL collected from the SAQDs in the planar region of our MTSQD sample is an 149 ensemble average of 3-6 different SAQDs. The collected PL from such ensemble shows PL emission around 910nm with FWHM ~23.8meV in Fig. 3.10 (b). The observed 872nm peak is from the 2D wetting layer of the InGaAs SAQDs in the planar region given its energy and the invariance of its emission position with excitation spot position in the planar region. The PL data are measured using 780nm CW excitation with excitation power density ~47W/cm 2 below the threshold for state-filling and multiexction events (~ I~10 4 W/cm 2 [3.39]). Under the measurement condition, the shown PL FWHM is a representation of the inhomogeneous broadening of the emission of individual SAQDs probed under the excitation beam. Figure 3.10. (a) Size distribution of the uncapped 4.25ML In 0.5 Ga 0.5 As SAQDs sample (RG130307) showing a bimodal size distribution with one around base 150 ~45nm, height ~3nm and the other around base ~60nm, height ~5nm from AFM data (b) PL data of the planar region of the 4.25ML In 0.5 Ga 0.5 As MTSQD sample (RG130916) collected with 780nm, CW excitation and spectra resolution of 2.5nm (3.5meV) at 77K. The excitation power is 591.3nW on mesa (power density ~47.1W/cm 2 ). (c) The plot of the summation of the four PL spectra collected at four different planar regions of the 4.25ML In 0.5 Ga 0.5 As MTSQD sample (RG130916). The red line shows the fit of two Gaussian peaks corresponding to the two observed bimodal QD size distribution in panel (a). The PL data of the 4.25ML In 0.5 Ga 0.5 As SAQDs in sample RG130916 have been collected from four different locations in the planar region (i.e. the region where no nanomesas were fabricated) separated by 50-200 μm from each other under the same excitation and detection conditions. All data show SAQD emission from 890-920nm similar to that in Fig. 3.10 (b). Figure 3.10 (c) shows the plot of the summation of these four PL data. The spectrum shown in Fig. 3.10 (c) represents an ensemble PL emission behavior of the estimated 12 to 24 different SAQDs in the four planar regions. The asymmetry of the SAQD PL spectra suggests the existence of two Gaussian type bimodal distribution (green line, fitted curve with Gaussian shape) of SAQDs sizes consistent with the AFM data shown in Fig. 3.10 (a).  Photoluminescence from Mesa-top Single Quantum Dot (MTSQD) Time-integrated PL measurements have been carried out on MTSQDs with excitation beam focused on the individual mesa top and at different values of the distance D from the mesa top as shown in Fig. 3.11 (a). Each MTSQD in the 5 × 8 array is identified by the row and column number of the MTSQD in the array. We 151 use MTSQD (3,5) as an example here to discuss the typical emission behavior. Figure 3.11 (a) Schematic of measurement geometry of the MTSQD sample. (b) PL data collected on MTSQD (3,5) with varying beam spot and mesa separation D (excitation beam moving along [110] direction).The QD is collected at 77.4K with 780nm CW excitation power 235nW (power density ~18.8W/cm 2 ) and spectral resolution of 1nm. (c) 930nm PL peak intensity as a function of D with excitation beam moving along [110] (black dot and line) and [1-10] (red dot and line) direction. The spectrum was collected with the 780nm excitation beam of diameter ~1.25μm from the CW laser at an excitation power of 235nW (power density ~18.8W/cm 2 ) focused on the mesa. The black curve in Fig. 3.11(b) shows the measured spectrum filtered with 900nm long pass filter and detected with spectral resolution of 1nm when the beam is focused on the mesa bearing the SQD (i.e D=0). The emission is 152 seen to be peaked near 930nm and spread over 920nm to 940nm. To confirm the spatial origin of the ~930nm peak, we collect the PL from MTSQD (3,5) while moving the excitation beam away from the mesa. The blue and red lines in Fig. 3.11 (b) show the collected PL spectra with D=1μm and D=2.5um, respectively, with the excitation beam moving along [110] crystal direction. The intensity of the 930nm peak drops rapidly to the APD (Picoquant τ-SPAD) dark count ~150/sec. Fig. 3.11 (c) shows the 930nm peak PL intensity with D increasing along [110] (black dots and line) and [1-10] (red dots and line) directions. The rapid drop of the PL peak intensity to the APD dark count indicates that the observed emission from 920 to 940nm is from a structure on the mesa top. This range is distinct from the SAQDs in the valley region between the mesas that show emission from 890 to 920nm, same as that in the planar region (Fig. 3.10 (b) and (c)). Compared to the SAQDs in the planar region of the sample with typical base ~50nm and height ~5nm, the MTSQDs with significantly smaller lateral size should have considerably larger in-plane quantum confinement that would increase the energy difference between the QD electron and hole ground state and should thus show higher QD emission energy (shorter wavelength than 920nm) for equivalent compressive strain as in the SAQDs. The observed shift to longer wavelengths thus indicates that the potential strain relaxation (as discussed in Chapter 2) on mesa top due to large surface area of the mesa can over compensate and decrease the quantum confinement potential that in turn reduces the QD emission energy. Indeed, as seen 153 for the InAs MTSQDs discussed in Chapter 2 the strain relaxation can cause the QD emission wavelength to red shift by as much as ~70 meV (~68 nm). To gain an understanding of the dominant features of the excitonic emission from the MTSQD in the observed 920-940nm region, systematic excitation power and temperature dependent PL studies have been carried out on MTSQD (3,5) employing 0.2nm spectral resolution, the highest resolution allowed by the instrumental setup, to uncover finer features not seen in Fig.3.11(b). All the PL and TRPL studies of the MTSQDs shown in this Chapter are carried out under above GaAs (barrier) gap excitation condition using primarily the 640nm 80MHz pulsed laser. The same excitation conditions are used for studying the single photon emission properties of MTSQDs discussed in Chapter 4. Moreover, to shed light on the origin of the finer peaks we examined the excitation power dependence of the PL emission. For MTSQD (3,5) the excitation power ranged from 4nW(~0.32W/cm 2 , ~160 photons per pulse) to 37.5nW ( ~3W/cm 2 , ~1500 photons per pulse). Fig. 3.12 (a) shows the PL at 77K from MTSQD (3,5) at three different excitation powers; 4nW, 10.9nW and 25.8nW. At the lowest excitation power, there is only one emitting peak labeled as X 1 at 929.2nm with a FWHM (full width at half maximum) ~1meV. With increasing excitation power, peaks labeled as X 2 and X 3 emerge at 927.4nm and 931.4nm respectively. With further increase of excitation power beyond ~15nW (1.2W/cm 2 , ~600 photons per pulse) another peak X 4 at 933.4nm appears. 154 Figure 3.12 (a) Power dependent PL from MTSQD (3,5) collected at 77.4 K with spectra resolution of 0.2nm and three different excitation power at ~4 nW (~0.32 W/cm 2 , ~160 photon per pulse), ~10.5nW (0.87W/cm 2 , ~420 photon per pulse) and ~25.8nW (2.06W/cm 2 , ~1030 photon per pulse) from 640nm 80MHz pulsed laser. (b) Power dependence of the MTSQD’s PL peak X 1 integrated intensity at 77.4 K. A fit (black line) to the data shows I~P 0.93 . The peak integrated intensity saturates around 18nW (1.44W/cm 2 , ~720 photon per pulse). (c) Schematic of the electron and hole configuration in the QD for the single neutral exciton. Here we focus on the origin of the dominant peak X 1 at 929.2nm, postponing the discussion of the other peaks to subsection §3.5. The integrated PL intensity of peak X 1 at 929.2nm grows linearly with excitation power, I~P 0.93 , and saturates around 18nW (P sat , ~1.44W/cm 2 , ~720 photon per pulse) excitation. It is to be noted that in the power dependence law, I=αP β , both α and β depend upon QD state filling (capture mechanism) [3.40]. Our finding of β close to one for the peak X 1 indicates 155 that this emission is dominated by the single exciton state, conventionally denoted as X 0 and schematically indicated in Fig.3.12(c). The small departure from unity is indicative of the impact of capture mechanism [3.40]. Detailed discussion of the carrier capture process in the MTSQDs is given in section §3.6. The saturation power indicates the power beyond which more than one exciton begins to be created in the QD per excitation pulse. At the lowest excitation power ~4nW, P/P sat =23%, and the number of excitons created in the QD per pulse is less than one. Thus the emission peak X 1 at 929.2nm is likely the neutral single exciton emission, conventionally denoted as X 0 . The observed linewidth of the neutral single exciton emission is limited by thermal broadening at 77K and Stark shift induced by possible charge fluctuations at the surface of the capping GaAs layer [3.41]. Detailed discussion of the origins of peaks X 2 , X 3 and X 4 are provided in sections §3.5 and §3.6. To confirm that the emission at 929.2nm is from the In 0.5 Ga 0.5 As MTSQD, two additional PL studies were carried out: (1) the excitation beam was stepped along the [110] and [1-10] directions from D= - 2.5 μm to D= 2.5 μm. (2) the temperature dependence of the PL peaks was measured. Fig. 3.13 below shows the measured PL peak intensity of peak X 1 at 929.2nm as a function of D (D=0 when the excitation beam center and mesa center are aligned along the [110] and [1-10] directions. The intensity drops rapidly to the APD dark count level when the excitation beam is moved away from the mesa along [110] (Fig. 156 3.13 (a)) and [1-10] (Fig. 3.13 (b)) directions. This confirms the spatial origin of the observed emission X 1 is from a 3D confined structure on the mesatop. Figure 3.13 Intensity of the X 1 peak of MTSQD (3,5) at 929.2nm as a function of D for excitation beam moving along (a) [110] and (b) [1-10] direction. Measurements were done at 77K with spectral resolution of 0.2nm and excitation power ~15.8nW (power density ~1.26W/cm 2 , ~630 photon per pulse).  Temperature Dependence of Peak X 1 Additional evidence that the spatial origin of the observed 929.2nm transition is from the InGaAs material grown on mesa top is provided by its temperature dependent behavior for excitation beam aligned with the mesa top. Fig 3.14 shows the measured shift in the emission energy of the peak X 1 of MTSQD (3,5) with temperature between ~8K and ~125K. It is found to follow the calculated temperature dependence of the bulk (thus unstrained) In 0.5 Ga 0.5 As band gap change (solid line, Fig. 3.14) obtained from the virtual crystal approximation model of an alloy which gives the following expression for bulk In x Ga 1-x As alloy band gap change with x [3.42, 3.43]: 157 ( , ) ( ) ( )(1 ) InAs GaAs E x T E T x E T x       (Eq.3.41) in which () InAs ET  and () GaAs ET  are the change (reduction) in the bandgap of the two components as a function of temperature given by the well-established Varshni form [3.44], 2 () T ET T     (Eq.3.42) with input parameters x=0.5, α InAs = -4.19x10 -4 eVK -2 , β InAs =271 K and α GaAs = - 5.8x10 4 eVK -2 , β GaAs = 300 K taken from Ref. [3.42] corresponding to the bulk (strain-free) InAs and GaAs, respectively. These findings further support the origin of the emission of peak X 1 as being from the InGaAs QDs. 0 25 50 75 100 125 150 -30 -20 -10 0 E(T) (meV) T (K) Peak X 1 Figure 3.14 Shift in the energy of PL peak X 1 of MTSQD (3,5) with temperature from 77.4K to 150K. The shift follows closely the calculated (see text) temperature dependence of the In 0.5 Ga 0.5 As bandgap change (black line). 158 §3.4.2 MTSQD Emission: Spectral Uniformity and Efficiency In the preceding subsection we established that (a) there is distinct emission from 3D confined InGaAs states at the mesatop and (b) the emission wavelength is longer than that of the InGaAs 3D island based SAQDs in the planar region of the sample in spite of its significantly smaller volume indicating that there also exists enough strain relaxation to overcompensate the increased quantum confinement. In this subsection we focus on the emission statistics of the entire 5 × 8 array (not just the illustrative individual MTSQD (3,5) discussed heretofore) with focus on the emission wavelength uniformity and emission efficiency of the array. Realizing sufficiently uniform and efficient emission from spatially ordered array of single quantum dots that can also act as triggered single photon emitters is, of course, at the core of the SESRE approach underlying this dissertation to enable steps towards on- chip integrated optical circuits as discussed in Chapter 1. We recall from Chapter 2 that the entire 5 x 8 MTSQD array occupies an area of the 20 × 35 um 2 of the sample and lies within a 1cm radius of the center of the substrate coincident with the center of the molyblock holding the substrate during growth. In our MBE growth chamber, with the substrate rotating, the flux uniformity within this 1cm radius is found to be > 98% (Details of flux uniformity are given in Appendix B). The high flux uniformity and the growth control on QD size and shape during substrate-encoded size-reducing epitaxy are the features that we expect will enable the high degree of spectral uniformity amongst MTSQDs needed for realizing optical circuits. Thus PL spectra were collected from all 40 MTSQDs in the 5 × 8 159 array and the observed emission peak wavelengths are depicted by the color-coded pixels in Fig.3.15 (a). Only two out of the 40 are non-emitting (marked as black boxes in Fig. 3.15 (a)). The rest 38 MTSQDs have single neutral exciton transition from 919nm to 952nm with an average emission wavelength of 935.5nm and a standard deviation, σ λ , of 8.3 nm which is <1% of the emission wavelength. Fig. 3.15 (b) shows the PL spectra from MTSQDs (3,5), (2,4), and (4,4). These three MTSQDs are the ones studied for their single photon emission behavior at liquid helium temperature in Chapter 4. Note that the three MTSQDs show similar emission pattern and linewidth ~1meV at 77K (Fig. 3.15(b)). The PL peak wavelengths of the all 38 emitting MTSQDs in the array are plotted as a histogram in Fig. 3. 15(c). Figure 3.15 (a) the single exciton PL peak wavelength of each MTSQD in the 5 × 8 array shown as color coded blocks. The two non-emitting MTSQDs are marked as black blocks with white outlines. Pairs of like-color circles identify MTSQDs with emission within the instrument resolution limit of 0.2nm. (b) PL spectra of MTSQD 160 (3,5), (2,4) and (4,4) collected at 77K with spectral resolution of 0.2nm and excitation power P/P sat ~23% under 640nm 80MHz excitation. The primary peaks shown are the single exciton transition from MTSQDs. (c) Histogram of PL peak wavelengths from the 38 emitting MTSQDs in the 5 × 8 array covering an area of 1000 μm 2 . The standard deviation of the wavelength is 8.3 nm. The red line represents the Gaussian fit to the distribution. The overall longer emission wavelengths of MTSQDs compared with the SAQDs in the planar region of the sample confirms again our expectation of at least partial strain relaxation in the In 0.5 Ga 0.5 As formed on the GaAs nanomesas in the array compared to the 4% lattice-mismatch strain that is the driving force for the formation of the 3D island SAQDs. Additionally, the observed spectral uniformity of the array (σ λ of 8.3 nm, and σ E of 11.3 meV) is a significant improvement over the InGaAs SAQDs in the planar region and the typical SAQDs (~50meV) [3.30] and NCQDs (nanocrystal QDs) (~100meV) [3.45]. The inherently built-in growth control on the QD size and shape (flat-top pyramidal shape with controlled crystallographic side planes) of the SESRE approach enables the observed high spectral uniformity of MTSQDs. Strikingly, there are pairs of MTSQDs marked with like-color circles (Fig. 3.15(a)) that emit at wavelengths that are within our current instrument resolution of 0.2nm. An example of the PL emission of such a pair, MTSQD (3,5) and MTSQD (3,1), at 77K and 8K is shown in Fig. 3.16 (a) and (b), respectively. 161 Figure 3.16. The normalized PL spectra of the MTSQD (3,5) (black dot and line) and the MTSQD (3,1) (red dot and line), the two MTSQDs marked by the red circles in Fig. 3.14(b), collected at (a) 77K and (b) 8K with spectral resolution of 0.2nm with 640nm, 80MHz excitation at excitation power P/P sat ~23%. The normalized PL is obtained by scaling the spectrum with respect to the peak value. The difference between the single exciton emission wavelengths of the two shown QDs is limited by the spectral resolution of the optical setup. The MTSQDs with emission at the shortest (919 nm) and longest (951 nm) wavelengths lie on the outer row and column in the 5 × 8 array. The dominant contribution to the observed variation in the MTSQD emission wavelengths most likely originates from the variation of the as-etched mesa size. Such mesa size variation causes variation in the starting QD base length (at the end of the homoepitaxy prior to InGaAs deposition) and thus variation in QD size (height and base) as well as indium composition owing to differing amounts of atom migration from side facet to (001) mesa top. Moreover, alloy disorder scattering also leads to broadening and inhomogeneity of the emission line. Indeed, these effects are 162 eliminated if binary InAs is used to form the MTSQDs such as we did in sample RG130625 discussed in Chapter 2. It shows spectral emission at λ=1122.5±1.2nm obtained over three randomly picked MTSQDs in the array (one example of PL from InAs MTSQD shown in Fig.2. 25(b)), a narrower spectral spread as expected [3.46]. No attempts were made to optimize these for the MTSQDs in any of the samples in these first studies. Through improved control on the starting as-etched mesa sizes and control on gallium and indium migration lengths by optimizing growth kinetics, spectral uniformity of the MTSQDs can be further enhanced. Efforts in these directions are underway. Overall, MTSQDs synthesized using SESRE approach provide not only spatially-ordered QDs but also considerable control on the spectral uniformity through the growth induced control on QD shape, size, and composition. This makes them suitable for optimization towards applications such as single photon source for optical circuits. The obtained spectral uniformity in the MTSQDs array and the closely separated pairs of emission from MTSQDs provides a compelling reason for pursuing investigations of the full potential of this class of SQD arrays for quantum optical circuits for quantum information processing. Another important figure of merit for the application of SQDs as light source in optical circuits is their emission efficiency. Thus the intensity of the 38 emitting MTSQDs of the 40 in the 5 × 8 array collected at 77K with the same excitation power of ~ 15.8nW (1.26W/cm 2 , 640nm, 80MHz excitation, ~630 photon per pulse) is shown color coded in Fig. 3.17 (a). It varies within a factor of 4 over the whole array. To assess the quality of MTSQDs with respect to other reported classes of 163 spatially ordered SQDs synthesized using different approaches, we looked at the power dependent PL behavior of MTSQD single exciton emission. As an illustration, in the MTSQD (3,5) power dependent PL data shown in Fig. 3.12(b), the single exciton transition saturates at an excitation power of ~ 18nW (1.44W/cm 2 , 720 photon per pulse), an order of magnitude lower saturation power compared to that reported for other classes of spatially-ordered QDs such as InGaAs/AlGaAs QDs in valleys (holes) in structurally patterned GaAs(111)B substrate (~100nW) [3.47], InAsP/InP nanowire QDs (~1μW) [3.48], and GaN/AlGaN nanopillar QDs (~10mW) [3.49]. The observed low saturation power is not unique to a particular MTSQD (3,5) shown here. Figure 3.16 (b) shows the measured saturation power of the single exciton transition from 7 different MTSQDs picked out of the 5 × 8 MTSQD array. The 7 MTSQDs shown have saturation power around 28.9±11.8 nW (~1150 photon per pulse), much lower than literature reported values obtained from the spatially ordered QDs discussed above. The saturation power reflects the carrier capture efficiency of the QD that is controlled by the exciton diffusion length, GaAs exciton lifetime, and the density of traps in the GaAs barrier. Due to the lack of information on details of the measurement conditions in the literature on the reported ordered QDs discussed above, we can infer only qualitatively that these MTSQDs synthesized through the SESRE approach may have high carrier capture efficiency guided by the saturation power data on our MTSQDs in comparison with other reported data on ordered QD array synthesized using different approaches discussed before. We note that the SESRE approach enables growth-controlled formation of 164 MTSQDs without any unintentional structures to compete with the QD for carrier capture, unlike the unintended parasitic heterostructures formed in nanowire QD structures [3.50]. The high carrier capture efficiency of MTSQDs also suggests good grown material quality of GaAs and In 0.5 Ga 0.5 As and in turn good quantum efficiency of the synthesized MTSQDs. Detailed discussion of MTSQD quantum efficiency and its single photon emission efficiency is provided in Chapter 4. Figure 3.17 (a) PL peak intensity of each MTSQD in the 5×8 array shown as color coded blocks. The two non-emitting MTSQDs are marked as black blocks with white outlines. Pairs of like-color circles identify MTSQDs with emission within the instrument resolution limit of 0.2nm as in Fig. 3.14 (b). (b) The saturation power of MTSQD single exciton transition obtained from 8 MTSQDs at 77K with 640nm, 80MHz excitation. The saturation power is ~28.9±11.8 nW (~1150 photons per pulse) for the shown MTSQDs. §3.5 MTSQD Excitonic Emission Types and Their Identification In the preceding section we focused on the single dominant peak (Fig.3.12(a)) in the PL from MTSQD at 77K and its emission wavelength uniformity and emission efficiency across the array. There are, however, several finer peaks seen in the MTSQD PL spectra and in this section we present a detailed study aimed at 165 identifying the origin of the multiple emission peaks (Fig. 3.12(a)) as this is central to assessing the potential use of MTSQDs as single photon source and, more importantly, entangled photon source. To this end, in the following subsections we establish their presence in the MTSQD region alone and exploit excitation power and polarization dependence of the emitted photons. §3.5.1 Multi-peak Emission from a MTSQD As seen in the preceding subsection, peaks labeled as X 2 and X 3 at 927.4nm and 931.4nm respectively in the PL from MTSQD(3,5) shown in Fig.3.12(a) start rising up with increasing excitation power besides the single exciton transition peak labeled as X 1 at 929.2nm. Further increase of excitation power leads to the emission from one more peak labeled as X 4 . These three additional peaks that show up at higher excitation powers are also from the MTSQD itself are checked by their excitation beam position dependent PL intensity. Fig. 3.18 below shows the measured PL peak intensity of peak X 2 , X 3 and X 4 as a function of D. The excitation beam has been stepped along [110] (Fig. 3.18 (a)) and [1-10] (Fig. 3.18 (b)) directions. The PL peak intensities of peaks X 2 , X 3 and X 4 all drop quickly to the APD dark count level when the excitation beam is moved out of the mesa along [110] and [1-10] direction. Thus we conclude that all the observed peaks X 2 , X 3 and X 4 are also from the mesa top region, i.e. from the MTSQD. The energy separation of the observed peaks from the single neutral exciton transition peak X 1 is from 2.5 to 6 meV, consistent with the typical observed energy separations between different 166 excitonic transitions reported for InGaAs SAQDs [3.28, 3.30, 3.51-3.54] that have been identified as charged excitons and biexciton. Figure 3.18 PL peak intensity of MTSQD (3,5)’s emission peak X 2 (at 927.4nm), peak X 3 (at 931.4nm), and peak X 4 (at 933.4nm) as a function of D with excitation beam moving along (a) [110] and (b) [1-10] direction measured at 77K with spectral resolution of 0.2nm and excitation power ~15.8nW (power density ~1.26W/cm 2 , ~630 photons per pulse). The observed multiple peaks from MTSQD (3,5) with energy separations within 10meV of the single exciton transition is not unique to this particular MTSQD shown in Fig. 3.12 (a). The MTSQDs in the array have in general similar multi-peak emission pattern. The PL emissions from MTSQD (4,4) and (2,4) are shown in Fig. 3.19 (a) and (b) respectively. In each panel of Fig. 3.19, PL spectra from the MTSQD at three different excitation powers, P/P sat ~23%, P/P sat ~60% and P/P sat ~200%, are shown where P sat is the saturation power for the single neutral exciton transition of the MTSQD. The peaks X 2 , X 3 and X 4 show up with increasing excitation power and have energy separation of <10meV between the single neutral exciton peak X 1 shown in Fig. 3.12 (a) and Fig. 3.19. The observed similar four 167 dominant emission peaks for different MTSQDs suggests that the observed emissions are the excitonic transitions from the MTSQDs. Figure 3.19 Power dependent PL from (a) MTSQD (2,4) and (b) MTSQD (4,4) collected at 77.4 K with spectral resolution of 0.2nm at three different excitation powers at ~ P/P sat ~23%, P/P sat ~60% and P/P sat ~200% from a 640nm 80MHz pulsed laser. The saturation power, P sat , for MTSQD (2,4) and MTSQD (4,4) are respectively 20nW (~1.6W/cm 2 , ~800 photons per pulse) and 16nW (~1.28W/cm 2 , ~640 photons per pulse). In the following subsections we discuss the systematic power dependent PL, polarization dependent PL, and time-resolved PL studies on the observed multiple emission peaks from MTSQDs to probe further their origin within the limited instrument resolution of 0.2nm. 168 §3.5.2 Power Dependent PL Studies: Excitonic Complexes in MTSQDs The basic behavior of PL from MTSQD (3,5) noted in the preceding subsections as an illustrative example is seen to reveal presence of emission from more than the neutral single exciton state that merited further studies to reveal their origins. The findings of these studies are discussed in this section. To facilitate understanding of these observations it is useful however to first capture briefly the simplest physical model and its theoretical predictions for the power dependence behavior of excitonic transitions in the QDs involving more than a single electron and hole. Multiple electrons (e) and holes (h) created in the QD can lead to the formation of excitonic complexes comprising multiplets made of different number of electrons and holes and their relative spin configurations. Usually biexcitons and trions (single exciton with one additional electron or hole) recombinations discussed in section §3.2 are observed in self-assembled as well as nanowire single QD emission experiments. The correct attribution of the different peaks in the PL from the MTSQDs is thus the first important step to achieve in order to control and manipulate the charge and the spin of the excitons to obtain high purity single photon emission from the desired excitonic state of the QD. One major approach to determine the nature of the observed multiple PL peaks is based on the determination of the excitation power law dependence of their intensity. As a consequence of the Pauli’s principle, it is well known that level filling of electron and hole states can be easily realized in the QD and different excitonic complexes can be generated [3.55] 169 with increased excitation power density. It has been shown that the power dependence behavior of different excitonic line intensities can be understood in terms of the random population theory based on a master rate equation model for each excitonic state [3.55]. Below we briefly recall the essential result.  Rate Equation Model for Excitonic State Population Following the original work in Ref. [3.55], we assume that M excitonic microstates plus a reservoir level with population N R is present for the QD where the non-resonant photoinjection is made. The population probability of the nth microstates is M n N . The capture of one more exciton is a loss term for the nth population and a pump term for the (n+1)th population. Assume that at any time only one well-defined number n of excitons may exist in the QD and electrons and holes are always captured as pairs into the QD, the master equation for M n N is [3.40] 11 ( 1) ( 1) M M M M M n n n R n R n r r c c dN n N nN N N N N M n M n dt M M              (Eq. 3.43) where r  and c  are the recombination and capture times for exciton. The last two terms are related to the capture of one exciton for the (n-1)th population and nth population. The capture probability dependents on the number of excitons in the QDs and is assumed to be linearly decreasing with the number of excitons: when n excitons fill the QDs the capture time is reduced by a factor of (1-n/M). For steady state conditions, we have 0 M n dN dt  . We can have the following expression for M n N [3.40]: 170 1 0 0 ( ) (1 ) ! M n M Rr n m c N N m N nM       (Eq. 3.44) Detailed discussion of the dynamics of exciton states accounting for the capture of single electrons and holes besides the exciton is captured in section §3.6.2. Considering the limit of M  ( M nn NN  , 1 0 (1 ) 1 n m m M     ) and the sum rule that 0 1 M M n n N    , n N is seen to follow a Poissonian distribution: exp[ ] ! n n n Nn n      (Eq. 3.45) where the average number of exciton <n> has been defined as / R R c nN    and is the average over several population/depopulation processes of the same single QD within the measurement time and under steady state excitation. Therefore, N 1, that represents the probability of finding a single exciton in the QD, increases linearly with <n> up to a maximum for <n>=1. Similarly, N 2 that represents the probability of finding a two exciton complex (biexciton) in the QD increases quadratically with <n> up to a maximum for <n>=2. Under the assumption of a linear pumping at rate x of the reservoir level, whose population can only reduce by capture of exciton into QDs, we can have r nx     [3.55]. It therefore follows that the single exciton and biexciton populations N 1 and N 2 are, respectively, linearly and quadratically dependent with the excitation power. The measured PL intensity from single exciton and biexciton state being proportional to the population of N 1 and N 2 are thus expected to be linearly and quadratically dependent on the excitation power. The 171 power dependence of the PL emission can thus be used as a tool to attribute the origin of different lines seen in single QD PL spectra. With this objective we discuss next the power dependence of the integrated PL peak intensity of peaks X 2 , X 3 and X 4 from MTSQD (3,5). The PL peak integrated intensity of the four major peaks X 1 -X 4 are obtained by fitting the PL spectra of MTSQD (3,5) at different excitation powers shown in Fig. 3.12 (a) with multiple Gaussian shape peaks. The QD’s intrinsic transitions have Lorentz shape as discussed in section §3.3. However, the thermal broadening and stark shift convolve with the intrinsic lineshape and turn its shape to Gaussian. Fig. 3.20 (a) shows the Gaussian fitted PL spectrum from MTSQD (3,5) collected at P/P sat ~23%, 60% and 150% (the spectrum without fitting is shown in Fig. 3.12(a)). The red line represents the multiple Gaussian peak fitting of the PL spectra. The green lines in each PL spectra show the Guassian peak fitting of peaks X 1 to X 4 . Fig. 3.20 (b) shows the obtained PL integrated intensity (peak area) of peaks X 1 to X 4 (color dots). The power law fitting of each peak’s PL integrated intensity is shown as colored straight lines. The power dependence behavior of peak X 1 shown in Fig. 3.20 (b), though discussed in section §3.4, is recaptured here for comparison with the rest three peaks. 172 Figure 3.20 (a) Multi-peak Gaussian fitting (red line) of the PL spectra of MTSQD (3,5) at excitation power of 4nW, 10.5nW and 25.8nW with each fitted individual Gaussian peak shown in green lines. (b) Power dependence of the MTSQD’s PL peak X 1 (black dot), X 2 (blue dot), X 3 (red dot) and X 4 (magenta dot) integrated intensity at 77.4 K obtained from multi-peak Gaussian fitting shown in panel (a). The fit of peak X 1 , X 2 , X 3 and X 4 shows I~P 0.93 (black line), I~P 1.19 (blue line), I~P 1.35 (red line) and I~P 2.27 (magenta line) respectively. From the power dependence of the PL peak intensity data shown in Fig. 3.20 (b), we find the following behavior for peaks X 1 through X 4 : (1) The integrated PL intensity of peak X 1 at 929.2nm (1.3345eV) grows near linearly with excitation, I~P 0.93 , and saturates around 18nW (P sat , ~1.44W/cm 2 , ~720 photons per pulse) excitation. (2) The integrated PL intensity of peak X 2 at 927.4nm (1.3371eV, ~2.6meV higher from peak X 1 ) grows near linearly with excitation, I~P 1.19 , and saturates around 37.5nW (~3W/cm 2 , ~1500 photons per pulse) excitation. 173 (3) The integrated PL intensity of peak X 3 at 931.4nm (1.3314eV, ~ -3.1meV below peak X 1 ) grows super-linearly with excitation, I~P 1.35 , and saturates around 27.5nW (~2.2W/cm 2 , ~1100 photons per pulse) excitation. (4) The integrated PL intensity of peak X 4 at 933.4nm (1.3285eV, ~ -6.0meV apart from peak X 1 ) shows up for excitation power > 10.5nW (~0.87W/cm 2 ) and grows nearly quadratically with excitation, I~P 2.27 , and saturates around 37.5nW (~3W/cm 2 , ~1500 photons per pulse) excitation. Peaks X 2 , X 3 and X 4 also show saturation behavior similar to the single neutral exciton transition peak X 1 , confirming the assessment of the spatial origin of these emission peaks to be from the MTSQD as discussed in section §3.4. The presence of peak X 4 at high power, ~60% of the P sat of the neutral exciton peak X 1 , and its quadratic PL intensity power dependence suggests that it comes from the biexciton transition of the MTSQD. The probable biexciton transition (peak X 4 ) has energy ~6.0meV lower the probable neutral exciton transition (peak X 1 ), indicating the biexciton binding energy 26 xx x xx B E E meV    . For InGaAs/GaAs QDs, mostly InGaAs/GaAs SAQDs, the reported biexciton transition of the QDs generally has lower energy compared to the neutral exciton transition, same as observed in our MTSQD case, but with smaller biexciton binding energy ~1 to 3meV [3.3, 3.52-55, 3.52-58]. The biexciton binding energy is related to the few-particle Coulomb interaction and exchange correlation energies that depend sensitively on the geometrical and chemical properties of the QDs [3.30, 3.51, 3.54]. In QDs which are less strained, have different or less developed facets, or 174 consist of lower piezoelectricity material, a positive large biexcton binding energy is favored [3.51, 3.54]. The large positive biexciton binding energy in the MTSQDs is consistent with the small facets and reduced strain in the MTSQD discussed in section §3.4. The peak X 2 and X 3 are probably coming from charged single exciton transitions, given the observed near linear PL intensity power dependence and the observed ~3meV higher energy of each of the two peaks with respect to the neutral single exciton transition (peak X 1 ). Figure 3.21 shows the PL emitting peak position and linewidth of peak X 1 (black dot), X 2 (blue dot) and X 3 (red dot). All three peaks blue shift with increasing excitation power. Peaks X 1 and X 2 blue shift ~100-120 μeV and the peak X 3 shifts ~460 μeV. The observed blue shift may come from the decrease of Coulomb interaction energy of the electrons and holes in the excitonic states due to the enlarged separation of electrons and holes induced by the increased local electric field that QD experiences with increasing excitation power. The peak X 2 and X 3 have larger linewidth compared to that of peak X 1 . The larger linewidth of the peak X 2 and X 3 may arise from the greater multiplicity of the excitonic state configuration. With increasing excitation power, the linewidths of peak X 2 and X 3 increase by 0.5-0.8meV. The observation of large peak linewidth and the increase of linewidth with excitation power suggest that the observed two peaks may come from charged exciton transitions [3.53, 3.59]. 175 0 1 2 3 4 5 6 7 8 929.0 929.1 929.2 929.3 929.4 927.2 927.3 927.4 927.5 927.6 931.1 931.2 931.3 931.4 931.5 931.6 FWH (meV) FWH (meV) FWH (meV) Peak X 3 Peak X 2 Wavelength (nm) Power Density (W/cm 2 ) Wavelength Peak X 1 Wavelength (nm) Wavelength Wavelength (nm) Wavelength 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 FWHM 2.0 2.5 3.0 3.5 4.0 FWHM FWHM Figure 3.21. The PL peak wavelength (black dot line) and FWHM (full-width at half maximum, red dot line) of peak X 1 (bottom panel), X 2 (middle panel) and X 3 (upper panel) from MTSQD (3,5) as a function of excitation power. All data are taken at 77K. Generally the excess carriers in the quantum dots that lead to the formation of charged excitons can have different origin. The first reason might be the creation of electrons and holes through above-gap excitation. Due to the different electron and hole mobilities, electrons and holes will have different carrier capture rates into the QD, leading to an imbalance of charge and the formation of charged excitons. The second contributing factor is the presence of deep levels in the GaAs barrier (defects and impurities) surrounding the QD. The different charged exciton configurations 176 that are formed in the QD depend on the relaxation rates of the optically injected carriers in the barrier material into the QD ground states. For the charged excitons with additional charges (electron or hole) being captured into the QD through the population of charge state in the GaAs matrix, the PL emission intensity can be reduced when optically pumping the QD with excitation energy below the GaAs bandgap [3.38, 3.60, 3.61]. To further gain insight into the nature of the observed transition peaks X 2 and X 3 , the power dependent PL study of MTSQD (3,5) has been carried out with subgap optical excitation at 850nm with pulsed frequency of 76MHz (E=1.459eV, below GaAs band gap of E g =1.489eV at 77K). Figure 3.22 (a) below shows PL collected at excitation power of 23% and 210% of the saturation power of the peak X 1 with excitation energy above GaAs band gap (640nm, black line) and below GaAs band gap (850nm, red line). The observed high PL emission intensity at the lowerst excitation power from peak X 1 is consistent with the assessment of it being from the neutral exciton transition. Emission intensity of Peak X 2 has reduced when excitons are created directly in the QD. The observed ratio of the integrated PL intensity of peak X 2 and peak X 1 has reduced by a factor of 2-3 (Fig. 3.22 (b)) suggesting that peak X 2 is a charged single exciton formed with the additional charge captured into the QD from the GaAs matrix. This observed charged exciton (peak X 2 ) is ~2.6meV higher in energy compared to the neutral exciton (peak X 1 ). Such a higher energy emission has been reported for InGaAs/GaAs SAQDs and has been identified as X + 177 exciton [3.53, 3.62]. This suggests that the observed PL peak X 2 is probably from a positively charged trion X + . The emission energy of the peak X 3 is ~3.1 meV lower than the neutral single excition peak X 1 at 1.3345eV. The negatively charged trion X - in the InGaAs/GaAs SAQDs is found to be of lower energy compared with neutral exciton X with by an energy around 2-6meV [3.52, 3.53, 3.62]. The observed linear power dependence of peak X 3 and the energy difference between peak X 3 and X 1 suggest that the emission peak X 3 might come from a negatively charged trion X - . An unchanged emission intensity found for peak X 3 for excitation energies above and below GaAs band gap (not shown) indicates that the additional electron captured into QD is probably from states in the GaAs bandgap, i.e. deep levels or donor type impurities. Figure 3.22 (a) PL spectra from MTSQD (3,5) collected at 77K with excitation power at 4nW (640nm 80MHz excitation, black line) and at 37.5nW (850nm 76MHz excitation, red line) shown in the lower panel and the PL collected with excitation power 37.5nW (640nm excitation, black line) and at 337nW (850nm excitation, red 178 line) shown in the upper panel. (b) The ratio of the integrated PL intensity of peak X 2 and X 1 as a function of P/P sat with 640nm (black dot line) and 850 (red dot line) excitation where P sat is the saturation power for peak X 1 . To summarize, the power dependent PL studies on MTSQD (3,5) with excitation energy above and below GaAs barrier band gap confirms that the peak X 1 at 929.2nm is from the neutral exciton (conventionally denoted X 0 ) decay. The peak X 2 at 927.4nm (~2.6meV higher than the neutral exciton) and the peak X 3 at 931.4nm (~3.1meV lower than the neutral exciton) are likely to be from positive trion (conventionally denoted X + ) and negative trion (conventionally denoted X - ) respectively. The peak X 4 at 933.4nm (~6meV lower than the neutral exciton) is likely from biexciton (conventionally denoted XX) decay. The observed deviation of power dependence of the single exciton (neutral and charged) and the biexcition from, respectively, linear and quardratic behavior, we propose, arises from the single carrier capture process. A detailed discussion and simulation of carrier capture process and the understanding of observed power dependence of different peaks is discussed in section §3.6. One additional thing to note from the PL data shown in Fig. 3.22 (a) is the asymmetry of the peak X 1, seen more clearly in the PL data collected with the excitation energy below the GaAs band gap (red line, in Fig. 3.22(a)). This suggests the existence of an additional peak ~928.6nm (labeled as X 1 *, ~860μeV higher in energy than the single neutral exciton transition from peak X 1 ). From power dependent PL data, this probable peak X 1 * shows integrated PL power dependence I~P 1.1 , closely linear suggesting that it likely comes from a single exciton state. 179 Indeed, the polarization dependent PL studies discussed next indicate that this peak is likely from a neutral exciton complex. §3.5.3 Polarization Dependent PL Studies of MTSQD: Excitonic Complexes in MTSQDs Besides the power dependence of the PL, polarization dependence of the emitted photon from different excitonic states can provide additional information on the associated electronic configuration to substantiate the assignment of the origin of the different transitions, X 1 ≡ X 0 , X 2 ≡ X + , X 3 ≡ X - , and X 4 ≡ XX. Importantly, the polarization of the emitted photons from the excitonic states is sensitive to the symmetry of the QD confinement potential as captured in section §3.2 and can thus be used as a probe for studying the symmetry of QD confinement potential. In this section, we focus on the polarization properties of emitted photons from different excitonic transitions in MTSQD (3,5) to assess the origin of the observed multiple emission peaks and the symmetry of the QD confinement potential. The polarization dependent PL spectra of the MTSQDs are collected in the back scattered geometry using our micro-PL setup discussed in section §3.3. A linear polarizer with extinction ratio of 10 4 :1 is added to the microscope in the optical detection branch to discriminate between the emitted photons collected into the objective with different in-plane polarizations. A schematic indicating the position of the linear polarizer in the setup is shown in Fig. 3.23 (a). Panel (b) of Fig. 3.23 shows the instrumentation of the home built microscope of the micro-PL setup with 180 installed polarizer position indicated. The x and y directional movement of the microscope controlled by the translation stage is used to label the direction of the microscope x and y axis as indicated in Fig.3.23 (b). The polarizer can be rotated in the x-y plane perpendicular the growth direction z as indicated in Fig. 3.23 (b). Figure 3.23 (a) A schematic and (b) detailed instrumentation of the micro-PL setup with the added linear polarizer (marked in red) that can be rotated in the x-y plane. The polarizer 0° (or 360°) direction is aligned with the cryptographic [1 -1 0] direction of the GaAs substrate with the aid of the triangular markers (discussed in Chapter 2) created on the sample. The alignment is achieved by aligning the base edge of the big triangular marker along [110] direction shown at the top in Fig. 3.24 (left most image) with respect to the y axis of the microscope indicated in Fig. 3.23 (b) with the aid of the CCD image of the sample projected on the monitor. The polarizer is then aligned with respect to y axis of the microscope to have the polarizer 181 0° (or 360°) aligned with the crystallographic [1 -1 0] direction. Guided by the SEM images (an SEM image example is shown in Fig. 3.24) based structural information of the presence of {101}, {103} and (001) facets on the mesa top before {103} pinch-off, the mesa top before {103} pinch off has a rhombus shape base with edges along <3 1 0> direction consistent with the {103} side facets. Upon deposition of InGaAs thus the QD base edge along [3 -1 0] direction, marked as red dot line in the schematic (Fig. 3.24, lower middle image), has an angle of α=22±4° with respect to the [1 -1 0] direction (polarizer 0°). Detailed structural information on the synthesized MTSQDs is in Chapter 2. We recapture here the above mentioned important information to facilitate discussion of the measured polarization data. Figure 3.24. The schematic (left most image) of the MTSQD sample with patterned mesa arrays labeled Si, i=0 to15, and the triangular markers on the sample to indicate cryptographic orientation. The MTSQD array under optical investigation in this dissertation (top middle image) is from the S11 region. A detailed geometry of a 182 mesa after GaAs buffer (just before InGaAs) growth is shown in the lower middle schematic image. The schematic of the mesa geometry and QD base shape follows the obtained SEM image of the mesa before the {103} plane dominated pinch-off. The polarization dependent PL data discussed here are collected from MTSQD (3,5) at 77K with instrument limited resolution of 0.2nm (~300μeV) and above GaAs bandgap excitation (640nm, 80MHz excitation). Photons emitted from MTSQD, passing through a linear polarizer with polar angle  with respect to the [1 -1 0] direction are collected in the micro-PL setup to acquire polarization discriminated time-integrated PL spectra. Polar angle  is adjusted from 0° to 360° with an increment of 10°. The measurements are carried out for excitation power P~10.5nW (~0.87W/cm 2 , 420 photons per pulse, ~ 60% of P sat of the neutral exciton) to avoid state filling while still being able to probe all four major peaks X 1 to X 4 . Figure 3.25 below shows the polar plot of the emission intensity (peak intensity) of (a) peak X 1 and (b) peak X 4 . The emissions from both transitions have an elliptical polar dependence. The major axis of the peak X 1 is at ~ 300° (Fig.3.25 (a)) where the crystallographic [1 -1 0] is taken as 0° (or 360°) and the minor axis is at 30°. The ratio of the amount of light with polarizer along the major and minor axis is ~1.5. The emission from peak X 4 has similar elliptical polar pattern. The major axis of the peak X 4 is at ~ 310° the minor axis is at 40° (10° rotated compared to that of peak X 1 ) with a smaller ratio of the amount of light with polarizer along the major and minor axis of ~1.3. Guided by the structural information on the shape (rhombus) of the QD base with its edges along <310> shown in red in Fig. 3.24, the ~[3 -1 0] QD edge indicated with dotted red line is ~20° with respect to the [1-10], marked as angle α in 183 Fig. 3.25 (a) and (b). The major axis of the elliptical polar pattern of peak X 1 and peak X 4 is thus found to be ~80° and 70° respectively with respect to QD edge orientation marked with red dotted line while the minor axis is ~10° and 20° to the edge. Given the 10° step size of the polar angle, the polar patterns from peak X 1 and peak X 4 have effectively almost the same elliptical major and minor axis. Since the biexciton emission is through the biexciton-exciton cascade process, the polarization properties of the biexciton emitted light is controlled by the single exciton emission process and thus biexciton emission should show the same polar pattern as the single neutral exciton. The observed same polar emission pattern from peak X 1 and peak X 4 confirms the preceding assignment of these two peaks as single neutral exciton and biexciton transitions based on the power dependent PL studies. Figure 3.25 Polar plot of the polarization dependent PL peak intensity (black dot) of (a) peak X 1 and (b) peak X 4 collected at 77K with non-resonant excitation (640nm, 80MHz, P=10.5nW, power density 0.87W/cm 2 ) and spectral resolution of 0.2nm. The black line represents the fit of the measured data using Eq. 3.50 with 0.125   for peak X 1 and 0.108   for peak X 4 . The base shape of the QD (red line) is captured here for easy reference to the QD structure. The QD edge [3 -1 0] marked 184 as red dotted line is 22±4° (angle α) with respect to the [1 -1 0] crystallographic direction. The emission pattern of the MTSQD single neutral exciton transitions collected in the x-y plane of the instrument geometry (Fig. 3.23) can be affected by two effects: (1) the symmetry of the confinement potential that affects the Bloch nature of the QD hole states and, (2) the geometrical anisotropy of the mesa top holding the MTSQD which affects the far field radiation pattern of the QD excitonic transition dipole. For a QD with D 2d rotational symmetry around the growth direction z ([001] direction), the ground hole state has only heavy hole Bloch nature. Under such condition, the two neutral bright exciton states |1, 1  and |1, 1  with total angular momentum J=1, following the notation in section §3.2, are degenerate in energy and their emissions are circularly polarized left (σ + ) or right (σ - ) in the QD x-y plane. The intensity of light from each bright exciton state () I  is 22 ( ) (sin cos ) I     where  represents the polar angle with respect to 0° ([1 -1 0] direction). When the QD symmetry has been reduced to C 2v such as with the presence of built in piezoelectric field and / or shape asymmetry, the electron-hole exchange interaction lifts the degeneracy between the two neutral bright exciton states. The energy separation between these two states is the so called fine structure splitting [3.28]. The bright eigenstates are now the two linear combinations 185 1 1 | (|1, 1 |1, 1 ) 2 hh        and 2 1 | (|1, 1 |1, 1 ) 2 hh        , the emission of which is linearly polarized along the crystallographic [110] and [1 -1 0] directions (details in section §3.2) with 2 1 ( ) sin hh I and 2 2 ( ) cos hh I polar dependence respectively. Collection of emission from these two states can lead to circular or elliptical polar dependence controlled by the relative ratio of the emission intensity of these states that are determined by the ratio of the oscillator strength, |d| 2 , with d being the transition dipole defined in Eq. 3.27. Given the reported fine structure splittings of 10-100 μeV in InGaAs SAQDs [3.28, 3.63] and the radiative life time of ~1ns of excitons [3.30], the photons from these two states do not have any phase correlation and hence cannot interfere. Therefore emission intensity from the two states can be represented as a direct summation of emission from the two orthogonal polarized states, resulting in either circular or elliptical emission polar pattern. It has been found that under non-resonant excitation, the emission intensity of both states is the same regardless of the difference in the oscillator strength [3.63]. Thus, the collection of individual 1 | hh  and 2 | hh  give rise to linear polar pattern and a circularly polarized pattern can be obtained from collecting both these two states under non-resonant excitation. When the QD symmetry is further lowered to below C 2v due to e.g. the in- plane shape asymmetry and anisotropic strain effects, the heavy hole (HH) and light hole (LH) state can mix, leading to the hole states 3 3 3 1 | | , | , 2 2 2 2 h E           186 captured in Eq. 3.37. The non-degenerate neutral exciton eigenstates become linear combinations of the elliptically polarized bright exciton states [3.64], 1 | (|1, 1 |1, 1 ) 2 X        , 1 | (|1, 1 |1, 1 ) 2 Y        (Eq. 3.46) where, 22 3 3 1 1 3 1 1 1 |1, 1 1 | , | , | , | , 2 2 2 2 2 2 2 2 i e                 (Eq. 3.47) Here  and  represent, respectively, the amplitude and phase of the mixing. The heavy hole, light hole and electron state follow the notation in section §3.2. Depending on the light hole ratio  and the angle  , the two eigenstates | X  and |Y  can be tilted with respect to the crystallographic [110] and [1 -1 0] direction, without being perpendicular to each other anymore. Under non-resonant excitation , assuming that the two states are equally populated, independent of the laser polarization and their relative oscillator strength, the normalized emission intensity of these two states are, 22 ( ) [ 1 sin sin( 2 )] 3 X I           (Eq. 3.48) 22 ( ) [ 1 cos cos( 2 )] 3 Y I           (Eq. 3.49) The total intensity from these two states is, 2 2 21 ( ) { 1 2 cos[2( )]} 33 tot I            (Eq. 3.50) 187 Thus the polar pattern, i.e. circular, elliptical or linear, can probe the confinement potential symmetry.  Impact of Nanomesa Size and Shape Enclosing the SQD The above discussion of the polar pattern of neutral excitonic transitions in QDs of different symmetry is based on the assumption that the transition dipole is in an infinite homogenous material. However, in our MTSQD case, as in the case of nanopillars and nanowires, the QD excitonic state transition dipole is imbedded in the small volume of the InGaAs SQD material that, while residing on the top of mesa, is buried in a volume of another material defined by specific shape of boundaries between different materials, i.e. InGaAs/GaAs and GaAs/air boundaries. To examine the impact of the geometrical anisotropy of the GaAs mesa on the polar emission pattern of the dipole transition, we have carried out finite element method based simulations of the polarization dependence of the emission pattern for a point dipole representing the QD transition dipole embedded in a mesa, as shown in Fig. 3.26 (a), with the dipole placed ~56nm down from the very top of the mesa and dipole transition energy ~1.333eV (930nm). The {103} and {101} type planes (Fig. 3.26 (b)) are reconstructed with their dimension and orientation consistent with the SEM images of the mesa (Fig. 3.24, details discussed in chapter 2). The dipole depth and energy are chosen to mimic the location and emission energy of the MTSQD. This analysis is courtesy of Mr Swarnabha Chattaraj in our group. 188 Figure 3.26. Schematic of the GaAs nanomesa (solid black lines) enclosing the quantum dot (red) used in finite element based simulations. Panels (a) and (b) show, respectively, the 60° tilted and top view of the nanomesa. Panel (c) captures the geometry of the calculation. The blue circle, ~3 μm away from the point dipole, represents the integration surface for the calculation of the overall Poynting vector. The red circle represents the instrument objective lens collection cone. The Poynting vector with electric field along the polar angle  (with respect to the [1-10] direction in the x-y plane) integrated over the objective collection area (red area in Fig. 3.26(c) defined by the collection cone angle of the microscope in the micro-PL setup) and integrated over the entire sphere (blue curves in Fig.3.26(c)) are calculated for a dipole in the x-y plane with equal magnitude of P x and P y component but with (1) π/2 phase difference (resulting in a TM 11 type radiation) and (2) –π/2 phase difference (resulting in a TM 1-1 type radiation) representing, respectively, the heavy hole dominated exciton transition states |1, 1  and |1, 1  in QDs of D 2d symmetry. The ratio of these two calculated results, i.e. the collection efficiency, as a 189 function of the polar angle  , for these two cases is shown below in Fig. 3.27 (a) and (b), respectively. The geometric anisotropy of the GaAs mesa, as expected, modifies the far field emission pattern and results in the seen elliptical polar pattern from circularly polarized dipoles. The elliptical polar pattern has major axis along 315° (Fig. 3.27(a)) and 45° (Fig. 3.27(b)), respectively, for the circularly polarized dipole case (1) and (2). Under the scenario that emission from the above mentioned two circularly polarized dipoles is collected simultaneously, the calculated polar pattern is found to be circular as shown in Fig. 3.27 (c). The calculated result in Fig. 3.27 (c) mimics collection of the emission from the two degenerate excitons from a QD possessing D 2d symmetry. The ellipticity of the polar pattern of a circular polarized dipole (Fig. 3.27 (a) and (b)) indicates non-equal collection efficiency of the light with electric field along X and Y directions (Fig. 3.26) due to the presence of the mesa. Thus in assessing the nature of QD confinement potential symmetry, this non- equal collection efficiency needs to be folded into the analysis of the polarization data from MTSQDs. 190 Figure. 3.27 Simulated results showing the polar plot of the polarization dependent collection efficiency of the emitted photons into the collection cone of the optical microscope for a dipole in the x-y plane with equal magnitude of P x and P y component but with (a) π/2 phase difference (resulting in a TM 11 type radiation) and (b) –π/2 phase difference (resulting in a TM 1-1 type radiation) mimicking respectively the exciton state |1, 1  and |1, 1  . Panel (c) shows the summation of the two transitions in (a) and (b). Taking into account the effect of QD symmetry and the effect of the size and shape of the GaAs mesa holding the MTSQD, the measured elliptical polar pattern of the emission from single neutral exciton decay suggests that the MTSQD confinement potential symmetry is below D 2d . For QD with D 2d symmetry, the collection of two degenerate circularly polarized light from heavy hole states results in a circular polar pattern from MTSQD on the GaAs mesa mimicked and shown in Fig. 3.27 (c). The deviation from circular pattern, i.e. the elliptical polar emission 191 pattern from MTSQD single neutral exciton transition, indicates the lower QD confinement symmetry. Given the 10-100 μeV [3.28, 3.64] reported fine structure splitting of the bright excitons for the InGaAs/GaAs SAQDs of sizes and shape similar to these MTSQDs, we are collecting both of the bright exciton transitions under the employed measurement condition with spectral collection window of 0.2nm (~300μeV). The emission from two non-degenerate linearly polarized bright excitons from a QD with C 2v symmetry should have a circular polar pattern as discussed before. However, the non-equal collection efficiency of the light with electric field along X and Y directions (Fig. 3.27 (a) and (b)) due to the GaAs nanomesa turns the circular pattern to elliptical. Similarly for a QD with symmetry lower than C 2v , the mixing of the HH and LH results in elliptical polar emission pattern for each individual non-degenerate exciton state. With the built-in non-equal collection efficiency of the light with electric field along X and Y directions, the combined emission from these two non-degenerate states also result in the elliptical polar pattern but probably with different major axis elliptical angle and degree of ellipticity compared with the QD of C 2v symmetry. Due to the lack of direct information on the vertical depth of the MTSQDs with respect to the mesa top and the omission of the InGaAs material into the finite element based simulations, no direct comparison of the elliptical major axis and ellipticity of the simulated result (Fig. 3.27) and the measured data (Fig. 3.25(a)) can be made. Hence, one cannot distinguish whether the QD is of C 2v symmetry or with symmetry lower than C 2v based on just the elliptical observed polar pattern of the single neutral exciton. 192 In the power dependent PL data from MTSQD (3,5) (Fig. 3.22 (a)), there is a weak secondary emission peak, labeled as X 1 * , at 928.6nm that is ~860ueV higher in energy compared to the peak X 1 . The presence of such a transition can also be seen in the polarization dependent PL data. Fig. 3.28 (a) shows PL collected from MTSQD (3,5) with the polarizer aligned along the 30 0 and 300 0 direction. The emission from peak X 1 * is more prominent when the polarizer is at 30 0 direction, opposite to that of the single neutral exciton peak X 1 . The PL polar dependence of the X 1 * peak, shown below in Fig. 3.28 (b), is of elliptical shape with the ratio of the amount of light with polarizer along the major and minor axis being ~1.3, similar to peak X 1 and X 4 , but with the major axis along 60 ° , rotated 120° compared to the single neutral exciton peak X 1 (Fig. 3.25 (a)). The near linear PL power law dependence, I~P 1.1 , of peak X 1 * and the large energy separation between it and the single neutral exciton peak X 1 compared with the 10 to 100 μeV [3.28, 3.64] reported fine structure splitting of the bright excitons for the InGaAs/GaAs SAQDs together suggest that it may come from the “dark” exciton of the MTSQD ground state. The “dark” exciton can acquire a “bright” component and emit photons due to the mixing of HH and LH states when QD has confinement potential of symmetry lower than C 2v . Such “dark” exciton state has low PL emission intensity due to low transition dipole moment. The mixing of HH and LH, in particular the z direction component of the dipole of the LH, plays an important role in controlling the polar pattern of its transition. The observed low PL emission intensity and the elliptical polar dependence of X 1 * transition are characteristics consistent with our tentative 193 assessment of it being from “dark” exciton complex of the MTSQD ground state exciton. Figure 3.28 (a) PL of MTSQD (3,5) with polarizer aligned at 30° (black line) and 300° (red line) with respect to the [1-10] direction in the x-y plane and (b) the Polar plot of the polarization dependent PL peak intensity (black dot) of peak X 1 * collected at 77K with non-resonant excitation (640nm, 80MHz, P=10.5nW, power density 0.87W/cm 2 ) and spectral resolution of 0.2nm. The black line represents the fit of the measured data using Eq. 3.50 with 0.11   . Thus the polarization data of peak X 1 * and peak X 1 suggest that the MTSQD studied might have QD confinement potential with symmetry lower than C 2v . Due to the limited instrument resolution and limited simulation results on the effect of GaAs mesa geometrical anisotropy on the far field emission pattern of QD dipole, it is not clear what is the relative strength of the role of the confinement potential and mesa geometry anisotropy on the observed ellipticity and direction of elliptical major axis of the polar emission pattern of MTSQD neutral exciton. As discussed in section §3.2, the polar pattern of different emission peaks can be used as a tool to assess the differing excitonic nature of the transitions. The polar 194 pattern of emission from peak X 4 having the same polar pattern as single neutral bright exciton peak X 1 confirms that it is from the biexciton (XX) decay. From previously discussed power dependent PL studies on MTSQD (3,5), the emission peak X 2 at 927.4nm and the peak X 3 at 931.4nm are inferred to be coming from X + and X - transitions, respectively. To assess this attribution, polarization dependent PL data have been obtained from these two peaks. Figure 3.29 shows the polar plot of the emission intensity (peak intensity) of (a) peak X 2 and (b) peak X 3 . The peak X 2 shows a near circular polar pattern with the major axis at ~ 300° (Fig.3.29 (a)) and the minor axis at 30°, same as peak X 1 . The ratio of the light intensity measured with the polarizer along the major and minor axis is ~1.15. The obtained  that defines the ellipticity of the pattern is 0.08   smaller than that for the behavior of the neutral and biexciton transitions ( 1.1 1.25  , in Fig. 3.25). The peak X 3 , however, shows a polar pattern very close to circular pattern (Fig. 3.29 (b)). The obtained  of this emission is 0.04   , much smaller than that for peaks X 1 , X 2 and X 4 . The near circular polar pattern from peaks X 2 and X 3 is consistent with the expected behavior for charged excitons. 195 Figure 3.29 Polar plot of the polarization dependent PL peak intensity (black dot) of (a) peak X 2 and (b) peak X 3 collected at 77K with non-resonant excitation (640nm, 80MHz, P=10.5nW, power density 0.87W/cm 2 ) and spectral resolution of 0.2nm. The black line represents the fit of the measured data using Eq. 3.50 with 0.08   and 0.04   for peak X 2 and X 3 respectively. The base shape of the QD (red line) is capture here for easy reference to the QD structure. The QD edge [3 -1 0] marked as red dotted line is 22±4° (angle α) with respect to the [1 -1 0] crystallographic direction. The charged excitons, owing to the spin singlet nature of the electron pair, are two fold degenerate. The emission from X + (X - ) in QDs with D 2d and C 2v symmetry and holes of HH Bloch nature is controlled by the transition 13 || 22        ( 31 || 22         ), following the notation of |J z > with J z representing the projection of the total angular momentum along z direction, resulting in the circularly polarized emission. The polar pattern from such transition is similar to that seen in Fig. 3.27 (c). When the QD symmetry has been lowered further below C 2v , heavy hole and light hole mixing occurs. The light hole admixture is associated with 196 circular polarization 11 || 22       , just opposite to heavy hole 3 | 2  so that each of the HH-LH mixed transition results in the charged exciton being in a state of |1, 1  (Eq. 3.47) and emitting elliptically polarized light. The collection of these two elliptically polarized light captured in Eq. 3. 47 should also result in a circular polar pattern. The observed circular pattern of peak X 3 is thus consistent with the assessment of it being a negatively charged trion, X - , and the assessment of QD symmetry below C 2v . The peak X 4 is near circularly polarized with a small deviation from the expected circular polar pattern. The deviation may come from the collection of background light due to the large 0.2nm spectral acceptance window and the possible acceptor near QD affecting the QD charge distribution. To summarize, from the polarization dependent PL studies on the multiple emission peaks from MTSQD (3,5), we infer that the MTSQD’s confinement potential symmetry is likely below C 2v with the participating QD hole state having mixed HH and LH Bloch character. The polarization pattern of emission from peaks X 2 , X 3 and X 4 is consistent with the assignment, respectively, of positively charged exciton X + , negatively charged exciton X - , and biexction XX. The 0.2nm measurement spectral resolution prevents us from resolving the fine structure splitting of the neutral exciton transition to clearly identify the nature of the secondary peak X 1 * and the degree of HH and LH mixing in the QDs. To further resolve the nature of the peaks and symmetry of the QD confinement potential, polarization dependent PL emission from different excitonic transitions with higher 197 resolution of < 20μeV capable of resolving the fine structure splitting of neutral bright exciton at liquid helium temperature is needed. Additionally, the effect of GaAs mesa on the far field polar pattern needs to be more carefully examined through simulations accounting for the position and volume of the InGaAs MTSQD on mesa top. The above mentioned measurements and simulations will enable the assessment of the relative strength of the confinement potential symmetry and GaAs mesa geometric anisotropy on the ellipticity and the direction of elliptical major axis of the emission pattern from MTSQD neutral exciton state to fully understand the nature of QD emission and confinement potential symmetry. §3.6 Dynamics of Excitonic Decay in MTSQDs: Time-Resolved PL Studies So far we have focused on the time-integrated PL spectra and assessed the origin of the observed multiple transition lines based on their power dependence and polarized emission pattern behavior. While the findings based on these studies provide information necessary to guide the studies of MTSQDs as single photon sources discussed in Sec.§4.3, the dynamics of exciton generation, thermalization, and decay control the dynamics of the nature of the emitted photons. As noted in Sec.§1.2, the exciton lifetime and emitted photon coherence times are critical characteristics defining the utility of the QD as a single photon source. It is thus important to examine the basic dynamics of exciton decay and in this subsection we discuss such studies. 198 Time-resolved PL (TRPL) data from MTSQDs are collected using the home built micro-PL setup discussed in section §3.3. The MTSQDs are excited with 640nm 80MHZ excitation laser or Ti:Sa mode-locked laser with 780nm (850nm) 76MHz excitation laser. The electrical signal representing the timing sequence of the optical excitation laser pulsed frequency is used as the “stop” signal in the multichannel timing electronics for TRPL measurement. The electrical signal from the APD detector representing the detection of emitted photon from MTSQDs is used as the “start” signal for the TRPL measurement. Due to finite timing resolution of the TRPL setup characterized by the instrument response function (IRF, shown in Fig. 3.9), the measured TRPL data is a convolution of the IRF and the actual measured TRPL intensity from the MTSQDs as shown below: 0 ( ) ( ) ( ) t I t IRF t g t t dt      (Eq. 3.51) where () It is the measured (convoluted) TRPL data and ) (t g is the actual measured TRPL response from MTSQDs. The deconvolution of the TRPL data is performed using a custom program (developed earlier by Dr. Siyuan Lu [3.22] of the group) based on the least square iterative reconvolution method [3.65, 3.66] and the commonly employed exponential form of the TRPL response ) (t g . For the undistorted TRPL response ) (t g with a known guessed functional form ) ... ; ( 1 n a a t G containing parameters n a a a ... , 2 1 to be determined, a set of fitting parameters are guessed and used to reconvolute ) ... , ; ( 2 1 n a a a t G with the IRF to obtain fitting 199 function 1 2 1 2 0 ( ; , ... ) ( ) ( ; , ... ) t nn y t a a a IRF t G t t a a a dt      . The fitting is done by adjusting the guessed n a a a ... , 2 1 parameter to minimize the weighted residue between the fitting function and the measured TRPL curve () ft Levenberg-Marquardt nonlinear optimization [3.67]. All the TRPL data obtained from MTSQDs shown in this section are deconvoluted and fitted using the above mentioned approach. §3.6.1 MTSQD Neutral Exciton PL Decay Dynamics To assess the decay time of QD neutral exciton, TRPL has been collected from MTSQD (3,5) with lowest excitation power (~4nW, 0.32W/cm 2 ) at a spectral resolution of 0.4nm. Figure 3.30 (a) and (c) show the measured TRPL data at 77K and 8K respectively from the neutral exciton transition (peak X 1 ). The TRPL response of the MTSQD neutral exciton is modeled using a three level system involving electron relaxation and exciton recombination [3.68] which leads to the following fitting function: 00 1 exp exp rd t t t t y C A tt                 (Eq. 3.52) in which r t and d t are the rise and decay times of the PL reflecting, respectively, the relaxation of electron to its ground level and the radiative recombination of the ground state exciton [3.68]. The red line in Fig. 3.30 (a) and (c) shows the reconvoluted fitted curve to the data using Eq. 3.52. The MTSQD (3,5) neutral exciton has a PL rise time of 0.2ns, limited by the instrument IRF whose rise time 200 (10%-90%) is ~150ps. The decay time of the transition at the lowest excitation power (~4nW, 0.32W/cm 2 ) from the fitting is found to be ~0.8ns at 77K as well as at 8K, comparable to that of typical InGaAs SAQDs [3.1, 3.4, 3.30]. With increasing excitation power, the QD decay time deviates from the observed 0.8ns to longer decay times as shown in Fig. 3.30 (b). This is due to carrier replenishment with increased number of carriers in the QD [3.69] that slows down the radiative decay process of the QD. Figure 3.30 TRPL of MTSQD (3,5) peak X 1 measured at (a) 77K and (c) 8K with spectral resolution of 0.4nm. Data shown in panel (a) are collected with 640n 80MHz excitation and excitation power P=4nW, power density 0.32W/cm 2 Data in panel (c) are collected with 780nm,76MHz excitation and excitation power P~12nW, power 201 density 1.28W/cm 2 . Panel (b) shows the PL decay time of peak X 1 measured at excitation powers from 23% to 215% of the saturation power of peak X 1 . The TRPL data obtained at the lowest excitation power represent the intrinsic decay time of the exciton in the QD ground state. The obtained intrinsic decay time d t represents the combined radiative and non-radiative decay rate of the QD shown below: 1 rad nrad d RR t  (Eq. 3.53) where rad R and nrad R are the radiative and non-radiative decay rate respectively. rad R depends on the dipole moment of the transition and does not change with temperature. The nrad R , on the other hand, depends strongly on the temperature with an activated form [3.69], exp( / ) nrad act B R E K T  with act E being the activation energy for the non-radiative decay channel. The observed same decay time at 77K and 8K indicates the radiative decay process dominates over the non-radiative decay, suggesting good quantum confinement of excitons in the MTSQD and good quantum efficiency. TRPL spectra have been collected from multiple MTSQDs in the 5 × 8 array and the extracted decay time of the X 1 peak of each is shown in Fig. 3.31. The data are collected with spectral resolution of 0.4nm and excitation power P~30% of the saturation power of each MTSQD. The examined MTSQDs show the average PL 202 decay time of 1.05 ± 0.26 ns, similar to those of typical InGaAs SAQDs [3.1, 3.4, 3.30]. The high measured PL intensity at 77.4 K at the low (nW) powers employed as discussed in section §3.4 together with the short (~1ns) PL decay lifetime of single excitonic transition in the MTSQDs indicate good quantum confinement of these excitons and that they can act as GHz light emitters using their neutral single exciton transition. 926 928 930 932 934 936 938 940 942 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Decay time (ns) Wavelength (nm) 77K (4,4) (3,5) (2,4) (3,8) Figure 3.31 The decay time of the X 1 peak obtained from the TRPL spectra of different MTSQDs studied at 77K with spectral resolution of 0.4nm and low excitation power P~30% of the saturation power of the transition of each MTSQD. The optical excitation used is the 640nm 80MHz excitation. The peak X 1 is identified with the neutral exciton (X 0 ) decay (see text below). §3.6.2 Dynamics of Excitonic Decays: Effect of Carrier Capture in MTSQD The detailed time-integrated PL data of the preceding subsections indicates that, in addition to the neutral single exciton decay (peak X 1 , hereafter denoted X 0 ), the MTSQD (3,5) likely has positively charged trion (peak X 2 , hereafter denoted X + ), negatively charged trion (peak X 3 , hereafter denoted X - ), and biexciton (peak X 4 , 203 hereafter denoted XX) emissions. To study the dynamics of the transitions underlying these different peaks from MTSQD (3,5), TRPL data have been collected at peaks X o , X + , X - , and XX at 77K with a spectral resolution of 0.2nm and above gap excitation (780nm, 76MHz) with excitation power P~30nW (2.4W/cm 2 , P/P sat =60%). Figure 3.32 below shows the measured TRPL data (black circles) from these four peaks. The red line in Fig. 3.32 shows the representative re-convoluted fitted curve of the data where Eq. 3.52 is used to represent the expected three-level form of TRPL signal from the QD. Figure 3.32 The measured TRPL spectra of (a) neutral exciton X 0 , (b) positively charged trion X + , (c) negatively charged trion X - and (d) biexciton XX from MTSQD (3,5) marked as open black circle collected at 77K with spectra resolution of 0.2nm 204 under optical excitation power P~30nW (2.4W/cm 2 , P/P sat =60% ). The red lines show the reconvoluted fitted curve of the data. All four peaks have PL rise time of 0.2ns, limited by the IRF whose rise time (10%-90%) is ~150ps. The neutral single exciton transition from peak X 0 has PL decay time of 1.39 ns, at the high power density of 2.4W/cm 2 , P/P sat =60%, consistent with the result obtained from the power dependent PL decay time in the preceding subsection (Fig.3.30(b)). The decay time of the biexciton (peak XX) is found to be 0.58ns from the fitting. At this higher power (needed to create sufficient biexcitons) the decay time of the single exciton peak X 0 is thus 2.4 times longer than that for the biexciton peak XX. Under the employed excitation power, the emission from peak XX just starts to rise. The occupancy probabilities of the exciton peak X 0 and biexction XX peaks are not the same. For a reliable comparison of the intrinsic lifetimes of the exciton and biexciton states the exciton lifetime of 0.8ns obtained at the lowest excitation power (Fig.3.30(a)) is better suited for comparison. In this case the intrinsic lifetime of exciton peak X 0 is ~1.4 times longer than that of the biexction peak XX. As biexcitons contain two excitons, they are observed with optical decay approximately twice as fast as excitons due to the existence of twice the number of radiative decay possibilities [3.4]. The observed ratio of the peak X 0 and XX decay times is consistent with theoretical expectation as well as the observation of 1.5 ratio for the neutral exciton to biexciton decays in InGaAs/GaAs SAQDs reported in the literature [3.70]. Therefore, for our MTSQDs, the observation of an exciton to biexciton decay lifetime ratio of ~1.5, together with the emission 205 power dependence behavior and emission polar pattern provides strong evidence that peak X 4 is from biexciton (XX) transition. The two other, probably charged trion emissions, have decay times ~1ns and 0.7ns, close to the observed decay time for the neutral exciton of the studied MTSQDs and consistent with the expected performance of charged excitons [3,4, 3.70].  Inter-Dependence of Emission from Multiple Types of Excitons: An Analysis The presence of the charged excitons in MTSQD indicates the capture of single carrier, electron or hole, into the QD in addition to the capture of electron-hole pair for excitation above the barrier band gap, as is the case for all the studies reported in the preceding. The description of the dynamical processes of exciton, trion and biexciton formation in the QD thus needs to be expanded beyond the simple model with only electron-hole pair capture discussed in section §3.5 to include capture of single carrier into the QD in order to understand the observed integrated PL power dependence for each excitonic transition. To describe our results quantitatively, we thus use a five level rate equation model and follow the analysis given in reference 3.71 and 3.72. The included five states, shown in Fig. 3.33, are the neutral SQD in its ground state (|0>), neutral exciton (|X>), double exciton (|XX>), negatively charged SQD (|e>), and negatively charged exciton (|X - >). The electron, hole, and exciton time-dependent capture rates are denoted by α(t), β(t) and x(t), respectively. The occupation probabilities of five QD states are represented 206 as a vector 0 ( ) [ , , , , ] c x cx xx n t p p p p p  , where i p is the probability of finding the QD in i th state at time t (Fig. 3.33). Figure 3.33. Scheme of energy levels and optical transitions considered in the model. The parameters α(t), β(t) and x(t) are the time dependent capture rates of electron, hole and exciton respectively. Constants x t , cx t and xx t are the decay time of respective states. The probability of finding the QD in i th state at time t is i p . Due to the low emission intensity of the positive trion X + (peak X 2 ) from MTSQD (3,5) in the PL spectrum shown in Fig. 3.20 (a), we neglect this charge state and only take into account the negative charged trion X - (peak X 3 , occupation probability p cx ). The possible transitions related to the capture of electron or an electron-hole pair (neutral exciton) as well as radiative recombinations included in the rate equation model are indicated in Fig. 3.33. The capture rate of single electron (hole) is represented as α (β). The capture rate of an exciton is labeled as x. The exciton capture rate for QD empty state (occupation probability p 0 ) is assumed to be same for QD single exciton state (occupation probability p x ). In the employed rate 207 equation model, fine structure of the excitonic states is neglected. Under these assumptions, for a system having the transition rates between different levels depicted in Fig. 3.33, the dynamics of the occupation vector is controlled by [3.71,3.72], () ( ) ( ) dn t R t n t dt  (Eq. 3.54) where () Rt is the transition rate matrix describing radiative decays, carrier caption and excitation processes: 1 ( ) ( ) 0 0 0 1 ( ) ( ) ( ) 0 0 11 ( ) ( ) ( ) ( ) 0 () 1 0 ( ) ( ) ( ) 0 1 0 0 ( ) ( ) x cx x xx cx xx t x t t t t x t t x t t t x t Rt tt x t t t t x t t t                             (Eq. 3.55) The excition, charged exciton, and biexciton radiative lifetimes are obtained from the TRPL result shown in Fig. 3.32 as fixed parameters with 1.1 x t ns  (average value of QD decay time before saturation), 1.0 , cx t ns  and 0.58 xx t ns  . The only free parameters in the model are α, β and x that are the electron, hole and exciton capture rates, respectively. The characteristic time scales of the electron, hole and exciton trapping are larger than the laser pulse width because of the relaxation process in the 208 GaAs barrier. We assume exponential time dependence of the three capture rates with a characteristic time 0.2 e t ns  , same as the TRPL rise time, in the following: 0 ( ) / exp( / ), ee t P t t t  0 ( ) / exp( / ) ee t P t t t  , 0 ( ) / exp( / ) ee x t x P t t t  (Eq. 3.56) where P is the optical excitation power. The rate equation in Eq.3.54 is solved numerically to compute () nt and the PL intensity ( ) / , / , / x x cx cx xx xx I t p t p t p t  for exciton, negatively charged trion and biexciton respectively. The steady state () It is obtained for t that satisfies ( ) ( ) rep n t n t T  where rep T is the time interval between excitation pulses. An arbitrary coefficient between the computed and the measured count rate for each excitonic transition is used to account for the photon detection efficiency. By fitting the observed PL power dependence data of the exciton X 0 (peak X 1 ), negatively charged exciton X - (peak X 3 ) and the biexciton transition XX (peak X 4 ) from MTSQD (3,5) shown in Fig. 3.20(b), we obtain the fitted parameters as 0 0.2   , 0 0.6   and 0 0.15 x  for the electron, hole and exciton capture that closely reproduce the observed PL intensity ratios between exciton, negatively charged exciton and biexciton as well as the observed PL power dependence behavior of the three transitions. From the rate equation model, the emission intensity of the neutral exciton transition follows I~P 0.91 while the negatively charged trion and biexciton follow I~P 1.56 and I~P 2.24 , respectively, as shown in Fig. 3.34 below. The capture of single carrier and exciton into the QD leads to the deviation 209 fromthe linear power and quadratic power dependence for the single exciton and biexciton decay, respectively. The model analysis uncovers the important finding that the capture of single electron is as probable as the capture of a single exciton in these MTSQD structures. 0.2 0.4 0.6 0.8 1 2 4 6 8 100 100 500 1000 5000 10000 50000 100000 X 0 : 929.2nm X - : 931.4nm XX: 933.4nm Integrated Intensity (c/sec) Power Density (W/cm 2 ) X 0 : I~P 0.91 X - : I~P 1.56 XX: I~P 2.24   =0.2    =0.6  x 0 =0.15 Figure 3.34. Shows the five level rate equation model based fitting (solid lines) to the measured data (same as in Fig.3.20(b)) of neutral exciton X 0 decay peak X 1 (black dots), negatively charged exaction X - decay peak X 3 (red dots) and biexciton XX decay peak X 4 (magenta dots) as a function of power. The near equal single carrier and exciton capture rates lead to deviation from the linear power law for single neutral and charged exciton decays. Finally, we note that the short exciton decay times of ~1ns, comparable with the self-assembled or nanowire quantum dots, make the MTSQDs suitable for acting as high frequency (~GHz) light emitters, especially when incorporated in a resonant cavity as discussed in chapter.5. 210 §3.7 Photoluminescence Excitation Spectroscopy: Electronic Structure of MTSQDs The PL and TRPL studies of neutral exciton, trions, and biexcitons reported in the preceding subsections involve only the QD ground manifold of electronic states of the MTSQD. In this subsection we present a study that probes the presence and energy of excited electron and hole states of the MTSQDs by scanning the excitation energy across the excited states (but remaining below the barrier bandgap) while detecting the luminescence from lowest energy transition, as schematically illustrated in Fig.3.35. In this approach, known as photoluminescence excitation (PLE) spectroscopy, the intensity of the detected luminescence mimics the density of states of the higher bound states in the QD electronic structure [3.73]. Figure 3.35. Schematic showing the PLE process in QDs with tunable excitation across QD excited electron and hole states and detection at QD ground state transition. 211 Unlike the relaxation to ground state and subsequent radiative decay in bulk semiconductors, the separation between the discrete energy levels characteristic of quantum dots (QDs) can impose limitations on energy conserving processes available to allow efficient relaxation following above barrier gap excitation. This has come to be known as the “phonon bottleneck” [3.68, 3.74, 3.75] as emission of multiple optical phonons, when possible, provides the most efficient process for energy matching and thus relaxation between discrete states. Thus QDs with photon bottleneck will also have long carrier relaxation times: in the many nanoseconds to microsecond regime. From the PL studies on the MTSQDs, the observed high PL yield at low optical excitation power and the fast carrier decay time (sub-ns) indicates that carriers can relax fast to the ground state without experiencing a phonon bottle neck effect. The PLE spectra collected from MTSQDs with optical pumping power below the power for state filling (hereafter referred to as “low” power) thus mimics the QD electronic density of states. The PLE spectra of the MTSQDs discussed in this section are collected with spectral resolution of 0.2nm with CW excitation from 850-930nm (Ti:Sa CW laser, spectraphysics 3900) which is below the GaAs barrier bandgap. The wavelength of the excitation light from the Ti-Sa CW laser is tuned from 850nm to the QD exciton emission wavelength with stepping size of 0.5nm. The excitation power is 5μW, ~30% of the saturation power of QD neutral exciton for below barrier bandgap excitation. (Recall that the corresponding 30% of saturation power for the above barrier bandgap excitation is 5nW). Figure 3.36(a) shows the collected PLE spectrum of 212 MTSQD (3,5) at 77K with the detection set at the neutral single exciton emission at 929.2nm (peak X 1 ), plotted as a function of the energy difference between the excitation and detected photon energy. Figure 3.36 PLE data from MTSQD (3,5) collected at 77K with a spectral resolution of 0.2nm and plotted as a function of the energy difference between excitation and detected photon energy with detection set at two different wavelengths within the 213 neutral exciton X 0 PL emission (peak X 1 of Fig.3.20): (a) at 929.2nm, the peak value of peak X 1 and (b) at 928.6nm, the higher energy end of peak X 1 . The excitation power is 5μW, ~30% of the saturation power for QD neutral exciton for below barrier bandgap excitation. Five prominent peaks are observed (labeled as P 0 through P 4 ) at position 8.9meV (P 0 ), 15.5meV (P 1 ), 25.8meV (P 2 ), 33.3meV (P 3 ) and 39.4meV (P 4 ). In PLE, peaks arising due to the variation in the density of states of the electron and hole excited states will shift with shift in the detection energy within the ground state emission line. For example, with the detection at the higher energy end of the density states of the ground state emission line, the peaks involving electron and hole ground states will correspondingly shift towards the smaller energy. Peaks arising from phonon inelastic relaxation, however, do not shift with shift in detection energy. To identify the origin of the observed peaks and distinguish the electron and hole related transitions from phonon assisted inelastic relaxation [3.77,3.78], PLE data were collected also for detection at a shorter wavelength of 928.6nm at the higher energy (~0.86meV) side of the neutral single exciton X 0 (peak X 1 ) and are shown in Fig.3.36(b). Five prominent peaks (labeled as P 0 through P 4 ) are also observed in the collected spectrum (Fig. 3.36 (b)). Positions of P 0 , P 1 , and P 4 shift towards smaller values by 0.6 to 0.8meV with the detection energy ~0.86meV higher than peak X 1 , consistent with the presence and involvement of excited hole (P 0 , P 1 ) and electron (P 4 ) states. The PLE data suggests that P 4 is the electron first excited state, ~40meV higher than the electron ground state and P 0 is the hole first excited state, ~10meV higher than hole ground state. The position of P 2 and P 3 (Fig. 3.36 (a) and (b)) are close to the expected InAs and GaAs LO phonon energy [3.76]. These 214 two peaks are also found to be independent of the detection energy (Fig. 3.36 (b) and thus are consistent with the expected behavior of InAs and GaAs optical phonon- assisted emission. Besides PLE spectroscopy, access to the higher energy bound electron states of the QDs can also be provided by examining PL at high power densities, readily for above barrier bandgap excitation. When the QD ground state exciton generation rate is made higher than the QD ground state exciton recombination times (~1ns), excitons begin to populate higher excited states and start to recombine, generating luminescence at excited state transition energies. At the typical high excitation powers at which luminescence from higher energy states is observed in the most well-studied quantum dots, the 3D island self-assembled quantum dots, many-body effects also come into play and the energies of the observed peaks get renormalized [3.30]. The observed peak positions are thus not as faithful to single particle derived transitions as the peaks observed in PLE at low excitation powers. Nevertheless, to gain a sense of the correspondence of the peaks in the PLE data and their high power density PL behavior we collected the PL data up to ~110W/cm 2 from MTSQD (3,5) at 77K and are shown in Fig. 3.37(a) below. With increasing excitation power and the increased populating of the QD ground state, an additional peak emerges at 895.6nm. This is ~51.3meV higher energy compared to the MTSQD ground state transition (1.3345eV), and is close to the sum of the first excited electron state (`40meV) and first hole state (10meV) energies noted above. Thus this peak may be coming from transition between the first excited states of the electron and hole. The 215 energy separation between this transition and the QD ground state transition is found to be consistent with the observed energy separation of the ground and first excited electron and hole states in the PLE data in Fig. 3.36 (a). Figure 3.37 (a) Power dependent PL data on MTSQD (3,5) collected at 77K with spectral resolution of 1nm and excitation power from 128nW to 1341.3nW (780nm, 216 CW excitation) (b) Temperature dependence of this MTSQD’s integrated PL intensity from 77.4 K to 150 K. The fit (black line) to the data reveals an exponential dependence on inverse temperature with an activation energy of 40±2 meV representing carrier (electron) escape from the first excited electron state of the QD. The ascertainment of the QD electronic structure (first excited electron and hole sates) from PLE and high power PL study is also supported by the temperature dependent integrated PL intensity (I) behavior of the MTSQD ground state exciton X 0 (peak X 1 ). The integrated PL intensity (I) plotted in Fig. 3.37 (b) in the range 77.4 K to 150 K shows an exponential dependence on the inverse of temperature, I~exp(E act /k B T), with an E act of 40±2 meV. This reveals a thermally activated electron escape through the first excited state about 40meV above the ground state providing a pathway contributing to the non-radiative decay. The matching of the activation energy E act and the energy separation between the ground and excited electron state P 4 ~40meV seen in PLE in Fig. 3.36 (a) substantiates the assignment of the first excited electron state at ~40meV in the MTSQD electronic structure. From the above discussed PLE data at 77K and the PL power and temperature dependent data, the MTSQD (3,5) can be reasonably inferred to have the first excited electron level ~40meV and hole level ~10meV from the ground electron and hole state, respectively. With increased temperature, the electron can escape from the QD through the first excited electron state. The peak P 2 and P 3 observed in PLE that do not change their position with detection energy are found to be close to the InAs and GaAs optical-phonon energies, suggesting phonon assisted relaxation in the QDs. 217  Temperature dependence of PLE The thermally activated carrier escape from QD electron and hole levels as well as the phonon assisted relaxation can magnify themselves in the temperature dependent PLE data as well. Fig. 3.38 below shows the PLE spectrum collected from MTSQD (3,5) at 90K with detection at neutral exciton X 0 peak X 1 and spectral window of 0.2nm. The five dominate peaks at ~11meV (P 0 ), ~16.2meV (P 1 ), ~25.8meV (P 2 ) and ~33.3meV (P 3 ) and ~39.8meV(P 4 ) are present. The positions of peak P 2 (~25.8meV) and P 3 (~33.3meV) do not change with temperature, confirming their origin as phonon-assisted transitions. The decreasing excitation efficiency and a slight blue shift for peaks P 0 , P 1 and P 4 at higher temperature compared to the 77K data is attributed to thermal escape of carriers from the excited QD states [3.79], confirming the assignment of P 0 , P 1 and P 4 to QD excited hole and electron states. Figure 3.38 The PLE data from MTSQD (3,5) peak X 1 collected at 90K with spectral resolution of 0.2nm as a function of the energy difference between excitation and detected photon energy at peak X 1. 218 To address the similarity of the electronic structure of different MTSQDs in the array, we studied the temperature dependent PL behavior of the neutral single exciton transition peak X 1 at 926.2nm (Fig. 3.15 (c), the middle panel) from MTSQD (2,4). The activation energy obtained in the temperature dependent studies reflects the energy separation between the ground and excited electron state as demonstrated from MTSQD (3,5). Similar to the behavior of MTSQD (3,5), the integrated PL intensity (I) of the MTSQD (2,4) ground state neutral exciton transition in the range of 77.4 K to 120 K (Fig. 3.39) shows an exponential dependence on the inverse of temperature with E act of 40.6±2 meV. The observed activation energy from MTSQD (2,4) suggests that the energy difference between the electron first excited and ground state is probably ~40meV, similar to that from MTSQD (3,5). This observation indicates similar electronic structures amongst the synthesized MTSQDs, a consequence of the growth controlled MTSQD size and shape, hence, the confinement potential. 6 7 8 9 10 11 12 13 14 15 1000 1000 2000 4000 6000 8000 10000 20000 T (K) Integrated intensity (c/sec) 1000/T (1/K) 160 140 120 100 80 MTSQD (2,4) Peak X 1 : 926.2nm Figure 3.39.Temperature dependence of MTSQD (2,4)’s ground state neutral exciton (peak X 1 at 926.2 in Fig. 3.14(a)) integrated PL intensity from 77.4 K to 120 K. The 219 fit (black line) to the data reveals an exponential dependence on inverse temperature with an activation energy of 40.5±2 meV representing carrier escape from the QD.  The deduced electronic level structure of the MTSQDs From the above discussed combined systematic PL, TRPL, and PLE studies, the deduced probable electronic structure of the synthesized MTSQDs is captured in Figure 3.40 below. The first excited electron (hole) state is ~40meV (~10meV) from the ground electron (hole) state. The large energy separations in the electron and hole state manifolds indicates good quantum confinement in these MTSQDs, consistent with the observed fast (~1ns) radiative decay time and high emission yield. The observed electronic structure of MTSQDs captured in Fig. 3.40 and the emission polar pattern of the MTSQD ground Figure 3.40. The schematic of the MTSQD geometry (from Chapter 2) and the schematic of QD electronic structure based on the PL/PLE spectroscopies results for the synthesized MTSQDs. 220 state (discussed in section §3.5.3) are a direct consequence of the QD shape, and the piezoelectric and strain fields in the QDs. Reported calculations of the InGaAs/ GaAs QDs with truncated pyramidal shape have been dominantly based on square base pyramidal QDs with edges along <100> direction and steep {101} type side facets, quite different from our rhombus base and shallow sidewall MTSQDs [3.19, 3.80, 3.81]. This limits the theoretical guidance that can be obtained from the existing literature. Besides the shape difference, the MTSQDs has smaller strain field (discussed in section §3.4) and experiences smaller piezoelectric potential due to the smaller side facets and edges compared with the typically calculated truncated pyramidal SAQDs. The degree of strain and piezoelectric field affects the charge distribution inside QDs, especially the hole charge distributions [3.19] and thus oscillator strengths for different transitions [3.81]. Detailed calculations of the electronic structure of MTSQDs capturing its shape shown in Fig. 3.40 and strain are needed to accompany the optical studies for a more quantitative understanding of the observed optical properties and the deduced QD electronic and excitonic structure (excitonic complexes and exciton fine structures) and QD ground state electron and hole charge distribution to relate the observed polarized emission patterns to the QD electronic and physical structure. §3.8 Summary Our systematic PL, TRPL, and PLE studies of SESRE grown mesa-top single quantum dots (MTSQDs) in a spatially regular 5 × 8 array reveal that the synthesized 221 GaAs(001)/InGaAs/GaAs MTSQDs with growth controlled size and shape exhibit a spectral uniformity of ~8nm, significantly improved over the 50 to 60nm variation found in 3D island SAQD ensembles [3.30] used for a large part of randomly picked single quantum dot for reported single photon studies. The low (~nW) saturation power of the neutral exciton PL decay for the MTSDQs in the array indicates good material quality. The ~40meV higher first excited electron state energy indicates good quantum confinement in these MTSQDs. The ~1ns decay lifetime of the neutral exciton transition suggests that the exciton decay can be used to act as ~1GHz light emitter, a good starting for exploration of its single photon emission property. Detailed studies of the single photon emission characteristics of the MTSQD neutral exciton transition are covered in the following chapter 4. Besides the neutral exciton emission, singly charged exction and biexciton emissions from the MTSQDs are observed in excitation power dependent time- integrated PL and in time-resolved studies of the PL dynamics. Employing a five- level rate equation analysis we found that MTSQDs have comparable single carrier and exciton capture efficiencies. The single carrier capture affects the dynamical process of photon emission from different excitonic states. This, in turn, affects the steady state charge and exciton distribution in the QDs, as manifest in the departure from simple linear and quadratic power law dependence of PL intensity from the neutral exciton and biexciton decays, respectively. The dynamics of single carrier capture process can also affect the timing sequence of the emission from different excitonic transitions that play an important 222 role in determining the suitability of MTSQD to act as entangled photon source. The cross-correlation measurements of photons from the exciton and biexction decays (discussed in Chapter 5) need to be carried out to obtain information on the trapping rates of single carriers and excitons that can affect biexction-exction cascade process. The biexciton in the MTSQD has a binding energy of ~6meV and a decay lifetime of 0.58ns. From the polarization dependent PL studies on MTSQDs, we deduce that the MTSQDs most likely have confinement potential symmetry lower than C 2v with the hole states having mixed HH and LH character. Under such a confinement potential, the bright excitons are expected to have a fine structure splitting. Regrettably, due to the spectral resolution limitation of 0.2nm and high measurement temperature of 77K in the current studies, the fine structure of the bright exciton could not be resolved. To assess the suitability of using the MTSQD biexciton- exciton cascade process to produce entangled photon pairs, the assessment of the fine structure splitting of the MTSQD exciton states is further addressed in Chapter 4 with the aid of the 8K PL and photon correlation measurements. Chapter 3 References: [3.1] S. Buckley, K. Rivoire and J. Vučković, “Engineered quantum dot single photon sources”, Rep. Prog. Phys. 75, 126503 (2012). [3.2] G. Shan, Z. Yin, C. H. Shek and W. Huang, “Single photon sources with single semiconductor quantum dots”, Front. Phys. 9, 170 (2014). [3.3] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs”, Nature Photon. 8, 224 (2014). 223 [3.4] R. M. Thompson, R. M. Stevenson, A. J. Shields, I. Farrer, C. J. Lobo, D. A. Ritchie and M. L. Leadbeater, “Single-photon emission from exciton complexes in individual quantum dots”, Phys. Rev. B. 64, 201302 (R) (2001). [3.5] X. Liu, H. Kumano, H. Nakajima, S. Odashima, T. Asano, T. Kuroda and I. Suemune, “Two-photon interference and coherent control of single InAs quantum dot emission in an Ag-embedded structure”, J. Appl. Phys. 116, 043103 (2014). [3.6] L. Chen, “Optical Properties of InGaAs/GaAs quantum well structures: Application to light modulators and switches”, PhD. Dissertation, University of Southern California , 1993. [3.7] J. M. Luttinger, W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields”, Phys. Rev. 97, 869 (1955). [3.8] J. M. Luttinger, “Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory”, Phys. Rev. 102, 1030 (1956). [3.9] E. O. Kane “Band structure of Indium Antimonide“, J. Phys. Chem. Solids, 1, 249 (1957). [3.10] J. Singh, Physics of semiconductors and their heterostructures, McGraw-Hill, Inc, 1993. [3.11] G. E. Pikus and G. L. Bir, Sov. Phys.-Solid State 1, 136 (1959). [3.12] S. B. Ogale, A. Madhukar, F. Voillot, M. Thomsen, W. C. Tang, T. C. Lee, J. Y. Kim and P. Chen, “Atomistic nature of heterointerfaces in III-V semiconductor- based quantum-well structures and its consequences for photoluminescence behavior”, Phys. Rev. B. 36, 1662 (1987). [3.13] L. W. Wang, J. Kim and A. Zunger, “Electronic structures of [110]-faceted self-assembled pyramidal InAs/GaAs quantum dots”, Phys. Rev. B. 59, 5678 (1999). [3.14] A. J. Williamson, L. W. Wang and A. Zunger, “Theoretical interpretation of the experimental electronic structure of lens-shaped self-assembled InAs/GaAs quantum dots”, Phys. Rev. B. 62, 12963 (2000). [3.15] G. Bester, S. Nair and A. Zunger, “Pseudopotential calculation of the excitonic fine structure of million-atom self-assembled In 1-x Ga x As/GaAs quantum dots”, Phys. Rev. B. 67, 161306 (R) (2003). [3.16] D. Gershoni, C. H. Henry and G. A. Baraff, “Calculating the optical properties of multidimensional heterostructures: application to the modeling of quaternary quantum well lasers”, IEEE J. Quantum Electron. 29, 2433 (1993). 224 [3.17] P. Ender, A. Bärwolff, M. Woerner and D. Suisky, “ p k    theory of energy bands, wave functions, and optical selection rules in strained tetrahedral semiconductors”, Phys. Rev. B 51, 16695 (1995). [3.18] M. Grundmann, O. Stier and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure”, Phys. Rev. B. 52, 11969 (1995). [3.19] O. Stier, M. Grundmann and D. Bimberg, “Electronic and optical properties of strained quantum dots modeled by 8-band p k    theory”, Phys. Rev. B 59, 5688 (1999). [3.20] G. Bastard, J. A. Brum, “Electronic states in semiconductor heterostructures”, IEEE J. Quantum Electron. 22, 1625 (1986). [3.21] G. Bastard, “Superlattice band structure in the envelope-function approximation”, Phys. Rev. B. 24, 5693 (1981). [3.22] S. Lu, “Some studies of nanocrystal quantum dots on chemically functionalized substrates (semiconductors) for novel biological sensing”, PhD. Disseratation, University of Southern California , 2006. [3.23] Y. H. Huo, B. J. Witek, S. Kumar, J. R. Cardenas, J. X. Zhang, N. Akopian, R. Singh, E. Zallo, R. Grifone, D. Kriegner, R. Trotta, F. Ding, J. Stangl, V. Zwiller, G. Bester, A. Rastelli and O. G. Schmidt, “ A light-hole exciton in a quantum dot”, Nature Phys. 10, 46 (2014). [3.24] J. Hohansen, S. Stobbe, I. S. Nikolaev, T. Lund-Hansen, P. T. Kristensen, J. M. Hvam, W. L. Vos and P. Lodahl, “Size dependence of the wavefunction of self- assembled InAs quantum dos from time-resolved optical measurements”, Phys. Rev. B 77, 073303 (2008). [3.25] A. V. Koudinov, I. A. Akimov, Y. G. Kusrayev and F. Henneberger, “ Optical and magnetic anisotropies of the hole states in stranski-krastanov quantum dots”, Phys. Rev. B 70, 241305 (2004). [3.26] Y. Léger, L. Besombes, L. Maingault and H. Mariette, “ Valence-band mixing in neutral, charged, and Mn-doped self-assembled quantum dots”, Phys. Rev. B 76,045331 (2007). [3.27] D. N. Krizhanovskii, A. Ebbens, A. I. Tartakovskii, F. Pulizzi, T. Wright, M. S. Skolnick and M. Hopkinson, “Individual neutral and charged In x Ga 1- xAs –GaAs quantum dots with strong in-plane optical anisotropy”, Physical Review B 72,161312 (2005). 225 [3.28] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke and S. N. Walck, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots”, Phys. Rev. B 65, 195315 (2002). [3.29] A. Messiah, Quantum mechanics, Dover, New York, 1999. [3.30] R. Heitz, “Optical Properties of Self-Organized Quantum dots”, Chapter 10 in Nano-Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. [3.31] C. F. Klingshirn, Semiconductor optics, Chapter 15, Springer Berlin Heidelberg New York, 2007. [3.32] R. Heitz, A. Kalburge, Q. Xie, M. Grundmann, P. Chen, A. Hoffmann, A. Madhukar, D. Bimberg, “Excited states and energy relaxation in stacked InAs/GaAs quantum dots”, Phys. Rev. B 57, 9050 (1998). [3.33] H. Benisty, C. M. Sotomayortorres, C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems”, Phys. Rev. B 44, 10945 (1991). [3.34] J. Y. Marzin, J. M. Gérard, A. Izraël, D. Barrier and G. Bastard, “Photoluminescence of Single InAs Quantum Dots Obtained by Self-Organized Growth on GaAs,” Phys. Rev. Lett., 73, 716 (1994). [3.35] M. Grundmann, J. Christen, N. N. Ledentsov, J. Böhrer, D. Bimberg, S. S. Ruvimov, P. Werner, U. Richter, U. Gösele, J. Heydenreich, V. M. Ustinov, A. Yu, A. E. Zhukov, P. S. Kop’ev and Z. I. Alferov, “Untranarrow luminescence lines from single quantum dots”, Phys. Rev. Lett. 74, 4043 (1995). [3.36] C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot”, Phys. Rev. Lett. 86, 1502 (2001). [3.37] J. Houel, A. V. Kuhlmann, L. Greuter, F. Xue, M. Poggio, B. D. Gerardot, P. A. Dalgarno, A. Badolato, P. M. Petroff, A. Ludwig, D. Reuter, A. D. Wieck and R. J. Warburton, “Probing single-charge fluctuations at a GaAs/AlAs interface using laser spectroscopy on a nearby InGaAs quantum dot”, Phys. Rev. Lett. 108, 107401 (2012). [3.38] D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel and Y. Yamamoto, “Photon antibunching from a single quantum-dot- microcavity system in the strong coupling regime”, Phys. Rev. Lett. 98, 117402 (2007). 226 [3.39] I. Mukhametzhanov, “Growth Control, Structural Characterization, and Electronic Structure of Stranski-Krastanov InAs/GaAs (001) Quantum Dots,” Ph.D. dissertation, Physics, University of Southern California, 2002. [3.40] M. Abbarchi, C. Mastrandrea, T. Kuroda, T. Mano, A. Vinattieri, K. Sakoda, and M. Gurioli, “Poissonian statistics of excitonic complexes in quantum dots”, J. Appl. Phys. 106, 053504 (2009). [3.41] M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In 0.60 Ga 0.40 As self-assembled quantum dots”, Phys. Rev. B. 65, 041308 (R) (2002). [3.42] S. Paul, J. B. Roy and P. K. Basu, “Empirical expressions for the alloy composition and temperature dependence of the band gap and intrinsic carrier density in Ga x In 1-x As”, J. Appl. Phys. 69, 827 (1991). [3.43] J. Zhang, S. Chattaraj, S. Lu and A. Madhukar, “Mesa-top quantum dot single photon emitter arrays: Growth, optical characteristics, and the simulated optical response of integrated dielectric nanoantenna-waveguide systems”, J. Appl. Phys. 120, 243103 (2016). [3.44] Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors”, Physica. 34, 149 (1967). [3.45] A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science 271, 933 (1996). [3.46] J. Zhang, Z. Lingley, S. Lu, and A. Madhukar, “Nanotemplate-directed InGaAs/GaAs single quantum dots: Toward addressable single photon emitter arrays,” J. Vac. Sci. Technol. B 32, 02C106 (2014). [3.47].H. Baier, E. Pelucchi, E. Kapon, S. Varoutsis, M. Gallart, I. Robert-Philip and I. Abram, “Single photon emission from site-controlled pyramidal quantum dots”, Appl. Phys. Lett. 84, 648 (2004). [3.48] G. Bulgarini, M.E. Reimer, T. Zehender, M. Hocevar, E.P.A.M. Bakkers, L.P. Kouwenhoven, and V. Zwiller, “Spontaneous emission control of single quantum dots in bottom-up nanowire waveguides”, Appl. Phys. Lett. 100, 121106 (2012). [3.49] M. J. Holmes, K. Choi, S. Kako, M. Arita, and Y. Arakawa, “Room- temperature triggered single photon emission from a III-nitride site-controlled nanowire quantum dot”, Nano Lett. 14, 982 (2014). [3.50] J. C. Harmand, F. Jabeen, L. Liu, G. Patriarche, K. Gauthron, P. Senellart, D. Elvira, and A. Beveratos, “InP 1-x As x quantum dots in InP nanowires: A route for single photon emitters”, J. Cryst. Growth. 378, 519 (2013). 227 [3.51] J. Shumway, A. Franceschetti and A. Zunger, “Correlation versus mean-field contribution to excitons, multiexcitons, and charging energies in semiconductor quantum dots”, Phys. Rev. B 63, 155316 (2001). [3.52] L. Landin, M. S. Miller, M. –E. Pistol, C. E. Pryor and L. Samuelson, “Optical studies of individual InAs quantum dots in GaAs: few-particle effects”, Science, 280, 262 (1998). [3.53] M. E. Ware, E. A. Stinaff, D. Gammon, M. F. Doty, A. S. Bracker, D. Gershoni, V. L. Korenev, S. C. Bădescu, Y. Lyanda-Geller and T. L. Reinecke, “Polarized fine structure in the Photluminescence excitation spectrum of a negatively charged quantum dot”, Phys. Rev. Lett. 95, 177403 (2005). [3.54] O. Stier, A. Schliwa, R. Heitz, M. Grundmann and D. Bimberg, “Stability of biexcitons in pyramidal InAs/GaAs quantum dots”, Phys. Stat. Sol. (b) 224, 115 (2001). [3.55] M. Grundmann and D. bimberg, “Theory of random population for quantum dots”, Phys. Rev. B. 55, 9740 (1997). [3.56] W. Chang, W. Chen, H. Chuang, T. Hsieh, J. Chyi and T. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities”, Phys. Rev. Lett. 96, 117401 (2006). [3.57] A. Schwagmann, S. Kalliakos, I. Farrer, J. P. Griffiths, G. A. C. Johes, D. Ritchie and A. J. Shields, “On-chip single photon emission from an integrated semiconductor quantum dot into a photonic crystal waveguide”, Appl. Phys. Lett. 99, 261108 (2011). [3.58] R. Winik, D. Cogan, Y. Don, I. Schwartz, L. Gantz, E. R. Schmidgall, N. Livneh, R. Rapaport, E. Buks and D. Gershoni, “On-demand source of maximally entangled photon-pairs using the biexciton-exciton radiative cascade”, arXiv:1703.04380v1. [3.59] P. Michler, Single semiconductor quantum dots, Springer-Verlag Berlin Heidelberg, 2009. [3.60] C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots”, Phys. Rev. B. 69, 205324 (2004). [3.61] M. Pelton, C. Santori, G. S. Solomon, O. Benson and Y. Yamamoto, “Triggered single photons and entangled photons from a quantum dot microcavity”, Eur. Phys. J. D 18, 179 (2002). 228 [3.62] R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots”, Phys. Rev. Lett. 95, 257402 (2005) [3.63] I. Favero, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira, C. Couteau, J. P. Poizat and J. M. Gérard, “Fast exciton spin relaxation in single quantum dots”, Phys. Rev. B. 71, 233304 (2005). [3.64] C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots”, Phys. Rev. B. 85, 155303 (2012). [3.65] A. E. McKinnon, A. G. Szabo, D. R. Miller, “Deconvolution of photoluminescence data”, J. Phys. Chem. 81, 1564 (1977). [3.66] D. V. O’Conner, W. R. Ware, J. C. Andre, “Deconvolution of fluorescence decay curves – Critical comparison of techniques”, J. Phys. Chem. 83, 1333 (1979). [3.67] J. T. Verdeyen, Laser Electronics, Prentice Hall, 1995. [3.68] R. Heitz, H. Born, F. Guffarth, O. Stier, A. Schliwa, A. Hoffmann, D. Bimberg, “Existence of a phonon bottleneck for excitons in quantum dots”, Phys. Rev. B 64, 241305 (2001). [3.69] E. C. Le Ru, J. Fack and R. Murray, “Temperature and excitation density dependence of the photoluminescence from annealed InAs/GaAs quantum dots”, Phys. Rev. B 67, 245318 (2003). [3.70] C. Santori, G. S. Solomon, M. Pelton, and Y. Yamamoto, “Time-resolved spectroscopy of multiexcitonic decay in an InAs quantum dot”, Phys. Rev. B. 65, 073310 (2002). [3.71] G. Sallen, A. Tribu, T. Aichele, R. André, L. Besombes, C. Bougerol, S. Tatarenko, K. Kheng and J. Ph. Poizat, “Excition dynamics of a single quantum dot embedded in a nanowire”, Phys. Rev. B. 80, 085310 (2009). [3.72] J. Suffczyński, T. Kazimierczuk, M. Goryca, B. Piechal, A. Trajnerowicz, K. Kowalik, P. Kissacki, A. Golnik, K. P. Korona, M. Nawrocki and J. A. Gaj, “Excitation mechnisms of individual CdTe/ZnTe quantum dots studies by photon correlation spectroscopy”, Phys. Rev. B 74, 085319 (2006). [3.73] J. I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1971 [3.74] U. Bockelmann, G. Bastard, “Phonon-scattering and energy relaxation in 2- dimensional, one dimensional and zero-dimensional electron gases”, Phys. Rev. B. 42, 8947 (1990). 229 [3.75] H. Benisty, C. M. Sotomayortorres, C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems”, Phys. Rev. B 44, 10945 (1991). [3.76] O. Madelung, Seminconductors: Data Handbook, Springer-Verlag Berlin Heidelberg, 2004. [3.77] R. Heitz, M. Veit, A. Kalburge, Q. Xie, M. Grundmann, P. Chen, N.N. Ledentsov, A. Hoffmann, A. Madhukar, D. Bimberg, V.M. Ustinov, P.S. Kop’ev and Zh.I. Alferov, “Hot carrier relaxation in InAs/GaAs quantum dots”, Physica E. 2, 578 (1998). [3.78] K. H. Schmidt, G. Medeiros-Ribeiro, M. Oestreich, P. M. Petroff and G. H. Dohler, “Carrier relaxation and electronic structure in InAs self-assembled quantum dots”, Phys. Rev. B. 54, 11346 (1996). [3.79] L. Brusaferri, S. Sanguinetti, E. Grilli, M. Guzzi, A. Bignazzi, F. Bogani, L. Carraresi, M. Colocci, A. Bosacchi, P. Frigeri and S. Franchi, “Thermally activated carrier transfer and luminescence line shape in self-organized InAs quantum dots”, Appl. Phys. Lett. 69, 3354 (1996) [3.80] F. Guffarth, R. Heitz, A. Schliwa, O. Stier, N.N. Ledentsov, A. R. Kovsh, V. M. Ustinov and D. Bimberg, “Strain engineering of self-organized InAs quantum dots”, Phys. Rev. B. 64, 085305 (2001). [3.81] R. Heitz, F. Guffarth, K. Pötschke, A. Schliwa, D. Bimberg, N. D. Zakharov and P. Werner, “Shell-like formation of self-organized InAs/GaAs quantum dots”, Phys. Rev. B. 71, 045325 (2005). 230 Chapter 4. Single photon emission from MTSQDs In the preceding two chapters we have discussed the synthesis of the ordered and spectrally uniform mesa-top single quantum dot (MTSQD) array (Chapter 2) and the systematic power and temperature dependent studies of their PL, PLE, and TRPL behavior (Chapter 3). Guided by the knowledge gained on the MTSQDs’ excitonic states -- neutral exciton, singly charged exciton (trions), and biexcitons - - identified within the spectral resolution limit of the instrumentation, we present and discuss in this chapter studies of the single photon emission characteristics of these MTSQDs comprising design and establishment of the Hanbury Brown and Twiss instrumentation- the basic approach to measurements of the second order intensity correlation function that reveals the photon emission statistics-, data acquisition, and their analysis aimed at assessing the potential of MTSQDs as on-chip single photon sources (SPSs). The chapter is organized as follows: In §4.1 we capture the theoretical underpinnings of: photon statistics for the three types of light sources - - thermal, coherent, and single photon; photon detection; and photon correlation functions. This is followed by a description of the Hanbury Brown and Twiss (HBT) approach to measurement of the characteristic signature of single photon emission. Then in §4.2 we recall the quantum theory of spontaneous emission process from QD exciton decay and the quantum nature of the emitted photon, followed by description of the biexciton-exciton cascade process that can produce polarization entangled photon pair. Following these theoretical background, in §4.3 we present and discuss the 231 single photon second order intensity correlation measurements, data analysis, and extraction of single photon emission purity. This is followed by a discussion of the emission rate and collection efficiency of our InGaAs MTSQDs array. Finally, in §4.4, we discuss the physical processes in the MTSQDs that affect and limit the measured MTSQD single photon emission purity. §4.1 Photon statistics, photon detection, and photon correlation As mentioned before, we discuss here the quantum theory of photon statistics of single photon sources (comparing it with typical thermal and laser sources), photon detection, and photon correlation function and its measurement that are the theoretical basis for the single photon emission characterization of the MTSQDs discussed in later sections §4.3 and §4.4. §4.1.1 Light Sources and Photon Emission Statistic A single photon source is a purely quantum mechanical entity whose uniqueness can be appreciated by contrasting its emission characteristics with those of the most familiar, the thermal light sources (light bulbs) and, now perhaps equally familiar, lasers. What follows here is thus a recapitulation of the most fundamental concepts and language describing the states of photons in terms of which the differing characteristics of the different types of light sources can be appreciated. The language describing the state of photons is based on the second quantization language of the radiation field where the radiation field is described by the harmonic oscillator instead of a classical electromagnetic wave. Details of the basis of the quantization of radiation field are captured in Appendix D. Below we use the second 232 quantization language to discuss different types of light source and its photon emission statistics. Consider a single mode of the radiation field of frequency ω having creation and annihilation operators a + and a, respectively. We denote by |n> the energy eigenstate with eigenvalue of E n . The following relation then follows from the harmonic oscillator representation of the radiation field: 1 | ( ) | | 2 n H n a a n E n         (Eq.4.1) When we apply the annihilation operator a from the left, we obtain, | ( ) | n Ha n E a n      (Eq.4.2) This means that the state | |1 an n n     is also an energy eigenstate with its corresponding eigenvalue 1nn EE    . Repetitively applying operator a n times, we have, 0 | 0 ( ) | 0 Ha E a n       (Eq.4.3) where |0> is vacuum state. Combining Eq. 4.1 and Eq. 4.3, we have 0 1 2 E   . Correspondingly, we obtain 1 () 2 n En   and also || a a n n n     . Therefore, |n> is not only the energy eigenstate of the field but also the eigenstate of the number operator n a a   . The state |n> can be now represented as, () | | 0 ! n a n n     (Eq.4.4) 233 The eigenstates |n> are called Fock states or photon number states. The energy eigenvalues, 1 () 2 n En   , can be interpreted as the presence of n quanta or n photons of energy  . The Fock states form a complete set of states, i.e. 0 | | 1 n nn       . Additionally, the Fock state |n> represents a mode with exactly n number of photons. It is a truly nonclassical light state and can be generated by a quantum light source, i.e. single photon source for Fock state |1>. The classical radiation field, on the other hand, can be resembled as nearly as quantum mechanics permits by a coherent state. The corresponding state vector is denoted here by |α>, which is the eigenstate of the annihilation operator of the field with eigenvalue α represented below: || a       (Eq.4.5) It can be shown [4.1, 4.2] that the coherent state can be written in terms of the number state |n> in the following form, 2 /2 || ! n n en n         (Eq.4.6) Thus the expectation value of the photon number operator in a state |α> can be calculated and the mean number of photons in the coherent state |α> turns out to be, 2 || aa        (Eq.4.7) From Eq. 4.6, one obtains that the probability of finding n photons in the coherent state |α> is given by, 234 2 ( ) | | !! n n nn e ne p n n n nn                 (Eq.4.8) where <n>=|α| 2 . This is the well-known Poisson distribution with the variance of photon number given by the mean (Δn) 2 =<n>. A unique feature of the coherent state is that it is a minimum-uncertainty state. Therefore, the coherent state best approximates the pure classical picture of waves with fully determined amplitude and phase. The conjugate coordinate q representation of |n> is given by, ( ) | n q q n     (Eq.4.9) It follows from the definition of the annihilation and creation operators (Appendix D, Eq.D.10) that, 11 ( ), ( ) 22 a q a q qq          (Eq.4.10) Using Eq.4.10, the equation | 0 0 a  can be represented as, 0 ( ) ( ) 0 qq q     (Eq.4.11) A normalized solution of Eq. 4.11 is [4.1], 1/4 2 0 ( ) exp( ) 2 q q        (Eq.4.12) Then the higher order eigenfunctions of annihilation operator in the coordinate representation can be obtained as,   00 1/2 1 ( ) ( ) ( ) ( ) (2 !) ! n nn n a q q H q q n n       (Eq.4.13) 235 where H n are the Hermite Polynomials. These are the well-known eigenfunctions of a harmonic oscillator. The expectation values of the generalized momentum and coordinate variables are listed below, 0 pq       (Eq.4.14a) 2 1 () 2 pn      (Eq.4.14b) 2 1 () 2 qn      (Eq.4.14c) From Eq. 4.14, the uncertainty product of momentum and coordinate variable is, 1 () 2 p q n     (Eq.4.15) Thus () n q  has the minimum uncertainty value for the ground state wave function 0 () q  . Therefore the coordinate representation of coherent states represents a minimum-uncertainty wave packet, resembling the classical field. Such coherent state of light can be generated from a laser. Besides the Fock state and coherent light state, another important state of light is the thermal state. The thermal field can also be represented by Fock state in the second-quantization language. A radiation field emitted by a source in thermal equilibrium at temperature T can be described by a canonical ensemble with density matrix represented as [4.1], exp( / ) [exp( / )] B B H k T Tr H k T     (Eq.4.16) 236 where k B is the Boltzmann constant and 1 () 2 H a a    is the free-field Hamiltonian. For simplicity, we restrict ourselves to a single mode of the field. By substituting the Hamiltonian into Eq. 4.16 and expanding it in the basis of Fock states, we obtain, [1 exp( )]exp( ) | | n BB n nn k T k T          (Eq.4.17) Correspondingly the expectation value of number of photons in the field is 1 ( ) [exp( ) 1] B n Tr a a kT         (Eq.4.18) Given Eq.4.18, we can rewrite Eq. 4.17 in terms of <n> and arrive at the following relation: 1 || (1 ) n n n n nn n           (Eq.4.19) The photon distribution probability in a thermal field can thus be represented as 1 (1 ) n nn n n P n        (Eq.4.20) This leads to the well-known result that photon distribution in a thermal field is described by Bose-Einstein distribution. All of the above mentioned different types of light fields have different photon statistics. To illustrated the difference of the photon statistics of the three different types of photon states, we capture here photon number probability distribution of these states for two different mean photon number <n>=1 and 10. Figure 4.1 below shows the photon number probability distribution for a Fock state, 237 a coherent state and a thermal field with only a single mode but two chosen different mean photon numbers <n>=1 and 10 mentioned before. The number distribution of Fock states (Fig. 4.1. (a)) has non-zero probability exclusively for the mean photon number <n>=n. While for coherent light and thermal light, the photon numbers in the single mode are in a distribution shown in Fig. 4.1 (b) and Fig. 4.1 (c), respectively. Considering the case of a single photon in a single emission mode, the non-classical Fock state cannot be achieved through attenuation of a light beam stream coming from a thermal or a coherent state as seen from the lower panel of Fig.4.1 (b) and Fig. 4.1 (c). It can only be achieved from a Fock state with n=1. Figure 4.1. Photon number probability distribution for single mode emission of (a) a Fock state, (b) a coherent state, and (c) a thermal state for two different mean photon numbers <n>=1 (lower panel) and 10 (upper panel) respectively . §4.1.2 Photon detection and photon correlation functions As noted in the preceding section, different types of light states have different photon statistics. The photon statistics can be studied by detecting the photons and measuring the photon correlation functions. A more general and detailed discussion of photon detection and correlation functions can be found in Ref. [4.1- 4.3]. We 238 only capture here important information to assist the discussion of characterization of MTSQDs as SPSs. Photons are typically detected by avalanche photodiodes. In such a case, the photon is absorbed and as a consequence the quantum state of light field is altered. Following the second quantized notation of the electric field (Appendix D, Eq. D.18 and Eq. D.19), the electric field operator ( , ) E r t can be expressed as the sum of two parts containing annihilation and creation operators separately:       , , , E r t E r t E r t   (Eq.4.21) where   ˆ , k ik r iw t k k k k E r t E a e      ,   ˆ , k ik r iw t k k k k E r t E a e      . In the photon detection event, only E  containing the annihilation operator k a contributes. The probability of photon detection (i.e. the detector to absorb a photon from the field at position r ), P 1 , between time t and t+dt is, 2 1 ~ | | ( , ) | | | ( , ) ( , ) | i i f i P f E r t i c i E r t E r t i            (Eq.4.22) where |i> is the initial state of the field before the detection process and |f> is the final state of the field after the detection process. Eq. 4.22 is based on a statistical description averaging over all the possible realization of the initial states as well as all the possible final states. We define the first-order correlation function of the field which is tied to photon detection event as, (1) 1 2 1 2 1 1 2 2 1 1 2 2 ( , ; , ) [ ( , ) ( , )] ( , ) ( , ) G r r t t Tr E r t E r t E r t E r t          (Eq.4.23) 239 where ρ is the density matrix operator of the field given as || i i c i i      . The correlation function depends on t 1 and t 2 only through the time difference τ=t 2 -t 1 . Therefore, (1) 1 2 1 2 ( , ; , ) G r r t t can be represented as (1) 12 ( , ; ) G r r  : (1) (1) 1 2 1 2 1 2 ( , ; , ) ( , ; ) G r r t t G r r   . The photon detection probability P 1 is given by (1) 1 ~ ( , ,0) P G r r . Next we consider the joint probability of detecting photons at two different photodetectors at position 1 r and 2 r . The joint probability P 2 of detecting one photon at one detector at position 1 r during time t 1 and t 1 +dt and the other photon at the second detector at position 2 r during time t 2 and t 2 +dt is represented as, 2 2 2 1 1 2 2 1 ~ | | ( , ) ( , ) | | | ( , ) ( , ) ( , ) ( , ) | if i i P f E r t E r t i c i E r t E r t E r t E r t i            (Eq.4.24) Similarly, we define the second-order quantum mechanical correlation function as, (2) 1 2 3 4 1 2 3 4 1 1 2 2 3 3 4 4 ( , , , ; , , , ) ( , ) ( , ) ( , ) ( , ) G r r r r t t t t E r t E r t E r t E r t        (Eq.4.25) Thus the joint probability of photon detection is governed by the second-order correlation function. In general, nth order correlation function can be defined as, () 1 2 1 2 1 1 1 1 2 2 ( ,... ; ,... ) ( , )... ( , ) ( , )... ( , ) n n n n n n n n n G r r t t E r t E r t E r t E r t         (Eq.4.26) Correlation functions of the field operators which are tied to any photon detection experiment based on destructive measurement of photon are in normal order. Correspondingly, the quantum mechanical first- and second-order degrees of coherence at the position r can be defined as, 240 (1) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) E r t E r t gr E r t E r t E r t E r t                   (Eq.4.27) (2) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) E r t E r t E r t E r t gr E r t E r t E r t E r t                       (Eq.4.28) where the field has been assumed to be statistically stationary. (1) ( , ) gr  and (2) ( , ) gr  are the normalized functions of first- and second-order correlation functions of the electric field at position r . Substituting the actual form of E  and E  into Eq.4.27 and Eq.4.28, the (1) ( , ) gr  and (2) ( , ) gr  functions can be expressed in terms of annihilation and creation operator, a and a  as the following: (1) ( ) ( ) () a t a t g aa          (Eq.4.29) (2 2 ( ) ( ) ( ) ( ) () a t a t a t a t g aa           (Eq.4.30) From Eq. 4.29, the first-order coherence (1) () g  is insensitive to the photon statistics of the field since it only depends on the average photon number ( ) ( ) n a t a t       . Therefore, the spectrally-filtered thermal light and coherent light of the same spectral width exhibit the same degree of first-order coherence. In contrast, the second order coherence (2) () g  in Eq. 4.30 allows distinguishing between the different types of light fields. The (2) () g  can be written in terms of photon number operator ( ) ( ) ( ) n t a t a t   as, (2 2 : ( ) ( ) : () n t n t g n        (Eq.4.31) 241 where :: denotes normal ordering. We capture below the second-order coherence for Fock state, coherent state (coherent light), and thermal state (thermal light). (1) For Fock state (photon number state), the second-order coherence at τ=0 is, (2) 22 | | ( 1) 1 (0) 1 || n a a aa n n n g n a a n n n           (Eq.4.32) For 1 n  , (2) (0) 1 g  . This type of light is called to be in sub-Poissonian light state. Especially, for single photon state where n=1, (2) (0) 0 g  . (2) Coherent state (coherent light), the second-order coherence at τ=0 is, 2 (2) 22 || (0) 1 || a a aa g aa              (Eq.4.33) (3) For thermal state (thermal light), the second-order coherence at τ=0 is, 2 (2) 2 () (0) 1 2 nn g n         (Eq.4.34) This indicates that photons have the tendency to be detected simultaneously at the two different photodectectors. The light with (2) (0) 1 g  is called to be super- Poission light state. In classical coherence theory, we always have the following relationship between intensity of light at position r at two different times with time difference τ: | ( , ) ( , ) | | ( , ) || ( , ) | I r t I r t I r t I r t         (Eq.4.35) Given that the intensity of light is proportional to photon number, ( ) ( ) I t n t , for classical light we thus always have (2) ( ) 1 g   following Eq. 4.31. Thus (2) ( ) 1 g   is 242 a nonclassical inequality that can only be satisfied by a quantum light state from a quantum emitter. Finally, a light whose second-order coherence satisfies the inequality (2) (2) ( ) (0) gg   is bunched light. The phenomenon is called photon bunching. Photons tend to distribute themselves preferentially in bunches rather than randomly. On the other side, a light whose second-order coherence satisfies the inequality (2) (2) ( ) (0) gg   is antibunched light. The phenomenon is called photon antibunching. Fewer photon pairs are detected close together than further apart. Classical light is always in the category of bunched light. Non-classical light, i.e. from Fock states, can show antibunched behavior. §4.1.3 Measurement of (2) () g  : Hanbury Brown and Twiss Approach The second order coherence function, (2) () g  , represents the probability of detecting one photon at time t 1 and another photon at time t 2 with t 2 =t 1 +τ and can be used to probe the statistical property of the emitted photons as discussed in the previous section. The direct method of measuring the second-order coherence function (2) () g  would be to measure the times of a single-photon detector’s counting events. The second-order coherence functions (2) () g  can be calculated based on the measured data given its definition. However, this method would limit the measurements to times scales longer than the dead time of the detector which is of several 10ns, typically ~50-100ns. To overcome this problem, detection schemes using two 243 independent detectors in a Hanbury Brown and Twiss (HBT) type setup [4.4] shown in Fig. 4.2 are usually used. The setup consists of two orthogonally arranged pathways centered around a nonpolarizing (50/50) beam splitter. Each arm of the HBT interferometer is equipped with a highly sensitive single-photon detector. Figure 4.2. The schematic of HBT setup with a 50/50 beam splitter and two detectors on the two output arm of beam splitter. In the HBT setup, we label the two detectors as detector 1 and 2 and use t 1 and t 2 to represent the time of detection event at detector 1 and 2, respectively. According to the quantum theory of photoelectron correlations discussed in section §4.1.2, the probability, P, of detecting a photon at detector 1 at t 1 and another one at detector 2 at t 2 is proportional to the second-order field correlation function, (2) 1 2 1 1 2 2 2 2 1 1 ( , ) ( ) ( ) ( ) ( ) P G t t b t b t b t b t      (Eq.4.36) where 1 () bt  and 2 () bt  are the creation operator of the field mode propagating to detector 1 and detector 2 respectively in the Heisenberg picture. Beam splitter relates the modes operator on the input arms and the modes on the output arms of the beam 244 splitter. Consider a beam splitter with transmission efficiency T and the input field incidenting on the two input arms with mode operator a and b shown in Fig. 4.3, the mode operator c and d of the two detection arm is related to that of the inputs through: 1 1 c T i T a db i T T                   (Eq.4.37) Figure 4.3. The schematic of the photon interference at a beam splitter with incident light of mode represented by a and b from two input arms of the beam splitter. The mode of outgoing light is represented by c and d. For a 50/50 beam splitter with only one input on the transmission arm of the beam splitter, the mode of two output arms 1 () bt  and 2 () bt  is related to that of the input arms () at  following Eq. 4.37 and can be written as 1 1 1 2 ( ) ( ) 2 b t a t   and 2 2 2 2 ( ) ( ) 2 b t i a t   . Therefore, the probability of detecting one photon at detector 1 at t 1 and subsequently another one at detector 2 at t 2 in the HBT setup is, (2) 1 2 1 2 2 1 ( , ) ( ) ( ) ( ) ( ) HBT P G t t a t a t a t a t      (Eq.4.38) 245 Now, by defining τ as 21 tt  , we have, (2) ( ) ( ) ( ) ( ) ( ) HBT P G a t a t a t a t           (Eq.4.39) The measured detection events of photon detection at two detectors with time difference τ is directly related to second order correlation function of the input light field. Therefore, the normalized second order correlation function or the second order degree of coherence, (2) () g  , of the light source is measured in HBT setup. The HBT setup that measures the second order correlation function of light field can also be viewed as a setup measuring the interference of photon numbers at two detection end. The correlation of intensity at two detectors is, 1 1 2 2 1 1 2 2 1 1 1 1 2 2 2 2 ( ) ( ) ~ ( ) ( ) ( ) ( ) ( ) ( ) I t I t n t n t b t b t b t b t         (Eq.4.40) Given the bosonic nature and commutation rule of 1 () bt and 2 () bt , we have, 1 1 1 1 2 2 2 2 1 1 2 2 2 2 1 1 : ( ) ( ) ( ) ( ) : ( ) ( ) ( ) ( ) b t b t b t b t b t b t b t b t          (Eq.4.41) Therefore, we obtain, (2) 1 2 2 1 ( ) : ( ) ( ) : ( ) ( ) HBT P G n t n t n t n t             (Eq.4.42) §4.2 Physics of Spontaneous Emission and Polarization Entangled Photon Pair Emission from QDs As discussed in previous section, single photon state is a Fock state |n> with photon number n=1. Photons in such states can be measured and quantified through the study of the second order correlation function and second order degree of coherence function of photons using the HBT setup. Single quantum dots (SQDs) are 246 known to emit single photons through the spontaneous emission of the QD exciton transitions [4.5-4.7]. In chapter 3 section §3.2 we recalled the theory of optical transitions in QDs and discussed the energy and polarization of the photon emitted from QD excitonic transition assuming that QD is interacting with classical radiation field. This is adequate in many situations and provides correct description of the optical transition rules, emission energy and transition polarizations, etc. Indeed, it is used to provide the theoretical basis for the photoluminescence studies on our MTSQDs discussed in Chapter 3. However, the full account of the spontaneous photon emission from exciton states and the state of emitted photon cannot be captured within the framework of QD interacting with classical radiation field. A fully quantum mechanical description of the radiation field with vacuum modes is needed to understand the spontaneous photon emission process and the state of the emitted photon. Therefore, we discuss in this section the interaction of the quantized radiation field (introduced in section §4.1) with QDs to facilitate understanding of the QD spontaneous photon emission process and the state of the emitted photons that underlie the experimentally observed single photon emission phenomena from single MTSQDs discussed in §4.3 and §4.4. §4.2.1 QD interaction with quantized radiation field The interaction of a radiation field E with a QD can be described by the following Hamiltonian in the dipole approximation: int QD F QD F H H H H H H er E       (Eq.4.43) 247 This Hamiltonian contains three parts: (1) the energies of the quantum dot, H QD (2) the energies of the radiation field, H F and (3) the interaction between the atom and field with dipole approximation with r being the position vector of the electron. In the dipole approximation, the field is assumed to be uniform over the QD. The radiation field is represented as quantized harmonic oscillators (appendix D) instead of electro-magnetic waves with vector potential A description used in Chapter 3. Thus, the energy of the field is given in terms of the creation and annihilation operators by, ,, , 1 () 2 F k k k k H a a       (Eq.4.44) where , k a   and , k a  are the photon creation and annihilation operators for wave vector k and polarization  . The Hamiltonian of the QD, H QD , and er can be expressed in terms of the quantum dot transition operators || ij ij     , where {| } i  represents a complete set of the QD energy eigenstates. The energy eigenstates satisfies the QD eigenvalue equation || QD i H i E i    . It then follows that the Hamiltonian of the QD independent of the presence of the field can be written as, || QD i i H E i i     (Eq.4.45) Similarly, er can also be represented in the complete bases set of {| } i  as, 248 ,, | | | | | | ij i j i j er e i i r j j i j           (Eq.4.46) where || ij e i r j     is the electric-dipole transition matrix element. We evaluate the electric field operator at the position of the QD. We have then the following: , , , , () k k k k k E a a         (Eq.4.47) where 1/2 0 () 2 k k V     , V is the volume of the box chosen in quantizing the electromagnetic field. By substituting Eq. 4.46-4.47 into Eq. 4.43, the Hamiltonian for the QD and field interacting system [4.1] is captured as, , , , , , , ,, () ij i ii ij k k k k k k i i j kk H E a a g a a                   (Eq.4.48) where , 1/2 ,, 0 ˆ ˆ () 2 ij k k ij k ij kk g V            . In Eq. 4.48, the zero-point energy from the field is omitted. Any dephasing processes that involve the interaction of the QD with its surrounding crystal environment are neglected in the discussion above. When electron and hole pairs are created in the QD through weak optical excitation (not saturating the QD lowest electron and hole energy), electrons and holes relax to the lowest states through phonon emissions [4.8, 4.9], the process referred to as thermalization. The electron-hole pair in its ground state decays through radiative and nonradiative recombination. Now, let us assume the QD is a perfect two-level system with a ground state |g> and an excited state |e> and also assume that ij  is a 249 real value. In this case, we have ge eg    and , , , ge eg k k k g g g     . The Hamiltonian of the two-level QD interacting with the electromagnetic field is, , , , , , ,, ( )( ) ge z k k k k k k kk H a a g a a                     (Eq.4.49) where ge e g EE   , | | | | z ee gg e e g g            , || eg eg       and || ge ge       . The σ z , σ + and σ - satisfy the Pauli matrices. The last item in Eq.4.49 represents the QD-field interaction which contains four terms. The term , k a     describes the process in which the atom is changed from excited state to ground state and a photon of wavevector k and polarization  is created or emitted. The term , k a    describes the opposite process that atom is excited from ground state to excited state while a photon of wavevector k and polarization  is annihilated or absorbed. The energy is conserved in both cases. The term , k a     describes the process that atom is taken from ground state to excited state and a photon of wavevector k and polarization  is created, resulting in the gain of energy about 2 ge  . Similarly, , k a    results in the loss of energy about 2 ge  . The energy non- conserving terms are considered to be a very small perturbation and can be neglected. Correspondingly the resulting simplified Hamiltonian is, , , , , , ,, () ge z k k k k k k kk H a a g a a                    (Eq.4.50) This Hamiltonian, capturing the interaction of a single two-level QD with a multi- mode field, is also known as the Jaynes-Cummings model [4.10,4.11]. 250 For photoluminescence, the initial state at time t=0 of the combined system of the quantum dot and the radiation field considering a bright exciton in the QD and zero photons, is thus represented as | | 0 e , where |0  denotes the vacuum state of the electromagnetic field. At finite time t, this state evolves into [4.1], , , , , | ( ) ( ) | | 0 ( ) | | 0 ge k it it a b k k k t c t e e c t e g a                 (Eq.4.53) The Schrödinger equation, with the Hamiltonian shown in Eq. 4.50, provides differential equations relating the complex functions () a ct and ,, () bk ct  that define the time evolution of the state. These equations can be solved in the Weisskopf- Wigner approximation, which assumes that k  varies little over the linewidth of the transition. The result of the calculation is [4.1]: /2 ( 0) t a c t e   (Eq.4.52) ( ) /2 /2 , , , 1 ( 0) ( ) / 2 ge k i t t t b k k ge k e c t g e i              (Eq.4.53) with 32 3 0 4 | | 1 43 ge eg c     , which is the intensity decay rate, and n is the index of refraction. For 1/ t    , the final state of the emitted photon is, ,, , | | 0 ( ) / 2 k it kk k ge k e ga i               (Eq.4.54) The spatial form of the photon wavepacket is obtained by taking the Fourier transform of the result in Eq. 4.54, using , , ik r r k r a e a      . In the far field, | || | 1 kR , Eq. 4.54 can be written as, 251 ( /2)( | |/ ) , , ( | |) | ( ) | 0 || ge i t R c lm R k ct R e R X a R                 (Eq.4.55) where  is the unit step function, lm X are the vector spherical harmonics [4.12] and R is the position vector of the observation. The angular dependence is the same as the electric field from a classical radiating dipole. The photon is in a pure quantum state, a Fock state, with the time dependence characterized by a single exponential decay controlled by  . The spontaneous emission rate  depends on the dipole moment eg  of the QD. Following the notation in Chapter 3, the dipole moment is 3* | | ( ) ( ) ij e h e i r j e d rr r r         3* ( ) ( ) e h cv e d rf r f r    (Eq.4.56) where 3* ,0 ,0 1 ( ) ( ) cv e h d rr r r       is a standard parameter of the material controlled by the electron and hole zone center Bloch functions. Here  is the volume of the unit cell. From Eq. 4.78, one can see that the QD dipole moments are controlled by the spatial overlap of the envelop function of electron and hole state. Given that / cv e  is ~1nm and can be approximated using 22 0 /2 cv p g e E m E   , where E g is the band gap and 24 p E eV  is a parameter that represents the energy of an free electron with momentum equaling the conduction and valance band transition dipole moment at Γ point in the Kane model [4.13], the expected spontaneous emission rate 252 for InGaAs MTSQDs is 1/ 0.28ns  . The observed decay lifetimes for our MTSQDs are typically around ~ 1ns or longer as discussed in Chapter 3. From the above discussion, a photon emitted from QD by spontaneous decay of exciton at a rate  is in a Fock state with wavevector k and polarization  . In photoluminescence experiments, the QD is optically excited to create at least an electron-hole pair that thermalizes to a bright exciton state | | 0 e . The optical excitation can be done in three major different ways: (1) above the GaAs (barrier) bandgap, (2) below GaAs bandgap but above the QD gap (non-resonant with QD ground state) (3) resonant excitation of the QD. The QDs are not isolated from environment as assumed in the theoretical discussion earlier but are connected to a thermal reservoir. In this section, we use the resonant excitation as an illustration to capture the theoretical framework including optical excitation of the QD and the interaction of the QD with the environment to examine the physical process of QD excitation and subsequent single photon emission through spontaneous decay.  Resonantly Driven QD in a Thermal Reservoir For describing a QD interacting with a resonant driving field and a thermal reservoir, we utilize the equation of motion for state density matrix operators in the interaction picture [4.1]: [ , ] i VL       (Eq.4.57) 253 with (| | | |) 2 R V e g g e        (Eq.4.58) representing the interaction of the QD with the resonant drive field and / R     . The second term is given by, [ 2 ] 2 L                      (Eq.4.59) and represents the interaction of the QD with the thermal reservoir. Detailed derivation of Eq. 4.59 can be found in Ref. [4.1, page 249-253 and 291-298] and is not captured here. The R  , related to electric field and QD dipole moment, is assumed to be a real number and is usually called Rabi frequency. The damping term is taken in the limit of 0 k n =0 (the average thermal boson number) with temperature T=0.  is the QD spontaneous decay rate. To find correlation functions of photons emitted from the QD, we need to find the expectation values of the dipole operators where ( ) ( ) , ( ) ( ) , ( ) ( ) ( ) ge ge i t i t ge eg z ee gg t t e t t e t t t                     . All higher order correlation functions can be determined from them. In the measurement of the second order correlation functions, photons of all frequencies are collected. The second order correlation function of the emitted photons from QD can be represented as ( ) ( ) ( ) ( ) t t t t               . Given ( ) ( ) 0 tt      and ( ) ( ) ( ( ) 1) / 2 z t t t           , the two-time correlation function ( ) ( ) ( ) ( ) t t t t               can be obtained and written as, 254 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 z t t t t t t t t t                                (Eq.4.60) Through solving Eq. 4.59 and substituting the obtained solutions into Eq. 4.60, we have, 3 2 2 4 22 3 ( ) ( ) ( ) ( ) ( ) [1 (cos sin ) ] ( 2 ) 4 R R t t t t e                               (Eq.4.61) where 1/2 2 2 16 R         . Therefore, the second-order degree of coherence can be written as, 3 (2) 4 2 ( ) ( ) ( ) ( ) 3 ( ) [1 (cos sin ) ] ( ) ( ) 4 t t t t ge tt                                 (Eq.4.62) From Eq. 4.62, the time dependent behavior of the (2) () g  depends on the relative value of R  and 1 4  . Figure 4.4 below shows the calculated second-order degree of coherence of the photons emitted from a QD that is interacting with the resonant driving field and thermal reservoir. Two different cases are shown in Figure 4.4: Case1: 1 20 R    , satisfies 1 4 R    under weak QD-field coupling , resembling the measured condition (above and below GaAs bandgap excitation) used in this dissertation; 255 Case 2: 20 R    , satisfies 1 4 R    under strong atom-field coupling. For both cases, we have (2) (0) 0 g  when 0   and (2) (2) (0) ( ) gg   for 0   which corresponds to the phenomenon of photon antibunching. For the strong atom-field coupling case, the (2) () g  shows an oscillatory behavior with the oscillation frequency R  , the Rabi frequency. The magnitude of oscillation reduces as τ is increased and (2) () g  reaches unity as   . Figure 4.4. Calculated second order correlation function (2) () g  plotted as a function of dimensionless number   for cases (a) 1 20 R    and (b) 20 R    . §4.2.2 Biexciton-Exciton Cascade: Polarization-Entangled Photon Pair Emission As discussed above, single photon emission can be obtained from QDs through QD exciton state spontaneous decay process following the optical excitation and creation of exciton inside the QD. With increased number of excitons inside a 256 QD, biexcitons can form in addition to single excitons. We now consider the possibility of generating polarization-entangled photons from biexciton-exciton cascade decay process in single QDs. In the class of QDs under investigation here, as discussed in Chapter 3, the single exciton states have fine structures containing two bright excitons and two dark excitons. In the two-photon cascade process, a single biexciton decays into one of the optically bright exciton states by emitting one photon, and then to the empty-dot state by emitting a second photon as was discussed in Chapter 1 and for handy reference is recaptured here as Fig.4.5. The polarizations of the photon pair emitted through these two decay paths are determined by the property of the bright exciton states controlled by the QD confinement potential symmetry, as discussed in detail in Chapter 3. For QDs with D 2d symmetry, the two bright single exciton states are degenerate and the two decay paths become “indistinguishable”, ideally producing polarization-entangled photons [4.14]. When QDs have lower symmetry, e.g. C 2v symmetry, the two bright excitons are split (Fig.4.5) in energy due to the electron-hole exchange interaction and results in photon emission with orthogonal linear polarizations, also discussed in detail in Chapter 3. It has been found that when the splitting is smaller than the radiative linewidth, the two decay paths can still be treated as “indistinguishable” producing polarization entangled photons with linear polarization [4.15]. We discuss here the two-photon cascade process and the effect of this bright exciton energy splitting on the polarization correlation properties of the emitted photons. 257 Figure 4.5. Schematic energy-level diagram for the cascaded decay of a biexciton from a QD with C 2v confinement potential symmetry. We follow the notation in Ref. [4.1] to discuss the emission of photon pairs from biexciton-exciton cascade process. Consider the highest, intermediate, and lowest states, labeled as |c>, |b> and |a> respectively in Fig.4.5. The state of emitted photon pair is: () ,, , | | 0 [ ( ) / 2][ ( ) / 2] i k q r c k b k q k kq q ca c q ba b k g g e aa ii                     (Eq.4.63) where , ck g and , bk g are the coupling coefficients between states |c> and |b> and the photon modes k and q ; k  and q  are the frequencies of photon modes k and q , k a  and q a  are the photon creation operators for modes k and q , and c  and b  are the decay rates of |c> and |b> due to the coupling coefficient , ck g and , bk g , respectively. When we consider the two decay paths shown in Fig. 4.5, each path produces a term similar to the right-hand side of Eq. 4.63 but with orthogonal polarization. Given the orthogonality of the polarization of the photon from two 258 paths and the irreversibility of QD decay, the terms corresponding to the two paths can be simply added. The complete two-photon state can thus be captured as, 1 2 1 2 12 , 1 2 2 2 1 | | 0 [ ( ) / 2] ( ) / 2 ( ) / 2 XX XX H X V X H H V V i i i                                     (Eq.4.64) where 1  and 2  are the frequencies of the emitted two photon, XX  is the biexciton state frequency, H  and V  are the frequencies of the linearly polarized single- exciton states, XX  is the decay rate of the biexciton state, X  is the decay rate of the single-exciton states, H   creates one photon of frequency  with a horizontally-polarized (shown in Fig.4.5) dipole radiation pattern while V   creates one photon of frequency  with a vertically-polarized (shown in Fig.4.5) dipole radiation pattern. These dipole radiation patterns produce horizontal and vertical polarizations, respectively, when measured along the z axis. In the optical measurements, photons collected are of all the allowed frequencies of the biexciton-exciton cascade emission process. The measured results can thus be described by a reduced polarization density matrix obtained from tracing over the frequencies of the two photon density matrix,||       . We get, * 1 {| | | | | | | |} 2 pol HH HH VV VV HH VV VV HH                (Eq.4.65) where |HH> and |VV> are two-photon states of indefinite energy with both photons being in dipole radiation patterns with horizontal or vertical polarizations, 259 respectively. The parameter χ in Eq. 4.65 defines the degree of entanglement, and is written as, 1 1 ( ) HV X i       (Eq.4.66) When there is no fine structure splitting, χ=1, and pol  is maximally entangled. On the other side, when the splitting is larger than the radiative decay rate of single- exciton state, 0   , and pol  becomes a statistical mixture of |HH> and |VV>. With our limited optical instrumentation, we could only study the single photon emission properties of our synthesized MTSQDs in this dissertation. The potential of MTSQDs acting as entangled photon sources is addressed with limited knowledge obtained on its single exciton energy splitting. §4.3 Single Photon Emission Characteristics of the MTSQD array In this section we discuss the measurements of the second order coherence function, (2) () g  , of the emitted photons from our MTSQDs to assess their potential use as single photon sources. The information on (2) () g  of the emitted photons from the studied MTSQD as well as the MTSQD photon emission rate can be obtained from the measured histogram of coincident counts between two detectors in the HBT setup. Followed by a brief discussion of the HBT setup, we discuss in detail the measurement of coincidence histogram, the physical representation of measured histogram and the single photon emission property of MTSQDs extracted from the 260 coincidence measurement in terms of its single photon emission purity, emission rate and collection efficiency. §4.3.1 Hanbury Brown and Twiss Instrumentation Given our objective in assessing the potential use of ordered spectrally uniform size- and shape-controlled MTSQDs as SPSs and EPSs, our micro- PL/TRPL/PLE setup discussed in Chapter 3 has been expanded with added Hanbury- Brown and Twiss setup to measure the second order correlation function of emitted photons from MTSQDs. Details of the setup is captured in Appendix C. We briefly capture here important features of the setup. Schematic drawing of the setup is shown in Fig. 4.6. For the (2) () g  measurement, a 640nm solid state diode laser with repetition rate of 80MHz (PicoQuant model LDH-P-C-640B) or an 532nm solid state laser (Coherent Verdi G8) pumped Ti:S laser (Mira 900D, tunable wavelength from 700-1000nm) laser with 76MHz repetition rate are used as the excitation source. Excitation from these pulsed lasers are coupled into multimode optical fiber with core 50 μm and NA 0.2, filtered by a 900nm short pass and a 900nm dichromatic filters and focused down to ~ 1.25μm diameter through a 40× NA 0.65 objective lens on to sample mounted inside a continuous flow cryostat (Janis ST-500, 4K to 360K). The emitted photons from an individual MTSQD is collected in the same vertical geometry using the same objective lens, filtered by a 900nm dichromatic and a 900nm long pass filter, coupled to a FC-PC adjustable collimator (Thorlab, CFC-2X-B) and focused into multimode optical fiber with core 25 μm and NA 0.1. Photoluminescence collected 261 through this optical fiber is focused by a pair of NIR-coated achromatic 1” f=25mm lens into the entrance slit of an f=0.3m single stage imaging spectrograph (Acton SP300i). Spectrally dispersed light with spectral window set by the spectrometer is collimated by a NIR-coated achromatic 1” f=75mm lens and directed to a NIR 50/50 beam splitter. The split light at both the transmitted and reflected direction of beam splitter is focused into silicon APDs (Excelitas model SPCM-NIR-14-FC and PicoQuant model τ-SPAD) by NIR-coated achromatic 1” f=50mm lens. The 50/50 NIR beam splitter together with the two silicon APD detector is the standard realization of the HBT setup. A 950nm OD 4 short pass filter is inserted in front of the APD at reflected arm of beam splitter to prevent detection of photons emitted by the APDs during the avalanche process and creating detector cross-talk [4.16, 4.17]. The timing of the TTL (Transistor-Transistor Logic) pulses from the two detectors is registered using two constant fraction differential discriminators (Ortec, model 9307) and fed to a time-to-amplitude (TAC) convertor (Ortec, model 457) as the start and stop signal. The stop signal is delayed by a delayer (Ortec, model 416A) before getting into TAC. The TAC outputs are read by a multi-channel analyzer (Ortec Trump-PCI) to generate a histogram of coincidence photon detection events. 262 Figure 4.6. The schematic of the HBT setup used for measuring second order correlation function of emitted photons from individual MTSQDs. The total timing uncertainty of the setup in measuring second order correlation histograms, Δτ, has three contributing parts: (1) the temporal dispersion during pulsed light propagation, Δτ opt , (2) the time jitter of APDs, Δτ APD , and (3) the timing uncertainty in the timing electronics, Δτ ele , as captured in the following equation: 2 2 2 2 2 2 2 22 2 2 2 opt APD ele opt APD c d                        (Eq.4.67) The uncertainty of the timing electronics as shown in Eq. 4.67 is controlled by two major contributing parts: the uncertainty of delay unit, Δτ d , ~220ps and the uncertainty of the timing of the constant fraction differential discriminators, Δτ c , ~25ps. Accounting for the APD time jitter ~350ps and the temporal dispersion of light in optical fibers ~20ps, the total timing uncertainty in the timing electronics 263 estimated using Eq.4.67 is ~550ps. Direct measurement of the timing resolution of the setup is done by measuring the instrument response function (IRF) of our setup. The IRF is operationally measured by directly using the 640nm 80MHz pulsed laser as the input light source into the HBT setup and measuring the photon correlation histogram of the pulsed light source. Given the pulse width of the pulsed laser ~93ps and the estimated timing resolution of the setup, the expected IRF FWHM is ~567ps. The measured IRF spectrum, i.e. the spectrum of the photon-correlation histogram for the input laser light, is shown in Fig. 4.7. As expected, we see series of peaks separated by 12.5 ns, the laser repetition period. The number labeled on top of each peaks in Fig. 4.7 is the normalized areas for each peak calculated by normalizing the peak area with respect to the average value of peak areas. And, as expected for pulses with a Poisson photon number distribution, the peaks all have normalized areas of about 1, including the central peak at τ = 0. The measured widths (FWHM) of the peaks are 745±30ps, much larger than the pulse width of the pulsed laser ~93ps, representing the timing resolution of our HBT setup. The discrepancy between the measured IRF FWHM and the estimated timing resolution may be contributed by the temporal dispersion of pulsed light in the spectrometer and possible timing uncertainty in the TAC and multichannel analyzers. 264 -30 -20 -10 0 10 20 30 0 50 100 150 200 1.03 0.98 0.97 1.01 Coincidence Counts  (ns) 1.01 Figure 4.7. A photon correlation histogram of 640nm 80MHz pulsed laser light. The numbers above the peak indicate the normalized peak areas with respect to the average peak areas of the shown peaks. §4.3.2 Physical representation of measured second-order correlation histogram As discussed in section §4.1.2, the second order degree of coherence, (2) () g  , can be written as (2 2 : ( ) ( ) : () n t n t g n        shown in Eq. 4.31 with () nt being the photon number operator. In the measurement of (2) () g  using HBT setup, the photons are detected by the two detectors at the two different light paths of the beam splitter. Following the standard quantum optics rules, (2) () g  can be expressed in terms of the photon numbers at the two detectors as, (2 2 1 2 1 22 ( ) ( ) ( ) (0) () n t n t n n g nn             (Eq.4.68) The time invariance and ordering is accounted for in arriving at Eq. 4.68. 1 () nt and 2 () nt are the photon numbers detected at time t at detector 1 and 2. When measuring 265 (2) () g  of a pulsed light source, an unnormalized photon correlation function, 21 ( ) ( ) n t n t     , integrated over t is obtained which contains a series of peaks separated by the repetition period of the source. Under this measurement scenario, a discretized version of the photon correlation function can be defined as [4.18-4.20] (2) 21 21 [ ] [0] [] n i n gi nn       (Eq.4.69) where [] k ni is the number of photons detected from pulse i at detector k. In arriving at Eq. 4.69, we assume that source properties are periodic such that the expectation values of [] k ni are independent of pulse number index. The center peak at i=0 corresponds to the events when two photons are detected from the same pulse. The side peaks at 0 i  correspond to the events when photons are detected from two different pulses separated by a time difference iT with T being the period of pulses.  Measured histogram and (2) () g  In the HBT setup, the timing differences of the electronic signals from the two photon detectors are recorded using the time-correlated single photon counting electronics. A histogram with a finite time-bin size is generated as a function of the relative delay 21 tt  between a photon detection event at detector 1 and 2 at times 1 t and 2 t respectively. In the following discussion we address why the histogram provides a good estimate for (2) () g  and the effect of photon detector dead time and dark counts on the measured histogram. 266 Consider the ideal case where the detectors do not have dead time and dark counts. The measured histogram, h(τ), that represents the number of recorded photon detection events in time intervals of length τ can be represented as, 12 1 ( ) ( ) ( ) N t h d t d t     (Eq.4.70) where ( ) {0,1 } k dt  represents the detector outputs from photon detector k. () k dt is only dependent on the number of incident photons on detector k ( () k nt ). N is total the number of samples or integration time. Therefore, the expectation value of the two-time correlation 21 ( ) ( ) d t d t     gives directly the estimate for (2) () g  in this ideal case. Now, considering the combined dead time, t d , of the detector and the processing electronics, a more accurate expression for h(τ) can be written as [4.21], max( , ) 1 1 2 1 111 ( ) ( ) [1 ( )] ( ) [1 ( )] dd tt N tjl h d t d t j d t d t l              (Eq.4.71) In Eq. 4.71, t d is assumed to be a constant. Any previous photon that gets in to detector 1 in less than the time t d is assumed to be recorded in the measurement. The product terms of (1-d) simply indicate that a photon cannot be recorded if a previous photon was detected on the same detector less than time t d ago. The condition that only one time interval can be recorded as a given start count on detector 1 has been imposed in Eq. 4.71. The expectation value of h(τ) then has the following form [4.21]: 267 1 2 1 1 2 11 max( , ) 1 2 2 1 max( , ) 1 1 2 1 11 ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] d d dd t N tj t l tt jj h d t d t d t d t j d t d t d t d t l d t d t j d t d t l                                     (Eq.4.72) Only the first term, the two-time correlation, contains information about (2) () g  . It can be expanded in terms of the incident photon numbers on the detectors as shown below: ( ) ( ) 1 2 1 2 1 2 00 ( ) ( ) Pr ( ( ) ( ) ) uv uv d t d t ob n t u n t v                (Eq.4.73) where () u k  represents the probability that detector k detects a photon given that u photons are incident. Correspondingly, (0) kd d   shows the dark count probability of the detector. If the original source is sufficiently attenuated with the number of photons getting into either detector being no more than one, Eq. 4.73 can be rewritten as, 2 12 1 2 1 2 1 2 1 2 ( ) ( ) ( ) Pr ( ( ) 1 ( ) 1) dd d t d t d n n d ob n t n t                  (Eq.4.74) where k n is the mean number of incident photon on detector k, 1  and 2  are the single photon detection efficiency of detector 1 and 2 respectively. From Eq. 4.69, we have (2) 12 12 ( ) Pr ( ( ) 1 ( ) 1) n n g ob n t n t       , and therefore Eq. 4.74 can be rewritten as , 2 (2) 1 2 1 2 1 2 1 2 1 2 ( ) ( ) ( ) ( ) dd d t d t d n n d n n g              (Eq.4.75) 268 Therefore, the information about (2) () g  is contained in 12 ( ) ( ) d t d t     but convoluted with dark count signal from detectors compared with the ideal case discussed before. Similarly, the first three-time correlation term in Eq. 4.72 are related to the three photon correlation function and can thus be represented in the form of 2 2 (3) 12 12 (0, , ) d t n n g j     . If the original source is sufficiently attenuated with 1 1 1 d tn   , three-time correlation term can be ignored. Therefore, the Eq. 4.72 can be simplified to be containing only the two-time correlation term when the number of photons expected to arrive in time intervals t d and τ are less than one with 1 1 1 d tn   and 2 2 1 n   . We can then have [4.21], 2 (2) 1 2 1 2 1 2 1 2 1 2 11 ( ) ( ) ( ) ( ) ( ) NN dd tt h t d t d t d n n d n n g                 (Eq.4.76) The photon correlation histogram measured using HBT setup can produce an accurate estimated of (2) () g  as captured in Eq. 4.76 when the probability of receiving a single photon in time interval t d and τ must be small and the source are sufficiently attenuated. In the measurement of correlation histogram of photons from MTSQDs, the recorded count rate of APD k , k kk rn   , under employed measurement condition with 80MHz pulsed excitation, is always less than 1000 c/sec and the probability of detecting one photon per pulse is thus less than 10 -4 . Given the dead time ~40ns of the used silicon APDs in the setup, the probability of detecting 269 one photon within 40ns is also a really small number. Therefore, the above two conditions are satisfied for our measurement on MTSQDs. The expected number of counts in a histogram with time bin of width t  can thus be written as, 2 (2) 1 2 1 2 ( ) [ ( ) ( )] m d d h T t r r r r rr g        (Eq.4.77) where T m is the integration time for the measurement, r d is the dark count rate, r 1 and r 2 are the photon count rate at two detectors. In the pulsed excitation scheme used in this dissertation, the discretized version of (2) () g  defined in Eq. 4.69 can be estimated from the measured histogram by accounting for the measured number of coincidence counts within each peaks shown in the histogram other than the counts in the time-bin of the multichannel analyzer: 2 (2) 1 2 1 2 [ ] [ ( ) [ ]] m d d h i T r r r r rr g i      (Eq.4.78) where  represents the time span covering entire peak i. Hence, (2) (0) g is obtained by subtracting the contribution of APDs dark counts and then estimating the ratio of the coincidence counts within the peak (integrated area of the peak) at τ=0 and that of the average coincidence counts with in the peaks at ,0 iT i  following Eq. 4.78. By measuring (2) [0] g , one can place an upper bound on the probability to generate two or more photons from MTSQDs in a single pulse 2 (2) 1 Pr ( 2) [0] 2 ob n n g  where n is the mean photon number per pulse.  Linewidth of measured histogram versus QD lifetime Besides the reliable estimation of (2) (0) g that can be extracted from the photon correlation histogram, it can also provide information on the QD emission 270 life time. Due to the limited timing resolution of the HBT setup, the measured histogram, h(τ), is a convolution of photon correlation histogram of the QD emission, f(τ) and the IRF as shown below: ( ) ( ) ( ) h IRF t f t dt    (Eq.4.79) The photon correlation histogram of the QD emission f(τ) is directly related to the correlation of photon numbers from QD at two different times t 1 and t 2 with 21 tt  : 12 ( ) ( ) ( ) f n t n t      (Eq.4.80) The number of photons from QD follows single exponential decay assuming QD as a simple two level system with electron and hole population fully correlated (details in Appendix A). The number of photons from QD, n k (t), at different time can be expressed as, ( ) exp( / ) QD n t c t   (Eq.4.81) where τ QD is the QD decay time. Given Eq. 4.79 -4.80, the variance of peaks in h(τ) can be represented as the sum of variance of f(τ) and the IRF: 2 2 2 2 2 ( ) ( ) ( ) 2 ( ) QD h f IRF IRF              (Eq.4.82) where σ 2 represents variance of the evolved funciton. Hence, the linewidth of peaks in h(τ) assuming a Gaussian shape of the peaks can be captured as, 22 ( ) 8ln 2 ( ) QD h IRF        (Eq.4.83) where () h   and () IRF   represents the linewidth (FWHM, full width at half maximum) of peaks in h(τ) and IRF respectively. The 8ln2 is the conversion factor from variance to FWHM. Therefore, the QD decay dynamics controls the shape of 271 peaks in the measured photon correlation histograms. The QD decay lifetime can be obtained by deconvoluting the line width of peaks in the measured photon correlation histograms using Eq. 4.83. §4.3.3 Single photon emission from MTSQDs: Measurements and Purity In this subsection we report on the single photon emission properties of MTSQDs determined by measuring the second order intensity correlation of emitted photons from MTSQDs using our HBT setup [4.18]. All data shown in this section are taken with 640 nm above gap excitation with 93 ps pulse at a repetition rate of 80 MHz at the lowest possible power and detected at each MTSQD’s PL peak wavelength with the spectrometer bandpass set at 0.4 nm. The bandpass is set to be the minimum it can go to with the aim of selecting single neutral exciton transition state given the limitation of low QD emission detection event on the detector. The photoluminescence from the MTSQD is spectrally filtered and directed through a 50/50 beam splitter to two Si APD detectors, one at each end of the two beams. The timing of the pulses from the two detectors is registered as the start and stop signal (stop signal delayed by a delayer) of the timing electronics. The outputs are read by a multi-channel analyzer to generate a histogram of coincidence photon detection events. Due to the limited resources and the prohibitive expense of liquid helium, studies of the single photon emission behavior of the MTSQDs at liquid helium temperature (~8K) had to be restricted to three different MTSQDs in the 5 × 8 array that are marked with green circles in the color coded figures 4.8 (a) and (b) of, 272 respectively, the MTSQD PL emission intensity and PL emission wavelength shown below. The coincident photon detection histogram is measured with time bin of width 32.5 t ps   set by the multichannel analyzer. Figure 4.9 shows the histograms of as-measured coincident detection of photon emission plotted with time bin width Figure 4.8. Color coded plot of (a) PL peak intensity and (b) PL peak wavelength of each MTSQD in the 5×8 array shown as blocks. The two non-emitting MTSQDs are marked as black blocks with white outlines. MTSQDs marked by green circles are those examined for g (2) (τ). 1.95 t ns   . Under pulsed excitation, the discretized version of the photon correlation function defined in Eq. 4.69 [4.19, 4.20] is measured and obtained from the measured histogram by accounting the measured number of coincidence counts within each peaks shown in the histogram using Eq. 4.78. Data plotted with time bin width larger than the bin width of the multichannel analyzer thus do not compromise the physics of interest and thus this is a standard practice in the literature [4.21, 4.22]. The histogram shown in Fig.4.9 are measured with excitation power / ~ 45% sat PP from MTSQD (3,5) (panel (a), excitation power P=8nW and power density 0.64W/cm 2 ), MTSQD (2, 4) (panel (b), excitation power P=10nW and power density 0.8W/cm 2 ) and MTSQD (4,4) (panel (c), excitation power P=6nW and power density 273 of 0.5W/cm 2 ). The histogram from MTSQD (3,5) and (2,4) are measured with integration over 8hrs while the one from MTSQD (4,4) are measured with integration over 2hrs limited by the liquid helium amount we had. Figure 4.9. Single Photon emission from MTSQDs. The 8 K as-measured coincidence count histogram of (a) MTSQD (3,5), (b) MTSQD (2,4) and (c) MTSQD (4,4) measured with 640nm, 80MHz laser at excitation power / ~ 45% sat PP . The intensity autocorrelation g (2) (0) values shown are extracted from the as-measured data and the values in parenthesis are extracted from data after detector dark count subtraction. The 8K as-measured data from MTSQD (3,5), (2,4) and (4,4) give normalized g (2) (0) of 0.2, 0.25 and 0.27 respectively, determined by the ratio of τ=0 274 peak area to the average of the nonzero peak areas from the measured histograms in Fig. 4.9. In the histogram measurements, the APDs used (Excellitas, model SPCM- NIR-14-FC) have dark counts ~50/sec. The dark counts of APDs contribute to a flat non-zero background of the measured histogram represented by the first two terms in [] hi of Eq. 4.78 that need to be corrected. The corrected g (2) (0) for MTSQD (3,5), (2,4) and (4,4) with detector dark count subtracted is 0.15, 0.19 and 0.20, respectively. The three MTSQDs examined at 8K show an average g (2) (0) ~ 0.19±0.03 extracted from data with the detector dark count subtracted. It suggests that the single photon emission purity of MTSQDs is around (2) 1 (0) 90 2% g    . To examine the MTSQDs potential use as single photon sources at higher temperatures, the photon correlation measurement has been carried out at 77K on four MTSQDs, including the three QDs examined at 8K. The one additional MTSQD (3,8) studied at 77K is marked as red circle in Fig. 4.8 (a) and (b). Figure 4.10 shows the histograms of as-measured coincident detection of photon emission plotted with time bin width 1 t ns   of MTSQDs (3,5), (2,4), (4,4), and (3,8). The data are all collected with excitation power / ~ 23% sat PP . Data on MTSQDs (3,5), (2,4) and (4,4) are collected using APDs from Excellitas (dark counts ~50/sec) while the last one on MTSQD (3,8) is collected using APDs from Picoquant (dark counts ~150/sec). The corresponding g (2) (0) extracted from as-measured and detector dark count subtracted data are 0.43 / 0.38, 0.38 / 0.32, 0.46 / 0.41, 0.36 / 0.29. The four MTSQDs examined at 77K give on average g (2) (0) ~ 0.35±0.06. The variation of the average peak counts in the histogram amongst the four MTSQDs is tied to the 275 difference of the QD emission efficiency and the integration time used. The single photon emission purity of MTSQDs at 77K is 81 2%  . Figure. 4.10. Single Photon emission from MTSQDs. The 77 K as-measured coincidence count histogram of (a) MTSQD (3,5), (b) MTSQD (2,4), (c) MTSQD (4,4) and (d) MTSQD (3,8) measured with 640nm, 80MHZ laser at excitation power / ~ 23% sat PP . The intensity autocorrelation g (2) (0) values shown are extracted from the as-measured data and the values in parenthesis are extracted from data after detector dark count subtraction. Thus these SESRE based MTSQDs provide single photon emission comparable to that of InGaAs SAQDs reported in the literature up to 77 K [4.23, 4.24] but with added control on the QD position and significantly improved spectral 276 uniformity over the typical used InGaAs SAQDs [4.25]. These attributes, and the ease of growing a planarizing overlayer for integrating with light manipulating elements, make SESRE MTSQDs well suited for nanophotonic on-chip integrable single photon source array for building photonic circuits. §4.3.4 Single photon emission rate and collection efficiency Besides single photon emission purity, single photon emission rate and emission collection efficiency are the two other important figures of merit for single photon sources. To obtain independent estimate of single photon emission rate of these MTSQDs, the lifetimes of exciton decay have been extracted using Eq. 4.105 and the measured photon correlation histograms at 77K and 8K for the four above mentioned MTSQDs. Figure 4.11 shows the extracted MTSQDs’ lifetime at (a) 77K and (b) 8K. The exciton lifetime extracted from the histogram data is of 1.1±0.1ns at 77K and is of 0.91±0.1ns at 8K. The extracted lifetime is consistent with the MTSQD lifetime obtained from the time-resolved PL measurements of MTSQDs [4.18, 4.26]. The short (~1 ns) excitonic lifetime in these MTSQDs suggests that the MTSQDs are capable of acting as single photon emission source with GHz operating frequencies. However, given the measured g (2) (0) and single photon emission purity of MTSQDs, the maximum emission rate of pure single photons that can be extracted with continuous pumping is limited to 800-900MHz. 277 Figure 4.11. MTSQD lifetime extracted from photon correlation histogram data at (a) 77K and (b) 8K. The numbers in the parenthesis indicate the MTSQD location in the 5 × 8 array.  Single photon emission collection efficiency In this section, we assess the collection efficiency of the single photons emitted from the MTSQD structure in our measurement setup. In an on-chip integrated system, efficient collection of the emitted photons via light manipulating device such as a nanoantenna that directs and feeds efficiently into ideally loss-less waveguide is the basic challenge [4.6, 4.27]. For horizontal architectures, selected single self-assembled quantum dot from a spatially random distribution have so far served as the photon source around which photonic 2D crystal based cavity and waveguide have been fabricated and studied [4.27]. Of course, the SESRE based MTSQDs completely eliminate the spatial location randomness and thus will enable not only the fabrication of the required light manipulating elements around each but, for the first time, allow for communication between quantum dots and thus enabling the first step towards quantum optical circuits. Thus assessing the photon collection 278 efficiency of these very first MTSQDs even without any light manipulating elements sets the baseline from which improvements would need to be incorporated as further discussed in subsection 4.3.5 below. The number of single photons emitted from the MTSQD and collected per second into the objective in our micro-PL setup labeled as n c is determined by the following relation: (2) det 1 (0) / c APD n n g     (Eq. 4.84) where n APD is the number per second of photon detection events recorded by the APD and η det is the detection efficiency of our micro-PL setup defined as the ratio of the number of photons detected by the APD and the number of photons collected in the microscope through the objective lens. Under the employed g (2) (0) measurement condition, the photon detection rate at Si APD is ~600-800/sec. Considering the detection efficiency of 1.13% of our micro-PL setup and the measured g (2) (0) ~0.2 for MTSQDs at 8K, the rate of single photons emitted from MTSQD and collected into the microscope objective in the micro-PL setup is ~4.55-6.35E4 /sec. Detailed discussion of setup detection efficiency is captured in Appendix C. Accounting for the transmission efficiency of the objective (~60%) and glass window of the cryostat (~90%), the overall number of single photons from MTSQD emitted within the collection cone angle of the objective is 8.4 E4-1.2 E5/ sec. With pulsed 80MHz excitation, the total number of photons that can be emitted from the MTSQDs is limited by the pulsed excitation frequency given the 1ns decay lifetime of MTSQDs. Thus, the total rate of photons emitted from 279 MTSQD is 8E7 /sec when QD has unity quantum efficiency and is optically pumped at QD saturation power, the power needed for creating one electron-hole pair per pulse. At the excitation power ~ / ~ 45% sat PP in the measurements discussed in previous section, the total number of photons emitted from the MTSQD is estimated to be ~ 3.6E7 /sec. Correspondingly, the total number of single photons from the MTSQD is ~3.2E7 /sec given the purity of single photons from the QD. Therefore, there is only 0.26%-0.38% of the total emitted single photons from MTSQDs propagating in the direction within the collection cone of the objective for subsequent detection in our measurement setup. This low fraction can have two contributing factors (1) the modification of QD emission pattern due to the geometry of the mesa that causes large fraction of photon to be directed to the substrate (2) the low quantum efficiency of the MTSQDs. To shed light on the cause of the low collection efficiency of photons from MTSQDs, calculation of Poynting vector of the electromagnetic field energy flux from the QD with QD represented as in-plane dipole with dipole moment of 1 Debye strength along [1-10] was carried out using the geometry shown in Fig. 4.12 (a) with the electromagnetic field calculated using COMSOL. Figure 4.12 (b) shows the calculated Poynting vector of the electromagnetic field energy flux from QD with an in-plane dipole along the crystal direction [1-10] representing the HH dominated bright exciton transition at 930nm. The calculated energy flux is a representation of photon number flux from the MTSQD. This analysis is courtesy of Mr Swarnabha Chattaraj in our group. Given the MTSQD growth structure discussed in Chapter 2, 280 the QD dipole (red arrow) is placed at ~56nm below the apex of the pinched mesa top accounting for the 200ML GaAs capping layer grown after the mesa-pinch off. The mesa with MTSQD on top is shown as thin black line in Fig. 4.12 (a). From the 3D polar plot of the Poynting vector shown in Fig. 4.12 (b), majority of the photons emitted represented as energy flux are directed downwards through the mesa towards the GaAs substrate underneath. The GaAs mesa holding the QD acts as a waveguide and directs most of the energy flux toward the GaAs substrate. Figure 4.12 (c) shows the calculated fraction of energy flux towards the collection cone of the objective, η obj , as a function of QD emission wavelength. η obj is found to be (0.28±0.025)% and is slightly decreasing with increasing MTSQD emission wavelength. From this simulation, we infer that the GaAs mesa guiding the emitted photons towards the substrate is the limiting factor for the observed low detection and collection count of the single photons from MTSQD in the backscattering measurement geometry. The closeness between the calculated η obj and the estimated fraction of photons from MTSQDs collected into measurement setup suggests that the quantum efficiency of our MTSQDs is near unity, consistent with the inference drawn based on the observed low saturation power discussed in Chapter 3. 281 Figure 4.12. The 3D polar plot of the calculated Poynting vector (panel b) of the electromagnetic field energy flux from the QD with QD represented as in-plane dipole with dipole moment of 1 Debye strength along [1-10]. The schematic representation of the calculation geometry is shown in panel (a). The corresponding ratio of emitted photons from QD into the collection cone (red circle) of the objective and the overall emitted photon (integrated over blue circle) is shown in panel (c) as a function of QD emission wavelength. §4.3.5 Enhancement of emission rate and collection efficiency The collection efficiency of single photon emission from MTSQDs and the single photon emission rate can be enhanced by integrating the MTSQDs with light manipulating elements (LME) through planarizing overgrowth discussed in Chapter 1. All classes of semiconductor quantum dots examined to-date have a natural spontaneous decay time of ~1ns and thus require external means of shortening the 282 decay time to enhance the emission rate to ~10GHz, a value thought to be the minimum needed for useful optical circuits [4.27]. The conventional approaches to achieving this is by placing the QD in a pillar or wire configuration in a cavity made of Bragg mirrors [4.6,4.23,4.28-4.30] or creating a 2D photonic crystal based defect- cavity around the QD [4.27,4.31-4.35]. Such studies have established enhancements of ~10, thus potentially making QDs a viable on-demand source of single photons. Of course, the photonic 2D crystal based light manipulating element approach is fully compatible with the SESRE based MTSQD arrays—indeed it has awaited such arrays. However, as noted in Chapter 1, our group has recently proposed a third approach that lends itself naturally to creating regular arrays of QD-LME units, the basic building block of optical circuits. Below we briefly discuss this new approach. The new QD-LME integration paradigm explored in the group uses all dielectric building block (DBB) based structures as LMEs. This alternative approach exploits the physics of the collective Mie resonances of interacting sub-wavelength sized DBBs to generate the needed multiple light manipulating functions noted above (i) SQD emission rate enhancement, (ii) directional emission, (iii) lossless propagation using a single collective mode of the whole system—a true light manipulating unit [4.16, 4.36]. An illustration of this new paradigm is shown in Fig.4.13. The simulated structure comprises all DBBs (spheres) in the top panel. Plotted in the three panels below the top one is the electric field E y in the x-y plane associated with the collective magnetic dipole mode of the entire structure (the LMU) designed to be at 980nm as a function of position along x axis. Three functions are 283 clearly visible: the directivity induced by the local antenna effect near the QD location, the lossless propagation implied by the non-decaying nature of the field along the chain. Additionally, at the QD location, at the modeled dipole emission wavelength of 980nm, there is a Purcell enhancement of the electric field of ~7. Indeed, recently these simulations have been extended to include splitting of the beam and combining two beams from two adjacent and parallel rows [4.37]. This suggests a pathway for implementing and studying photon interference in between different MTSQD-DBB LME units. The work on creating MTSQD-DBB integrated structure is an on-going effort in the group and is discussed in detail in Chapter 5. Figure 4.13 Shows the interacting dielectric building blocks (spheres) based structure whose collective single Mie resonance (here the magnetic dipole mode at 980nm 284 arising from GaAs spheres) provides the shown spatial distribution (lower three panels) of the E y component in the x-y plane as a function of position along the x axis. The directivity induced by the nanoantenna effect is unmistakable in the second panel. The lossless propagation is manifest in the constant electric field in the propagation region. Not visible is the enhancement of the Purcell factor at the SQD (pyramid) location by ~7. Note that all three spatial-region-dependent functions arise from a single (collective) mode of the structure made of the interacting dielectric building blocks which we therefore dub light manipulation unit (LMU). §4.4 Physical processes affecting single photon emission from MTSQDs In this section we focus on the physical processes affecting the single photon emission purity of the synthesized MTSQDs to understand the origin of the observed non-zero g (2) (0) from MTSQDs. From the study of photon correlation histogram measured from MTSQDs at 8K and 77K, it is found that the probability of emitting two or more photons in one pulse from MTSQS is reduced to 0.1 to 0.2 times compared to that for a Poisson source of the same intensity. The remaining two-photon events we observed from MTSQDs may have several possible origins. First, it is possible that the quantum dot could be re-excited after it emits a photon. If a new electron-hole pair is added to the dot in the same light pulse after a one-exciton photon has already been emitted, a second photon can be emitted at the one-exciton wavelength, and no amount of spectral filtering can protect against this. The attainable degree of two-photon suppression is limited by the ratio of the duration of the excitation process to the spontaneous-emission lifetime. Given the laser pulse used is 93 ps in duration, and the relaxation of the generated electron-hole pair to the lowest-energy state takes tens of picoseconds or less [4.22], such re-excitation events might have showed up. 285 Second, it is possible that we collect some emission that is not from the a single one- exciton line given the limitation of our spectral window of the measurement 0.4nm (~700 μeV around MTSQD emission region). The asymmetry of the confinement potential due to QD shape asymmetry, piezoelectric and strain effect, can break the degeneracy of heavy hole states and cause mixing of heavy hole and light holes states resulting in four non-degenerate finely separated exciton lines with energy separation of a few to hundreds of micro-electron volts as discussed in Chapter 3. The collection of more than one of such lines in our instrument can lead to the observed two-photon emission events from MTSQDs. Finally, the multi-exciton emission, especially biexciton emission, can cause the collection of two-photon event from the QDs. We address the effect of the above mentioned factors below.  Re-excitation of QD The probability of re-excitation of the QD in the same light pulse is related to the relative time scale of pulse duration, the relaxation time, t 1 , for an electron-hole pair to get to the QD lowest energy state and the spontaneous decay time, t d . The relaxation time and QD decay time, t d , can be measured through time-resolved PL measurement discussed in Chapter 3. From the time-resolved PL data collected with 640nm excitation (beyond GaAs bandgap) shown in Chapter 3, we find that the relaxation time for excitons in the GaAs matrix to get to MTSQDs lowest confined energy is shorter than the IRF time (~200ps) while the MTSQD decay time is ~1ns. No clear guidance on the relaxation time of MTSQDs can be obtained. The large 286 lifetime of 1ns suggests that the probability of re-excitation event with 93ps pulsed excitation should be less than 5%. To check the probability of re-excitation event in MTSQDs and its effect on single photon emission purity, the photon correlation histogram of MTSQD (3,5) picked out of the four MTSQDs studied at 77K is measured using 850nm (below GaAs band gap,~833nm at 77K) 76MHz pulsed excitation with pulse width of ~3 ps. Given the laser pulse is only 3 ps in duration, the attainable degree of two-photon suppression is higher in this case compared with the 93 ps pulsed used for data shown in Fig. 4.9 and Fig. 4.10. Figure 4.14 shows the histograms of as-measured coincident detection of photon emission plotted with time bin width 1 t ns   of MTSQDs (3,5) collected with excitation power / ~ 23% sat PP , same as the one used for data shown in Fig. 4.10. The corresponding g (2) (0) extracted from the as- measured and after detector dark count subtraction data is 0.35 / 0.31. Compared with the g (2) (0) value from the data collected with 93 ps duration, the two-photon detection probability ( (2) 1 1 (0) g  ) has dropped from 21.2% to 16.9%. The probability of re-excitating QD in the same light pulse is a very small probability, ~5% and QD re-excitation is not the major contributing factor liming MTSQDs single photon emission with g (2) (0)~0.3-0.4 at 77K. The observed g (2) (0)~0.3 with 3ps excitation pulse and the small contribution of re-excitation of QD to the non-zero g (2) (0) also suggests that the relaxation time for excitons inside the QD is probably of the order of tens of picosecond. 287 Figure 4.14. The 77 K as-measured coincidence count histogram of MTSQD (3,5), measured with 850nm, 76MHz laser with excitation power / ~ 23% sat PP . The intensity autocorrelation g (2) (0) values shown are extracted from the as-measured data and the values in parenthesis are extracted from data after detector dark count subtraction.  Multistate emission The nonzero g (2) (0) value at low excitation powers ( / 45% sat PP  ) and the minimal effect of re-excitation of QDs suggests that there is more than one initial state of the electron participating in the creation of the emitted photons and in turn limits the g (2) (0) to ~0.3. Indeed, the PL behavior of all investigated MTSQDs at 8 K, shown here for MTSQDs (3,5), (2,4) and (4,4) in Fig.4.15 (a), (b) and (c), respectively, reveals a lineshape that requires fitting of at least two Lorentizian peaks separated by ~ 0.25 nm (363 μeV,) and a peak area ratio ~5-7 within the detection 288 window marked as dotted line. The peaks are not clearly separated and revealed due to the instrument resolution ~0.2nm (290 μeV). Figure 4.15 Photoluminescence (PL) spectra of (a) MTSQD (3,5) , (b) MTSQD (2,4) and (c) MTSQD (4,4) collected at 8 K with 0.25 nm spectral resolution and ~5 nW excitation power (power density ~0.4W/cm 2 , ~23% of P sat ). Black dots represent the raw data and the green line shows a fit with two individual Lorentzian peaks ( 1 and 2 ) shown as the red curves. Now consider a simple case of two single photon emitters with emitter response function as delta function, emission from these two emitters are both collected into the HBT setup for the measurement of g (2) (0). The state of light coming from these two emitters into the beam splitter in HBT setup can be written as 1 1 1 2 1 1 1 2 | | 0 |1 |1 |1 |1 in k k k k p p p p            (Eq. 4.85) 289 where 11 11 , ss nn pp FF  . n 1 and n 2 represents the number of photons from emitter 1 and 2 per second. F s represents the excitation laser repetition rate. Eq. 4.85 can be written in terms of vacuum state and field creation operators as 1 2 1 2 1 2 1 2 | (1 ) | 0 in k k k k p a p a p p a a            (Eq. 4.86) The output photon state after the beam splitter, using Eq. 4.59 and Eq. 4.86 can be written as 1 2 1 2 1 2 1 2 1 2 1 2 | [1 ( ) ( ) ( )( )]| 0 2 22 out p p p p b ib b ib b ib b ib                    (Eq.4.87) where 1 () bt  and 2 () bt  are the creation operator for the field mode propagating to detector 1 and detector 2 respectively. From Eq. 4.69, (2) (0) g can be written as the ratio of the probability that both detectors 1 and 2 detect a photon and the probability that only one of the detectors detects a photon. Thus, the (2) (0) g of the output photon state shown in Eq. 4.87 is (2) 1 2 2 1 1 1 2 2 22 1 2 1 2 2 2 2 2 2 2 1 2 1 2 || (0) | | | | 2( / 2) 2( ) [( ) / 2] ( ) out out out out out out b b b b g b b b b p p p p p p p p                (Eq.4.88) When p 1 and p 2 both are not zero, (2) (0) 0 g  and the largest (2) (0) g for such a output state is (2) (0) 0.5 g  when 12 pp  . The existence of more than one single 290 photon emission states within the spectral detection window can cause the observed non-zero (2) (0) g . To check the consistency of the observed intensity ratio of the two finely separated QD emission and the observed non-zero (2) (0) g value, we consider the photon state emitted from QD optical transitions. Following Eq. 4.54, the photon states, 1 |   and 2 |   coming from two transitions with frequency 1  and 2  , respectively, can be written as 1 1 1 1 11 1 | | 0 ( ) / 2 k it k k k e Aa i             and 2 2 2 2 22 2 | | 0 ( ) / 2 k it k k k e Aa i             where 2 1 || A and 2 2 || A is proportional to total number of photons for all the possible wavevectors 1 k and 2 k . Thus, the input photon states from the collected QD emission that contains two transitions can be written as 12 12 12 12 12 12 | (1 )(1 ) | 0 ( ) / 2 ( ) / 2 kk i t i t in kk kk kk ee A a A a ii                     (Eq. 4.89) Following similar calculation approach for emitters with emitter response function as delta function, we find, 00 00 12 22 1 2 1 2 1 2 2 1 2 2 2 2 12 12 | | | | || 2 ( ) / 4 ( ) / 4 | || | 2 k k k k out out k k k k kk A A dk dk b b b b II                          (Eq. 4.90) 291 where 0 0 1 2 1 11 22 1 | | | | ( ) / 4 kk kk k dk IA         and 0 0 2 2 2 22 22 2 | | | | ( ) / 4 kk kk k dk IA         . The 0 k represents the detection position for the measurement, 2 k  represents the detection window. Correspondingly, I 1 and I 2 represent the overall integrated intensity (number) of photons coming from the two finely separated peaks. Similarly, we have 1 2 1 2 1 1 2 2 | | | | | | | | | | | | ( )( ) 22 out out out out I I I I b b b b            (Eq. 4.91) Thus, (2) 12 2 12 2 | || | (0) (| | | |) II g II   (Eq. 4.92) Using the intensity ratio of 5-7 for the two peaks extracted from the data of Fig.4.14, a (2) (0) g of ~ 0.3 is obtained from Eq.4.92, consistent with the measured (2) (0) g values. From the PL studies on MTSQDs at 77K discussed in Chapter 3, we infer that the peaks shown in Fig. 4.15 (a)-(c) are most probably the neutral exciton transition from MTSQDs. The two finely separated peaks may be, thus, from the different recombination channels from one neutral exciton. To provide further evidence, we have performed excitation-power dependent PL measurements on MTSQD (3,5) at 8K and evaluated the emission intensity ratio of the two observed finely separated peaks as a function of excitation power. Fig. 4.16 shows the power dependent PL integrated area intensity of the two finely separated peaks (a) 1 and 292 (b) 2 and (c) their ratios 1 / 2 . The integrated area intensity of 1 and 2 peaks, I 1 and I 2 , both increase near linearly with excitation power with I 1 ~P 0.94 and I 2 ~P 0.75 as shown in Fig. 4.16 (a) and (b) respectively. In addition, the intensity ratios between these two peaks keep practically constant in the full range of excitation powers. This suggests that the observed two peaks are from different recombination channels from one neutral exciton [4.38]. Figure 4.16 Excitation power dependent integrated PL intensity of the two peaks (a) 1 and (b) 2 from MTSQDs (3,5) collected with 640nm, 80MHz excitation. The black line is a fit of measured data showing (a) I 1 ~P 0.94 and (b) I 2 ~P 0.75 . Panel (c) shows the ratio of the integrated area intensity of 1 and 2 . The two finely separated peaks are likely from the neutral excitons with QD ground hole state having HH-LH mixed character. We note that two bright excitons 293 from QDs with C 2v symmetry without HH-LH mixing have same transition dipole moments with an energy splitting, the so called the fine-structure splitting, typically on the order of 10-100 μeV [4.39] reported for the InGaAs/GaAs SAQDs of sizes and shape similar to these MTSQDs. The observation of two peaks with intensity ratio of two indicates that the two transitions have different dipole moments and suggests that they are not likely to come from finely separated bright excitons with HH dominated hole state character. The MTSQDs most likely have in-plane asymmetry in the confinement potential and a symmetry lower than C 2v . The energy splitting between the two bright excitons and the two dark excitons arising from the hole state with HH character is in the range of 200-300 μeV, similar to our observed energy separation [4.40, 4.41]. In the case of QD with confinement potential symmetry lower than C 2v , the hole state can have HH and LH mixed character. The HH-LH mixing makes the otherwise “dark” exciton of HH related exciton states to become “bright” excitons [4.42]. Following the notation in Chapter 3, the QD hole state can be represented by, 3 3 3 1 | | , | , 2 2 2 2 h E           (Eq. 4.93) The two bright excitons are 11 | | , 22 h ep     and 11 | | , 22 h ep      with dipole moments 11 | | , ( ) ( ) 2226 h x y x y e p e ie e ie E           . The two otherwise “dark” exciton are 11 | | , 22 h ep      and 11 | | , 22 h ep     with 294 dipole moments 1 1 2 | | , 22 6 hz e p e E          , different from the bright excitons. The observed energy separation and the intensity ratio of the emission between peak 1 and 2 suggest that these two peaks may be the “bright” and “dark” excitons coming from transition with HH-LH mixed character consistent with the observed elliptical polarization dependent PL of the combined behavior of these peaks at 77K. Further measurements of polarization dependent PL emission of the observed peaks and 2 at higher instrument resolution (~50μeV or less) need to be carried out to verify the nature of these two peaks. The energy separation between the two “bright” excitons is much smaller than the instrument resolution and cannot be observed. From the PL spectra shown in Fig. 4.15, we can infer from the line widths of peak 1 and 2 that the energy separation between them is probably less than 100 μeV. Further measurements at higher instrument resolution (~50μeV or less) need to be carried out to identify those transitions.  Multiple photon Emission from the Biexciton State Given the near constant ratio of the PL intensity of 1 and 2 peaks (shown in Fig. 4.16 within the detection window 0.4nm for the g (2) (0) measurement, the g (2) (0) of MTSQD (3,5) should not change as the excitation power P . However, the g (2) (0) of MTSQD (3,5) at 77K and 8K increases with excitation power. Figure 4.17 shows the background corrected g (2) (0) of MTSQD (3,5) measured at 77K 640nm 80MHz excitation and 0.4nm spectral window. With increasing excitation power, the 295 measured g (2) (0) approaches 0.5 when excitation power is about 9nW (2.25W/cm 2 ) with / ~ 50% sat PP . At even higher excitation power, the MTSQD emitter approaches the classical behavior with g (2) (0) ~0.7. 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 g (2) (0) 77.4K 0 4 8 12 16 20 24 28 32 36 Power (nW) 0.0 0.4 0.8 1.2 1.6 2.0 Figure 4.17 Measured g (2) (0) dependence on excitation power for MTSQD (3,5) under 640nm 80MHz excitation with detection window of 0.4nm. The red solid line shows the fitted results based on Eq. 4.99. This observed behavior indicates the expected departure from single photon emission resulting from multiphoton emission from biexciton or multi-exciton states. From the PL studies discussed in Chapter 3, the probable biexciton emission around 933.4nm shows up and grows quadratically with excitation power when / 50% sat PP  . With increased possibility of creating biexcitons in the QDs, the biexciton can decay through a two-photon cascade process and can produce photons at energies within the spectral detection window that includes the single exciton 296 emission. Consider a QD with two excitonic states, |X> and |XX>, that can be excited with excitation pulse. The effect of a single excitation pulse is described by, 11 22 1 1 pp ab pp ab                   (Eq. 4.94) where 1 p and 2 p are, respectively, the probability for the QD to be in |X> or |XX> states, and a and b are the transition rates from |X> to |XX> and |XX> to |X>. If a photon was detected due to pulse 0, the system evolves as, 1 2 [] 1 (1 ) [] 1 m pm a a a a b pm b a b a b                    (Eq. 4.95) where p i [m] is the probability of the system to be in state i after pulse m. The probabilities a and b can be obtained considering the dynamical process in the QD captured schematically in Fig. 4.18. The system evolves in two steps. First, a strong optical field is applied for a short-time duration Δt, during which upward transitions are induced. Two of these transitions with rates α and β, change the total charge of the dot from neutral to charged and from charged to neutral respectively. Figure 4.18. Energy levels and transitions for the model described in the text. 297 These transitions correspond to the physical process of capturing single electrons and holes from the surrounding region into QD. The third transition with rate x, brings the QD directly to exciton state through capture of one electron-hole pair. During the second step, the system relaxes down to the ground level |0> through electron and hole recombination. Solving the dynamical equations for such a system [4.43], the following relations are obtained: ( ) ( ) (1 ), (1 ) tt a e b e                     (Eq. 4.96) and () 12 1, xt a e b          (Eq. 4.97) where 1  and 2  represent the probability of emitting a photon immediately after an excitation pulse given the QD was in state |X> or |XX> before the pulse. Then, the g (2) (0) value can be expressed as the probability that one photon emitted from the optically excited |X> state and the other from the cascade process of |XX> state: (2) 1 1 2 2 1 1 (0) ~ [ (0)] [ (0) (0)] g A p p p     (Eq. 4.98) where A is an empirical parameter capturing the detection efficiency of the setup and the MTSQD intrinsic g (2) (0) value at excitation power P=0. Given the empirical linear power dependence of the neutral exciton transition, () xt  can be assigned as ( ) / sat x t P P     . Then, combing the above relations, we have, / (2) 2 (0) ~ (1 ) sat PP g B e      (Eq. 4.99) 298 where 3 4 () BA     . The red line in the Fig. 4.17 above shows the fitting of g (2) (0) as a function of P/P sat with the fitting parameters B=0.786 and    =0.353. From the fitting, it is clear that multiphoton emission from biexciton states lead to the deviation of single photon emission from QDs at low excitation powers. From the above noted studies in this section, we conclude that the SESRE MTSQDs act as single photon emitters with g (2) (0) ~ 0.19±0.03 indicating single photon emission purity of around (2) 1 (0) 0.9 0.02 g    at 8K. Such ordered spectrally uniform MTSQDs’ single photon emission behavior holds at 77K with g (2) (0) ~ 0.35±0.06 and single photon emission purity ~ 0.81 0.02  . The low 0.26%- 0.38% of the total emitted single photons from MTSQDs being in directions within the collection cone angle of the optical measurement setup is due to the previously noted effect of the nanomesa guiding the photons through the mesa into the GaAs substrate as revealed in the simulations. This effect limits the number of single photons detected at the detectors to be <1000/sec. Thus the MTSQDs need to be integrated with features that prevent loss into the substrate such as a high refractive index discontinuity with an underlying layer combined with horizontal waveguiding. This will enhance photon collection efficiency necessary for the use of MTSQD single photon emitters in integrated QIP systems. The purity of single photon emission from MTSQDs is currently limited by the collection of two finely separated exciton states, probably from an exciton complex with mixed HH-LH character owing to the large (>0.2nm) spectral collection window imposed by the current 299 spectrometer in the instrumentation. As expected, the multiphoton emission from biexcitons plays a major role in affecting the robustness of MTSQDs as single photon emitters at high excitation powers close to the saturation power and leads MTSQDs to behave as classical emitter. In this dissertation work we regrettably could not test MTSQDs as entangled photon source. Given the limited instrument resolution, the energy splitting of the fine structure of bright excitons could not be determined but could only be inferred to be smaller than ~100μeV. Further high resolution spectroscopy studies need to be carried out to study the fine structure splitting and assess the use of MTSQD as entangled photon source. Given the knowledge of InGaAs SAQDs, the fine structure splitting is expected to be in the same range of 10-100 μeV. When QD lift time is shortened to be 10-100ps, the fine structural splitting would be smaller than the radiative linewidth. MTSQDs can have the potential of acting as entangled photon source using biexciton-excion cascade process. The potential use of MTSQDs in integrated QD-LME structures as emitter for entangled photon pair and its characterization is discussed in Chapter 5 under future work. Chapter 4 References: [4.1] M. O. Scully and M. S. Zubairy, Quantum optics, Cambridge University Press, 1997. [4.2] R. Loudon, The quantum theory of light, Oxford Science, 2000. [4.3] C. Santori, D. Fattal and Y. Yamamoto, Single-Photon Device and Application, Weinheim: Wiley-VCH , 2010. 300 [4.4] R. Hanbury Brown and R. Q. Twiss, “A Test of a New Type of Stellar Interferometer on Sirius”, Nature. 178, 1046 (1956). [4.5] R. M. Thompson, R. M. Stevenson, A. J. Shields, I. Farrer, C. J. Lobo, D. A. Ritchie and M. L. Leadbeater, “Single-photon emission from exciton complexes in individual quantum dots”, Phys. Rev. B. 64, 201302 (R) (2001). [4.6] S. Buckley, K. Rivoire and J. Vučković, “Engineered quantum dot single photon sources”, Rep. Prog. Phys. 75, 126503 (2012). [4.7] G. Shan, Z. Yin, C. H. Shek and W. Huang, “Single photon sources with single semiconductor quantum dots”, Front. Phys. 9, 170 (2014). [4.8] R. Heitz, M. Grundmann, N. N. Ledentsov, L. Eckey, M. Veit, D. Bimberg, V. M. Ustinov, A. Y. Egorov, A. E. Zhukov, P. S. Kop'ev and Zh. I. Alferov, “Multiphonon-relaxation processes in self-organized InAs/GaAs quantum dots”, Appl. Phys. Lett. 68, 361 (1996). [4.9] F. Adler, M. Geiger, A. Bauknecht, F. Scholz, H. Schweizer, M. H. Pilkuhn, B. Ohnesorge and A. Forchel, “Optical transitions and carrier relaxation in self- assembled InAs/GaAs quantum dots”, J. Appl. Phys. 80, 4019 (1996). [4.10] E. T. Jaynes and F. W. Cummings, “Comparison of Quantum and Semiclassical Radiation Theories with Application to the Beam Master”, Proc.IEEE .51, 89 (1963). [4.11] B. W. Shore and P. L. Knight. “The Jaynes-Cummings Model”, J. Mod. Opt. 40, 1195 (1993). [4.12] J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons ,1975. [4.13] S. L. Chuang, Physics of Optoelectronic Devices. New York: John Wiley & Sons ,1995. [4.14] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs”, Nature Photon. 8, 224 (2014). [4.15] G. Juska, V. Dimastrodonato, L. O. Mereni, A. Gocalinska and E. Pelucchi, “Towards quantum-dot arrays of entangled photon emitters”, Nature Photon. 7, 527 (2013). [4.16] C. Kurtsiefer, P. Zarda, S. Mayer and H. Weinfurter, “The breakdown flash of silicon avalanche photodiode- backdoor for eavesdropper attacks?” J. Mod. Opt. 48, 2039 (2001). 301 [4.17] W. Becker, Advanced time-correlated single photon counting techniques, Springer-Verlag Berlin Heidelberg, 2005. [4.18] J. Zhang, S. Chattaraj, S. Lu and A. Madhukar, “Mesa-top quantum dot single photon emitter arrays: Growth, optical characteristics, and the simulated optical response of integrated dielectric nanoantenna-waveguide systems”, J. Appl. Phys. 120, 243103 (2016). [4.19] M. Koashi, K. Kono, T. Hirano and M. Matsuoka, “Photon antibunching in pulsed squeezed light generated via parametric amplification”, Phys. Rev. Lett. 71, 1164 (1993). [4.20] E. Waks, C. Santori and Y. Yamamoto, “Security aspect of quantum key distribution with sub-Poisson light”, Phys. Rev. A. 66, 042315 (2002). [4.21] C. M. Santori, “Generation of nonclassical light using semiconductor quantum dots”, Ph.D. dissertation, Applied Physics, Stanford University, 2003. [4.22] P. Michler, Single semiconductor quantum dots, Springer-Verlag Berlin Heidelberg, 2009. [4.23] X. M. Dou, X. Y. Chang, B. Q. Sun, Y. H. Xiong, Z. C. Niu, S. S. Huang, H. Q. Ni, Y. Du, and J. B. Xia, “Single-photon-emitting diode at liquid nitrogen temperature,” Appl. Phys. Lett. 93,101107 (2008). [4.24] L. Dusanowki, M. Syperek, A. Maryński, L. H. Li, J. Misiewicz, S. Höfling, M. Kamp, A. Fiore, and G. Sęk, “Single photon emission up to liquid nitrogen temperature from charged excitons confined in GaAs-based epitaxial nanostructures,” Appl. Phys. Lett. 106, 233107 (2015). [4.25] M. Grundmann, Nano-Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. [4.26] J. Zhang, Z. Lingley, S. Lu, and A. Madhukar, “Nanotemplate-directed InGaAs/GaAs single quantum dots: Toward addressable single photon emitter arrays,” J. Vac. Sci. Technol. B 32, 02C106 (2014). [4.27] P. Lodahl, S. Mahmoodian and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures”, Rev. Mod. Phys. 87, 347 (2015). [4.28] N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gónez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G.White, L. Lanco and P. Senellart, “Near-optimal single-photon sources in the solid state”, Nature Photon. 10, 340 (2016). 302 [4.29] J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lelanne and J. M. Gerard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire”, Nature Photon. 4, 174 (2010). [4.30] M. E. Reimer, G. Bulgarini, N. Akopian, M. Hovevar, M. B. Bavinck, M. A. Verheijen, E. P.A.M. Bakkers, L. P. Kouwenhoven and V. Zwiller, “Bright single- photon sources in bottom-up tailored nanowires”, Nature Commun. 3, 737 (2012). [4.31] M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide”, Phys. Rev. Lett. 113, 093603 (2014). [4.32] A. Laucht, S. Pütz, T. Günthner, N. Hauke, R. Saive, S. Frédérick, M. Bichler, M.-C. Amann, A.W. Holleitner, M. Kaniber and J.J. Finley, “A waveguide-coupled on-chip single photon source”, Phys. Rev. X. 2, 011014 (2012). [4.33] A. Faraon, A. Majumdar, D. Englund, E. Kim, M. Bajcsy and J. Vučković, “Integrated quantum optical networks based on quantum dots and photonic crystals”, New J. Phys. 13, 055025 (2011). [4.34] A. Schwagmann, S. Kalliakos, I. Farrer, J.P. Griffiths, G. A. C. Jones, D. A. Ritchie and A. J. Shields, “On-chip single photon emission from an integrated semiconductor quantum dot into a photonic crystal waveguide”, Appl. Phys. Lett. 99, 261108 (2011). [4.35] S. Laurent, S. Varoutsis, L. L. Gratiet, A. Lemaître, I. Sagnes, F. Raineri, A. Levenson, I. Robert-Philip and I. Abram, “Indistinguishable single photons from a single-quantum dot in a two-dimensional photonic crystal cavity”, Appl. Phys. Lett. 86, 163107 (2005). [4.36] S. Chattaraj and A. Madhukar, “Multifunctional all-dielectric nano-optical systems using collective multipole Mie resonances: toward on-chip integrated nanophotonics”, J. Opt. Soc. Am. B. 33, 2414 (2016). [4.37] S. Chattaraj, J. Zhang, S. Lu and A. Madhukar, “On-chip quantum optical networks comprising co-designed and spectrally uniform single photon source array and dielectric light manipulating elements”, presented at IEEE summer topicals meeting, San Juan, Puerto Rico, July 10-12, 2017, Abstract number TuF1.4. [4.38] Y. H. Huo, B. J. Witek, S. Kumar, J. R. Cardenas, J. X. Zhang, N. Akopian, R. Singh, E. Zallo, R. Grifone, D. Kriegner, R. Trotta, F. Ding, J. Stangl, V. Zwiller, G. Bester, A. Rastelli and O. G. Schmidt, “ A light-hole exciton in a quantum dot”, Nature Phys. 10, 46 (2014). 303 [4.39] R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots”, Phys. Rev. Lett. 95, 257402 (2005). [4.40] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke and S. N. Walck, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots”, Phys. Rev. B. 65, 195315 (2002). [4.41] G. Bester, S. Nair and A. Zunger, “Pseudopotential calculation of the excitonic fine structure of million-atom self-assembled In 1-x Ga x As/GaAs quantum dots”, Phys. Rev. B. 67, 161306 (R) (2003). [4.42] A. V. Koudinov, I. A. Akimov, Y. G. Kusrayev and F. Henneberger, “Optical and magnetic anisotropies of the hole states in Stranski-Krastanov quantum dots”, Phys. Rev. B. 70, 241305 (R) (2004). [4.43] C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots”, Phys. Rev. B. 69, 205324 (2004). 304 Chapter 5. Conclusion and Future Work This dissertation was motivated by and contributes to the goal of exploring the potential of semiconductor single quantum dot (SQD) as viable single photon sources for enabling quantum optical circuits designed for quantum information processing (QIP). To this end, a holistic view of and approach to realizing nanophotonic systems comprising the SQD as the SPS, light manipulating elements (LMEs) (cavity, waveguide, etc.), and detectors for QIP is essential. However, much of the pertinent literature to-date is largely based on fabricating a cavity (mostly) and a waveguide around a single 3D island self-assembled quantum dot (SAQD) picked out of a large random ensemble of spectrally and spatially non-uniform SAQDs [5.1- 5.3]. For getting spatial regularity of the SQDs, the dominant work, however, is largely based on a SQD formed in nanowires fabricated in an array [5.4]. Such architectures enable vertical integration [5.4, 5.5] with cavity and waveguide within the nanowire but are essentially unsuited for creating interconnections between these basic building units needed for on-chip scalable quantum optical circuits. To facilitate movement towards the implementation of the needed on-chip scalable integrated quantum optical circuits, we envisioned and proposed [5.6] a scalable architecture (Fig. 5.1) comprising ordered single quantum dot based single photon source (SPS) realized via an on-chip scalable approach and integrable with either the currently employed photonic 2D crystal based light manipulating elements (cavity and waveguide typically) or, as we propose, with dielectric building block (DBB) based multifunctional light manipulation unit (LMU) that exploit the physics 305 of the collective Mie resonances of DBB assemblies. To this end, this dissertation has explored and demonstrated the realization of the spatially ordered arrays of spectrally uniform SQDs that act as single photon sources (yellow pyramids) and around which light manipulating units can be readily integrated for building scalable on-chip nanophotonic QIP systems as symbolized in Fig. 5.1. Figure 5.1 Schematic of the envisioned paradigm for realizing photonic quantum optical circuits based on an array of spatially ordered mesa-top quantum dots (pyramids) on-chip integrated with lossless dielectric building block (DBB) based light manipulating unit (LMU; blue elements) that provide multiple functions (resonant cavity/nanoantenna/waveguide) and beam splitters to direct light to detectors (purple elements). In this concluding chapter we summarize the main findings of the dissertation and provide some potential future research directions. The summary captures the inter-connected elements of the findings distributed over chapters 2, 3, and 4. Specifically it highlights: the design and growth of on-chip integrable single quantum dots in spatially regular arrays utilizing the substrate-encoded size-reducing epitaxy (SESRE) approach first introduced by this group some three decades ago (Ch.2); examination of the optical emission behavior of these mesa-top SQDs to establish the significantly improved spectral uniformity achieved in very first cut of the SESRE approach (Ch.3); examining the polarization characteristics of the emitted photons to narrow down, within the 0.2nm resolution limit of the current 306 instrumentation, the underlying electronic states involved in the observed multiple emission lines reflecting the presence of neutral single exciton, singly charged excitons, and the biexciton (Ch.3); setting up the instrumentation for examining the single photon emission characteristics and, through measurements of the photon emission statistics on selected four MTSQDs out of 40 in a 5×8 array, establish their single photon emission characteristics up to 77K and over 90% purity at 8K (Ch.4). The following provides a brief recapitulation of specific findings for each of the above noted four categories. §5.1 Realizing spatially regular array of spectrally uniform Single Quantum Dots:  The Substrate-Encoded Size-Reducing Epitaxy Approach Substrate-encoded size-reducing epitaxy (SESRE) is an approach to synthesizing quantum nanostructures in spatially-ordered arrays based on growth on substrates with pre-patterned arrays of nanoscale mesas. It exploits the nanomesa surface curvature induced surface stress gradients to preferentially drive adatoms, during growth, to migrate from the mesa sidewalls to the top to realize spatial selective formation of QDs on mesa tops. We implemented the SESRE approach employing solid source molecular beam epitaxy to synthesize- under reproducible growth conditions guided by our unique approach of utilizing the substrate surface reconstruction phase diagram- SQDs on the top of GaAs (001) nanomesas in 5×8 arrays fabricated using electron beam lithographically. The mesas have edges along 307 <100> direction with vertical side wall and lateral size of 50 to 500nm. The mesa orientation is chosen to enable adatom migration from side facets to the mesa top for the mesa top size-reducing growth. Through scanning electron microscope (SEM) and limited transmission electron microscope (TEM) studies of growths on such mesa structures, we found: (1) The profile of the growth front influenced by controlled sidewall-to-mesa-top inter-facet migration undergoes the (001) mesa-top size reduction with two pinch-off stages: first bounded by {103} facets; then, upon continued growth re-appearance of (001) mesa-top with the {103} sidewall facets evolving into {101} sidewalls; finally, with further deposition the {101} sidewall facets grow to cause the second pinch-off (Sec. 2.3.3). (2) The growth rate on the (001) facets varies as a function of deposition amount (Sec. 2.3.4). The mesa top growth rate is found to decrease with decreasing mesa top size. Such a variation of growth rate with reducing mesa top size is related to the change of net lateral adatom migration and possible changes in the local effective growth conditions (particularly “pressures”) on such small (<20nm) mesa top sizes induced by the lateral adatom migration itself. Interestingly, this variation may potentially self-limit the quantum dot size fluctuations arising from variations in the starting mesa size in the array and thus contribute to high spectral uniformity. Based on the varying growth rate, we found that the Ga atom migration length is of ~80-660nm and ~20-180nm along the {101} 308 and {103} surface (Sec. 2.3.4), respectively, before the {103} dominated mesa-top pinch off. §5.2 Ordered spectrally uniform mesa top single quantum dot array The two different mesa top pinch-off stages uncovered for SESRE growth on as-patterned <100> edge oriented GaAs (001) mesas were exploited to synthesize mesa top single quantum dot (MTSQD) with {103} side facets or {101} side facets. (1) From SEM and TEM studies we established that the synthesized truncated pyramidal MTSQDs with {103} side facets have a rhombus base with base edges along ~[1 -3 0] and [3 -1 0] directions (~22°±4° with respect to [1 -1 0] direction) (Sec. 2.3.3). By contrast, the MTSQDs with {101} side facets have square base with edges along <100> directions (Sec. 2.3.3). The former served as the main focus of the dissertation as deposition of In 0.5 Ga 0.5 As created MTSQDs that emit in the 920nm to 950nm range, the focus of this work owing to the limitation imposed by the Si APD detector efficiency being too low for wavelengths longer than ~980nm. The latter allowed creating binary InAs MTSQDs with the steeper {101} sidewalls and thus longer wavelengths near 1120nm. These are well suited for extension to the 1300nm and potentially 1550nm regimes of central importance to current optoelectronic communication technology platforms. In both cases, the growth controlled shape defined by the crystallographic facets and edges enables great control of the spectral uniformity of the optical 309 response from such SESRE QDs. Indeed, as recapitulated below, for the binary InAs, the emission uniformity across the 5×8 array is found to be over a factor of two better than the InGaAs alloy MTSQDs which, in part at least, is owing to the expected lack of alloy composition fluctuation. (2) Towards the objective of assessing the potential of such MTSQDs as SPSs and EPSs, we synthesized and systematically studied a 5 × 8 array of 4.25ML truncated pyramidal In 0.5 Ga 0.5 As MTSQD with {103} side facets of base length ~13nm and height ~3nm whose targeted QD emission wavelength is less than 950nm (Sec. 2.3.4). As noted above, the wavelength regime was chosen to allow the Si APD detectors in the Hanbury-Brown and Twiss (HBT) setup of our home-built micro-PL (photoluminescence) system to measure single photon emission with reasonable signal to noise over a practicable data acquisition time (fighting particularly against drift of the optical beam with respect to the MTSQD). The PL of all 40 MTSQDs in the 5 × 8 array was measured. Only 2 MTSQDs are dead while the rest 38 are bright with high emission yield. These show average emission wavelength ~935.6nm at 77K with spectral uniformity (standard deviation, σ λ ) of 8.3nm (Sec. 3.4.2), significantly improved over the typical SAQDs and comparable with the dominant nanowire based ordered QDs [5.7]. These MTSQDs represent our first attempt to synthesize ordered QDs using the SESRE approach. No attempt has been made to optimize the spectral uniformity. Nevertheless, strikingly, pairs of MTSQDs with emission wavelength difference within the 0.2nm instrument resolution are present in the synthesized array. This is highly encouraging for the 310 potential of MTSQDs providing a deterministic sufficiently spectrally uniform SPS array suitable for the creation of quantum optical circuits. With further optimization of the growth condition, the spectral uniformity can be further improved, especially through the use of binary InAs material for even the smaller volume {103} sidewall bounded MTSQDs. Anticipating the essential need for on-chip integration with light manipulating elements (LMEs), it is important to emphasize that the ordered and spectrally uniform MTSQDs can be overgrown with a planarizing GaAs layer, making them readily integrable with LMEs in a horizontal scalable architecture for creating interconnected QD-LME units (Fig. 5.1) that overcomes the limitation of integration faced by nanowire based ordered QDs with comparable uniformity. §5.3 Emission Characteristics and Implications for MTSQD Electronic States To gain insight into the nature of the electronic structure of the In 0.5 Ga 0.5 As MTSQD and its carrier dynamics we undertook systematic study of the PL, PLE (PL excitation) and TRPL (time-resolved PL) behavior. The time-integrated PL and PLE studies revealed the following regarding the electronic structure of MTSQDs: (1) The temperature and detection wavelength dependent PLE studies revealed that the MTSQD’s first excited electron and hole states are, respectively, around 40meV and 10meV higher than their QD ground states (Sec. 3.7). The sum of these PLE observed energy separations is consistent with the observed additional transition 311 at ~51meV higher than the ground state in PL studies at high power optical excitation saturating the QD ground state (Sec. 3.7). (2) The electrons in the QD ground state escape via thermal activation to the excited states thus contributing to the non-radiative decay with an activation energy of 40meV, matching the energy difference between excited and ground electron state (Sec. 3.7). (3) MTSQDs emitting at slightly different wavelengths reveal the same activation energy of ~40meV for thermally activated escape through the first excited state, indicating uniformity of the electronic structures of the MTSQDs in the array (Sec. 3.7). The observed QD electronic structure indicates good quantum confinement in the synthesized MTSQDs of the array, a fundamental requirement for the QDs to act as single photon emitters at cryogenic and even elevated temperature. Additionally, the carrier dynamics of the QD also plays an important role in the potential use of the QD as SPS. Thus from combined PL and TRPL studies we found the following regarding the carrier dynamics of MTSQDs: (1) The PL intensity dynamics of the MTSQD neutral exciton gives a decay time of ~1ns (Sec. 3.6.1). (2) For above the barrier bandgap excitation (640nm), the MTSQDs exhibit nearly equal intensity for neutral exciton and singly negatively charged exciton (Sec.3.5.2). This important finding could be understood through an analysis of a rate equation model of relaxation dynamics between five competing electronic states 312 (Sec.3.6.2) that includes the exciton and single carrier capture processes. We found that the nearly equal intensities are a result of nearly equal exciton and single electron capture rates. (3) For sub-gap excitation also we found comparable PL intensities for the neutral exciton and singly negatively charged excitons (Sec. 3.5.2). This indicates the presence of charge fluctuation in the quantum dot local environment, the charges arising from the population of impurities or deep levels in the GaAs barrier (sec. 3.5.2). The ~1ns decay lifetime of the neutral exciton transition suggests that the exciton transition can be used to act as 1GHz light emitter, a good starting light source to explore its single photon emission property. The probable charge fluctuation experienced by the QD may affect the purity of the single photon emission from the QD. §5.4 Single Photon Emission from the MTSQDs The above summarized findings of the optical studies indicated that the neutral excitons in the MTSQDs exhibiting good quantum confinement and fast decay lifetime are suitable for generating, at high frequency (~1GHz), on-demand single photons. To demonstrate single photon emission from MTSQDs and study processes that affect it, photon intensity correlation measurements were carried out using the HBT setup of our micro-PL system. Through these studies, we successfully demonstrated that: 313 (1) MTSQDs act as single photon emitters with average g (2) (0) ~ 0.19±0.03 indicating single photon emission purity of around 0.9 0.02  at 8K (statistics limited to 3 MTSQDs owing to the prohibitive cost of LHe). (2) MTSQDs’ single photon emission behavior holds up to 77K with average g (2) (0) ~ 0.35±0.06 and single photon emission purity ~ 0.81 0.02  (statistics obtained over 4 MTSQDs). The high temperature single photon emission from the InGaAs MTSQDs reflects the high degree of three dimensional quantum confinement realized through control on growth. The observed low temperature g (2) (0) is not as good as the reported an order of magnitude lower values for SAQDs and nanowire based QDs. The MTSQD g (2) (0) is limited by (a) the 0.2nm (~300μeV) instrument resolution that prevents the collection of single exciton state from the MTSQD, and (b) charge fluctuations around the QD under above gap (640nm) excitation. The collection from two exciton states with energy difference ~360 μeV, revealed by the 8K PL studies, under the employed g (2) (0) measurement condition limits the g (2) (0) to ~0.3, given the PL intensity ratio of the two transitions collected. The measured g (2) (0) values thus reflect lower limit on the purity of single photon emission from MTSQDs as, with improved resolution measurements, one can expect finding lower g (2) (0) values. The single photon emission studies also revealed that only 0.26% to 0.38% of the total emitted single photons from the MTSQDs are in directions within the collection cone angle and thus collected into the optical measurement setup, limiting 314 the single photons detection count to <1000/sec under the employed measurement condition. This low single photon collection efficiency is due to the guiding of photons through the mesa into the GaAs substrate as revealed by our finite element simulations of emission from an in-plane dipole imbedded in the GaAs mesa of the same shape as the mesa holding MTSQD. The integration of the MTSQDs with features that prevent loss into the substrate such as a high refractive index discontinuity with an underlying layer combined with horizontal waveguiding layer can further enhance photon collection efficiency necessary for the use of MTSQD single photon emitters in integrated QIP systems. The demonstration of the single photon emission from the ordered spectrally uniform MTSQDs grown following the SESRE approach indicates the potential of such QDs for building on-chip integrated quantum optical circuits for QIP. Further investigations of the full potential of this class of SQD arrays for building quantum optical circuits are worthy of pursuing. §5.5 Potential of MTSQDs acting as EPSs The building of quantum optical circuits also requires entangled photon sources (EPSs) in addition to SPSs. It has been generally known that the biexction- exciton cascade photon emission process of QDs can be utilized to generate polarization entangled photon pairs [5.2]. When the fine structure splitting of the QD is less than the decay life time of exciton and biexction, the photons emitted from biexction-exciton cascade are polarization entangled. To assess the potential of 315 MTSQD acting as EPSs, knowledge of the fine structure splitting of the neutral exciton complex and the biexciton decay dynamics is critical [5.2]. The QD confinement potential symmetry controls the fine structure splitting of the neutral exciton complex. Thus polarization dependent PL based studies of the polar emission pattern of neutral exciton, charged exciton, and biexciton emission were carried out at 77K to probe the symmetry of MTSQD confinement potential. We found that (1) Neutral exciton and biexciton emissions show elliptical polar pattern while the charged exciton shows near circular polar pattern obtained with 0.2 nm (~300 μeV) instrument resolution. Taken into account the effect of geometry anisotropy of the profile of the post-growth mesa on the far field emission pattern of the QD dipole transition as obtained from finite element based simulations, the elliptical polar patterns of the neutral exciton and biexciton suggest that the symmetry of the confinement potential of the MTSQDs is likely less than C 2v with the QD hole state having mixed heavy hole (HH) and light hole (LH) character. The bright excitons are thus probably non-degenerate with non-zero fine structure splitting. The two finely separated emission peaks with linewidth ~150μeV, energy separation ~360 μeV and integrated PL intensity ratio of 2 obtained from the 8K PL studies indicates again that the confinement potential of QD is probably less than C 2v . The observed energy separation ~360 μeV is probably the energy separation between the bright and emitting “dark” excitons due to the mixture of HH and LH. The fine structure splitting of bright exciton is thus probably less than 150μeV, not resolved under our 316 instrument resolution. Further high resolution spectroscopy studies need to be carried out to study the fine structure splitting of the ground state neutral exciton from the MTSQD. (2) The MTSQDs biexciton emission with a biexction binding energy of ~6meV showed a lifetime of 0.58 ns (~2μeV), ~1.5 times shorter compared to neutral exciton. Through integration of MTSQDs with cavities to shorten the QD decay time to become comparable to the fine structure splitting of QD, the biexciton- exciton cascade process of the biexciton emission in the MTSQDs can be used to generate entangled photons. The degree of polarization entanglement of the photons from the biexciton-exciton cascade process of MTSQDs can be then examined towards the application of MTSQDs as EPSs which can be explored in future work. §5.6 Outlook: MTSQD based integrated optical circuits for information processing In the preceding subsections of this concluding chapter, we have covered the following four key ingredients towards the objective of creating the needed ordered spectrally uniform SPS and EPS array: (1) the synthesis of InGaAs/GaAs based spatially ordered mesa-top single quantum dot (MTSQD) array with growth controlled size and shape, (2) the first optical study of all the QDs in the array demonstrating highly efficient emission from such ordered MTSQDs and spectral uniformity of 8.3nm, 317 over five times better than the typical self-assembled quantum dots in prevalent use for SPS, (3) the systematic optical studies of the electronic and the excitonic structure of the ordered MTSQDs using PL/PLE/TRPL technique to assess its potential use as SPSs and EPSs, (4) demonstration of single photon emission from such ordered and highly uniform SQD arrays up to 77K with >80% purity (~90% purity at 8K). The realization of the on-chip integrable and scalable ordered spectrally uniform MTSQDs based SPSs array provides the needed QD platform to enable the first real attempt towards building on-chip interconnected optical circuits for information processing. The full potential of this class of SQD arrays as SPSs and EPSs however requires further investigation as discussed below. (1) Single MTSQD Studies  Establishing the Nature and Magnitude of Fine Structure Splitting The systematic optical studies of the MTSQDs carried out so far have been limited by the instrument resolution of 0.2nm (~300μeV) and measurements at the liquid nitrogen temperature. Such a resolution and high temperature has prevented us from stablishing the nature of the electronic states underlying excitonic states which, in turn, prevents knowing the exact confinement potential symmetry, the nature of the QD hole state Bloch part and correspondingly the value of the fine structure splitting of the MTSQD ground state excitons. These also limit the measured single 318 photon emission purity of ~ 90% at 8K and prevent a clear and clean assessment of the potential of MTSQDs acting as a source for polarization entangled photon pairs. To assess the full potential of the MTSQD as SPSs and EPSs, the following optical studies are needed:  Unambiguous identification of Excitonic States High resolution ( < 50 μeV) and cryogenic temperature (~8-10K) power and polarization dependent PL and PLE studies for sub-barrier bandgap excitation and SQD resonant excitation to resolve, in detail, the neutral and charged excitonic complexes, the biexciton complex, and the underlying QD electronic structure.  Single Photon Emission Purity Study of the single photon emission using MTSQD resonant excitation scheme [5.1, 5.8, 5.9] to eliminate the effect of charging and reveal the true nature of single photon emission purity of the ground state exciton transition.  Photon Emission Density Matrix Study of the cross-correlation of photons emitted from single neutral exciton state and negatively charged state as well as photons from single neutral exciton and biexciton state to reveal the timing sequence of the photons from different excitonic states to assess the effect of single charge capture on the dynamical process of the photon emission.  Biexciton-Exciton Photon Cascade for EPSs Study of two-photon biexciton excitation [5.2] scheme based high resolution PL and single photon emission characteristics of the QD neutral exciton and 319 biexciton emission to assess the potential of using biexciton-exciton cascade process in the MTSQDs to generate entangled photon pairs. (2) Integrated Single MTSQD with LME To demonstrate QD coherence time, realize interference of photon, and study the degree of entanglement of photons from the biexciton-exciton cascade process, the MTSQDs need to be integrated with cavities. To this end, the following further growth syntheses are needed:  Planarizing Overlayer Growth Explore the growth conditions for the MTSQD planarizing overgrowth (Fig. 5.2 (a)) that is optimal for balancing the competition between facet-dependent incorporation kinetics and interfacet migration kinetics driven by the evolving facet area-dependent “capillarity” effect while influencing the underlying quantum dots the least. The atom dynamics are controlled by the growth temperature and relative flux ratio of Group III and Group V atoms to realize the desired planarizing growth preserving the high quality of the MTSQD. Figure 5.2. Schematic of the planarization of the MTSQD surface morphology through the growth of an overlayer on the MTSQD (pyramids) array (panel a) resulting in the buried ordered array of MTSQDs (panel b) around which light manipulating units can be lithographically created within the designed footprint (broken white lines). 320  Fabrication and Characterization of Integrated MTSQD-Cavity-Waveguide Structures The planarized MTSQDs (Fig. 5.2 (b)) can enable the integration with LMEs, i.e cavities, to control QD decay rate. Towards such integration, the approach introduced by our group is to use dielectric building block (blue blocks in Fig. 5.1) based multifuncitional light manipulating units [5.6, 5.10] that uses Mie resonance to create on-chip interconnected optical circuits schematically captured in Fig. 5.1. The demonstration of the planarizing overgrowth of MTSQD will enable the effort of constructing the first simple MTSQD-DBB integrated units [5.6] (an example captured in Fig. 5.3) through post growth lithography to assess the potential of such QD-DBB integrated approach towards the goal of creating the interconnected circuits (Fig. 5.1) to study the interference of photons from different QDs. Figure 5.3. Schematic of the simplest MTSQD-DBB integrated unit structure to be fabricated and studied towards the goal of optical circuits in Fig.5.1. The basic character of the MTSQDs i.e. the coherence time of the QD and interference and entanglement of photons from the same QD, need to be studied and understood to guide the construction of optical circuits captured in Fig. 5.1. With the above mentioned further studies accomplished, all of the essential aspects relating to the MTSQDs single photon sources would have been covered. 321 The gained understanding of MTSQDs will assist in the efforts towards their integration with the well-established photonic 2D crystal based or our newly proposed DBB based light manipulating elements with predictable control of the performance of the MTSQDs. This, in-turn, will enable basic experiments for examining the interference and entanglement effects between photons from different single quantum dots, a necessary step towards the creation of on-chip scalable architectures of quantum optical circuits as schematically captured in Fig. 5.1. Chapter 5 References: [5.1] P. Lodahl, S. Mahmoodian and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures”, Rev. Mod. Phys. 87, 347 (2015). [5.2] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs”, Nature Photon. 8, 224 (2014). [5.3] D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel and Y. Yamamoto, “Photon antibunching from a single quantum-dot- microcavity system in the strong coupling regime”, Phys. Rev. Lett. 98, 117402 (2007). [5.4] M. E. Reimer, G. Bulgarini, N. Akopian, M. Hovevar, M. B. Bavinck, M. A. Verheijen, E. P.A.M. Bakkers, L. P. Kouwenhoven and V. Zwiller, “Bright single- photon sources in bottom-up tailored nanowires”, Nature Commun. 3, 737 (2012). [5.5] N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gónez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G.White, L. Lanco and P. Senellart, “Near-optimal single-photon sources in the solid state”, Nature Photon. 10, 340 (2016) [5.6] J. Zhang, S. Chattaraj, S. Lu and A. Madhukar, “Mesa-top quantum dot single photon emitter arrays: Growth, optical characteristics, and the simulated optical response of integrated dielectric nanoantenna-waveguide systems”, J. Appl. Phys. 120, 243103 (2016). 322 [5.7] Y. Chen, I. E. Zedeh, K. D. Jöns, A. Fognini, M. E. Reimer, J. Zhang, D. Dalacu, P. J. Poole, F. Ding, V. Zwiller, and O. G. Schmidt, “ Controlling the exciton energy of a nanowire quantum dot by strain field”, Appl. Phys. Lett. 108, 182103 (2016). [5.8] J. Houel, A. V. Kuhlmann, L. Greuter, F. Xue, M. Poggio, B. D. Gerardot, P. A. Dalgarno, A. Badolato, P. M. Petroff, A. Ludwig, D. Reuter, A. D. Wieck and R. J. Warburton, “Probing single-charge fluctuations at a GaAs/AlAs interface using laser spectroscopy on a nearby InGaAs quantum dot”, Phys. Rev. Lett. 108, 107401 (2012). [5.9] C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot”, Phys. Rev. Lett. 86, 1502 (2001). [5.10] S. Chattaraj and A. Madhukar, “Multifunctional all-dielectric nano-optical systems using collective multipole Mie resonances: toward on-chip integrated nanophotonics”, J. Opt. Soc. Am. B. 33, 2414 (2016). 323 Bibliography M. Abbarchi, C. Mastrandrea, T. Kuroda, T. Mano, A. Vinattieri, K. Sakoda, and M. Gurioli, “Poissonian statistics of excitonic complexes in quantum dots”, J. Appl. Phys. 106, 053504 (2009). F. Adler, M. Geiger, A. Bauknecht, F. Scholz, H. Schweizer, M. H. Pilkuhn, B. Ohnesorge and A. Forchel, “Optical transitions and carrier relaxation in self- assembled InAs/GaAs quantum dots”, J. Appl. Phys. 80, 4019 (1996). I. Aharonovich, S. Castelletto, D. A. Simpson, C-H. Su, A. D. Greetree and S. Prawer, “Diamond-based single-photon emitters”, Rep. Prog. Phys. 74, 076501 (2011). J.E. Allen, E.R. Hemesath, D.E. Perea, J.L. Lensch-falk, Z.Y. Li, F. Yin, M.H. Gass, P. Wang, A.L. Bleloch, R.E. Palmer, and L.J. Lauhon, “High-resolution detection of Au catalyst atoms in Si nanowires,” Nature Nanotech. 3, 168 (2008). A.P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science 271, 933 (1996). Y. Arakawa and A. Yariv, “Quantum well lasers—Gain, Spectra, dynamics”, IEEE J. Quantum Electron 22,1887 (1987). M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide”, Phys. Rev. Lett. 113, 093603 (2014). P. Atkinson, S. Kiravittaya, M. Benyoucef, A. Rastelli, and O.G. Schmidt, “Site- controlled growth and luminescence of InAs quantum dots using in situ Ga-assisted deoxidation of patterned substrates”, Appl. Phys. Lett. 93, 101908 (2008). H. Baier, E. Pelucchi, E. Kapon, S. Varoutsis, M. Gallart, I. Robert-Philip and I. Abram, “Single photon emission from site-controlled pyramidal quantum dots”, Appl. Phys. Lett. 84, 648 (2004). G. Bastard, “Superlattice band structure in the envelope-function approximation”, Phys. Rev. B. 24, 5693 (1981). G. Bastard, J. A. Brum, “Electronic states in semiconductor heterostructures”, IEEE J. Quantum Electron. 22, 1625 (1986). 324 M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In 0.60 Ga 0.40 As self-assembled quantum dots”, Phys. Rev. B. 65, 041308 (R) (2002). M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke and S. N. Walck, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots”, Phys. Rev. B 65, 195315 (2002). W. Becker, Advanced time-correlated single photon counting techniques, Springer- Verlag Berlin Heidelberg, 2005. H. Benisty, C. M. Sotomayortorres, C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems”, Phys. Rev. B 44, 10945 (1991). C. Bennett, F. Bessette, G. Brassard, L. Salvail and J. Smolin, “Experimental quantum cryptography”, J. Cryptol. 5, 3 (1992). O. Benson, C. Santori, M. Pelton and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot”, Phys. Rev. Lett. 84, 2513 (2000). G. Bester, S. Nair and A. Zunger, “Pseudopotential calculation of the excitonic fine structure of million-atom self-assembled In 1-x Ga x As/GaAs quantum dots”, Phys. Rev. B. 67, 161306 (R) (2003). U. Bockelmann, G. Bastard, “Phonon-scattering and energy relaxation in 2- dimensional, one dimensional and zero-dimensional electron gases”, Phys. Rev. B. 42, 8947 (1990). L. Brusaferri, S. Sanguinetti, E. Grilli, M. Guzzi, A. Bignazzi, F. Bogani, L. Carraresi, M. Colocci, A. Bosacchi, P. Frigeri and S. Franchi, “Thermally activated carrier transfer and luminescence line shape in self-organized InAs quantum dots”, Appl. Phys. Lett. 69, 3354 (1996). S. Buckley, K. Rivoire and J. Vučković, “Engineered quantum dot single photon sources”, Rep. Prog. Phys. 75, 126503 (2012). G. Bulgarini, M. E. Reimer, M. B. Bavinck, K. D. Jöns, D. Dalacu, P. J. Poole, E. P. A. M. Bakkers and V. Zwiller, “Nanowire waveguide launching single photons in a Gaussian mode for ideal fiber coupling”, Nano Lett. 14, 4102 (2014). G. Bulgarini, M. E. Reimer, T. Zehender, M. Hocevar, E.P.A.M. Bakkers, L.P. Kouwenhoven, and V. Zwiller, “Spontaneous emission control of single quantum dots in bottom-up nanowire waveguides”, Appl. Phys. Lett. 100, 121106 (2012). 325 W. K. Burton, N. Cabrera and F. C. Frank, “The growth of crystal and equilibrium structure of their surfaces”, Trans. Royal Soc. London 243, 299 (1951). W. Chang, W. Chen, H. Chuang, T. Hsieh, J. Chyi and T. Hsu, “Efficient single- photon sources based on low-density quantum dots in photonic-crystal nanocavities”, Phys. Rev. Lett. 96, 117401 (2006). S. Chattaraj and A. Madhukar, “Multifunctional all-dielectric nano-optical systems using collective multipole Mie resonances: toward on-chip integrated nanophotonics”, J. Opt. Soc. Am. B. 33, 2414 (2016). S. Chattaraj, J. Zhang, S. Lu and A. Madhukar, “On-chip quantum optical networks comprising co-designed and spectrally uniform single photon source array and dielectric light manipulating elements”, presented at IEEE summer topicals meeting, San Juan, Puerto Rico, July 10-12, 2017. L. Chen, “Optical Properties of InGaAs/GaAs quantum well structures: Application to light modulators and switches”, PhD. Dissertation, University of Southern California ,1993. Y. Chen, I. E. Zedeh, K. D. Jöns, A. Fognini, M. E. Reimer, J. Zhang, D. Dalacu, P. J. Poole, F. Ding, V. Zwiller, and O. G. Schmidt, “ Controlling the exciton energy of a nanowire quantum dot by strain field,” Appl. Phys. Lett. 108, 182103 (2016). K. Choi, M. Arita, S. Kako and Y. Arakawa, “Site-controlled growth of single GaN quantum dots in nanowires by MOCVD”, J. Cryst. Growth. 370, 328 (2013). S. L. Chuang, Physics of Optoelectronic Devices. New York: John Wiley & Sons ,1995. J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lelanne and J. M. Gerard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire”, Nature Photon. 4, 174 (2010). D. Dalacu, K. Mnaymneh, J. Lapointe, X. Wu, P.J. Poole, G. Bulgarini, V. Zwiller, and M.E. Reimer, “Ultraclean emission from InAsP quantum dots in defect-free wurtzite InP nanowires,” Nano Lett. 12, 5919 (2012). S. Deshpande, T. Frost, A. Hazari, and P. Bhattacharya, “Electrically pumped single- photon emission at room temperature from a single InGaN/GaN quantum dot,” Appl. Phys. Lett. 105, 141109 (2014). X. Ding, Y. He, Z. –C. Duan, Niels Gregersen, M.-C. Chen, S. Unsleber, S. Maier, C. Schneider, M. Kamp, S. Höfling, C. Lu and J. Pan, “On-demand single photons with high extraction efficiency and near-unity indistinguishability from a resonantly driven quantum dot in a micropillar”, Phys. Rev. Lett.116, 020401 (2016). 326 X. M. Dou, X. Y. Chang, B. Q. Sun, Y. H. Xiong, Z. C. Niu, S. S. Huang, H. Q. Ni, Y. Du, and J. B. Xia, “Single-photon-emitting diode at liquid nitrogen temperature,” Appl. Phys. Lett. 93,101107 (2008). L. Dusanowki, M. Syperek, A. Maryński, L. H. Li, J. Misiewicz, S. Höfling, M. Kamp, A. Fiore, and G. Sęk, “Single photon emission up to liquid nitrogen temperature from charged excitons confined in GaAs-based epitaxial nanostructures,” Appl. Phys. Lett. 106, 233107 (2015). P. Ender, A. Bärwolff, M. Woerner and D. Suisky, “ p k    theory of energy bands, wave functions, and optical selection rules in strained tetrahedral semiconductors”, Phys. Rev. B 51, 16695 (1995). A. Faraon, A. Majumdar, D. Englund, E. Kim, M. Bajcsy and J. Vučković, “Integrated quantum optical networks based on quantum dots and photonic crystals”, New J. Phys. 13, 055025 (2011). I. Favero, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira, C. Couteau, J. P. Poizat and J. M. Gérard, “Fast exciton spin relaxation in single quantum dots”, Phys. Rev. B. 71, 233304 (2005). O. Fedorych, C. Kruse, A. Ruban, D. Hommel, G. Bacher, and T. Kümmell, “Room temperature single photon emission from an epitaxially grown quantum dot,” Appl. Phys. Lett. 100, 061114 (2012). M. Felici, P. Gallo, A. Mohan, B. Dwir, A. Rudra, and E. Kapon, “Site-controlled InGaAs quantum dots with tunable emission energy,” Small. 5, 938 (2009). D. Gershoni, C. H. Henry and G. A. Baraff, “Calculating the optical properties of multidimensional heterostructures: application to the modeling of quaternary quantum well lasers”, IEEE J. Quantum Electron. 29, 2433 (1993). J. W. Gibbs, The collected works of J. W. Gibbs, Yale University, New Haven, (1948). V. Giesz, O. Gazzano, A. K. Nowak, S. L. Portalupi, A. Lemaȋtre, I. Sagnes, L. Lanco and P. Senellart, “Influence of the Purcell effect on the purity of bright single photon sources”, Appl. Phys. Lett. 103, 033113 (2013). N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, “Quantum cryptography”, Rev. Mod. Phys. 74, 145 (2002). N. Gisin and R. Thew, “Quantum communication”, Nat. Photonics 1, 165 (2007). 327 L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack”, Phys. Rev. Lett. 79, 325 (1997). M. Grundmann, Nano-Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. M. Grundmann and D. Bimberg, “Theory of random population for quantum dots”, Phys. Rev. B. 55, 9740 (1997). M. Grundmann, J. Christen, N. N. Ledentsov, J. Böhrer, D. Bimberg, S. S. Ruvimov, P. Werner, U. Richter, U. Gösele, J. Heydenreich, V. M. Ustinov, A. Yu, A. E. Zhukov, P. S. Kop’ev and Z. I. Alferov, “Untranarrow luminescence lines from single quantum dots”, Phys. Rev. Lett. 74, 4043 (1995). M. Grundmann, O. Stier and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure”, Phys. Rev. B. 52, 11969 (1995). F. Guffarth, R. Heitz, A. Schliwa, O. Stier, N.N. Ledentsov, A. R. Kovsh, V. M. Ustinov and D. Bimberg, “Strain engineering of self-organized InAs quantum dots”, Phys. Rev. B. 64, 085305 (2001). S. Guha, “MBE Growth of InGaAs and AlGaAs on patterned and nonpatterned GaAs(100): A study of inter-facet migration, morphology and defect formation”, Ph.D. dissertation, Materials Science, University of Southern California, 1991. S. Guha and A. Madhukar, “An explanation for the directionality of interfacet migration during molecular beam epitaxial growth on patterned substrates”, J. Appl. Phys. 73, 8662 (1993). S. Guha, A. Mahukar, K. Kaviani, L. Chen, R. Kuchibhotla, R. Kapre, M. Hyugachi and Z. Xie, “Molecular beam epitaxical growth of Al x Ga 1-x As on non-planar patterned GaAs (100)”, Mat. Res. Soc. Symp. Proc. 145, 29 (1989). R. Hanbury Brown and R. Q. Twiss, “A Test of a New Type of Stellar Interferometer on Sirius”, Nature. 178, 1046 (1956). J. C. Harmand, F. Jabeen, L. Liu, G. Patriarche, K. Gauthron, P. Senellart, D. Elvira, and A. Beveratos, “InP1-xAsx quantum dots in InP nanowires: A route for single photon emitters,” J. Cryst. Growth. 378, 519 (2013). R. Heitz, “Optical Properties of Self-Organized Quantum dots”, Chapter 10 in Nano- Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. 328 R. Heitz, H. Born, F. Guffarth, O. Stier, A. Schliwa, A. Hoffmann, D. Bimberg, “Existence of a phonon bottleneck for excitons in quantum dots”, Phys. Rev. B 64, 241305 (2001). R. Heitz, A. Kalburge, Q. Xie, M. Grundmann, P. Chen, A. Hoffmann, A. Madhukar, D. Bimberg, “Excited states and energy relaxation in stacked InAs/GaAs quantum dots”, Phys. Rev. B 57, 9050 (1998). R. Heitz, M. Grundmann, N. N. Ledentsov, L. Eckey, M. Veit, D. Bimberg, V. M. Ustinov, A. Y. Egorov, A. E. Zhukov, P. S. Kop'ev and Zh. I. Alferov, “Multiphonon-relaxation processes in self-organized InAs/GaAs quantum dots”, Appl. Phys. Lett. 68, 361 (1996). R. Heitz, F. Guffarth, K. Pötschke, A. Schliwa, D. Bimberg, N. D. Zakharov and P. Werner, “Shell-like formation of self-organized InAs/GaAs quantum dots”, Phys. Rev. B. 71, 045325 (2005). R. Heitz, T. R. Ramachandran, A. Kalburge, Q. Xie, I. Mukhametzhanov, P. Chen and A. Madhukar, “Observation of Reentrant 2D to 3D Morphology Transition in Highly Strained Epitaxy: InAs on GaAs”, Phys. Rev. Lett. 78, 4071 (1997). R. Heitz, M. Veit, A. Kalburge, Q. Xie, M. Grundmann, P. Chen, N.N. Ledentsov, A. Hoffmann, A. Madhukar, D. Bimberg, V.M. Ustinov, P.S. Kop’ev and Zh.I. Alferov, “Hot carrier relaxation in InAs/GaAs quantum dots”, Physica E. 2, 578 (1998). C. Herring, “Diffusion viscosity of a polycrystalline solid”, J. Appl. Phys. 21, 437 (1950). J. Hohansen, S. Stobbe, I. S. Nikolaev, T. Lund-Hansen, P. T. Kristensen, J. M. Hvam, W. L. Vos and P. Lodahl, “Size dependence of the wavefunction of self- assembled InAs quantum dos from time-resolved optical measurements”, Phys. Rev. B 77, 073303 (2008). M.J. Holmes, K. Choi, S. Kako, M. Arita and Y. Arakawa, “Room-temperature triggered single photon emission from a III-nitride site-controlled nanowire quantum dot”, Nano Lett. 14, 982 (2014). J. Houel, A. V. Kuhlmann, L. Greuter, F. Xue, M. Poggio, B. D. Gerardot, P. A. Dalgarno, A. Badolato, P. M. Petroff, A. Ludwig, D. Reuter, A. D. Wieck and R. J. Warburton, “Probing single-charge fluctuations at a GaAs/AlAs interface using laser spectroscopy on a nearby InGaAs quantum dot”, Phys. Rev. Lett. 108, 107401 (2012). T. Hsieh, J. Chyi, H. Chang, W. Chen, T. Hsu, and W. Chang, “Single photon emission from an InGaAs quantum dot precisely positioned on a nanoplane”, Appl. Phys. Lett. 90, 073105 (2007). 329 Y. H. Huo, B. J. Witek, S. Kumar, J. R. Cardenas, J. X. Zhang, N. Akopian, R. Singh, E. Zallo, R. Grifone, D. Kriegner, R. Trotta, F. Ding, J. Stangl, V. Zwiller, G. Bester, A. Rastelli and O. G. Schmidt, “ A light-hole exciton in a quantum dot”, Nature Phys. 10, 46 (2014). J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons ,1975. C. Jarlov, A. Lyasota, L. Ferrier, P. Gallo, B. Dwir, A. Rudra and E. Kapon, “Exciton dynamics in a site-controlled quantum dot coupled to a photonic crystal cavity”, Appl. Phys. Lett. 107, 191101 (2015). E. T. Jaynes and F. W. Cummings, “Comparison of Quantum and Semiclassical Radiation Theories with Application to the Beam Master”, Proc.IEEE .51, 89 (1963). T. Jennewein, C. Simon, G. Weihs, H. Weinfurter and A. Zeilinger, “Quantum cryptography with entangled photons”, Phys. Rev. Lett. 84, 4729 (2000). D. E. Jesson, S. J. Pennycook, J. –M. Baribeau and D. C. Houghton, “Direct imaging of surface cusp evolution during strained-layer epitaxy and implicaitons for strain relaxation”, Phys. Rev. Lett. 71, 1744 (1993). K. D. Jöns, P. Atkinson, M. Müller, M. Heldmaier, M. Ulrich, O.G. Schmidt, and P. Michler, “Triggered indistinguishable single photons with narrow line widths from site-controlled quantum dots”, Nano Lett. 13, 126 (2013). G. Juska, V. Dimastrodonato, L. O. Mereni, A. Gocalinska and E. Pelucchi, “Towards quantum-dot arrays of entangled photon emitters”, Nature Photon. 7, 527 (2013). M. A. Kakeev, R. K. Kalia, A. Nakano, P. Vashishta and A. Madhukar, “Effect of geometry on stress relaxation in InAs/GaAs rectangular nanomesas: multimillion- atom molecular dynamics simulations”, J. Appl. Phys. 98, 114313 (2005). S. Kako, C. Santori, K. Hoshino, S. Gtzinger, Y. Yamamoto and Y. Arakawa, “A gallium nitride single-photon source operating 200 K”, Nat. Mater. 5, 887 (2006). E. O. Kane “Band structure of Indium Antimonide“, J. Phys. Chem. Solids 1, 249 (1957). E. Kapon, J. P. Harbison, C. P. Yun and L. T. Florez, “Patterned quantum well semiconductor laser arrays”, Appl. Phys. Lett. 54, 304 (1989). M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther, “Continuous generation of single photons with controlled waveform in an ion-trap cavity system”, Nature 431, 1075 (2004). 330 C. F. Klingshirn, Semiconductor optics, Chapter 15, Springer Berlin Heidelberg New York, 2007. E. Knill, R. Laflamme and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001). M. Koashi, K. Kono, T. Hirano and M. Matsuoka, “Photon antibunching in pulsed squeezed light generated via parametric amplification”, Phys. Rev. Lett. 71, 1164 (1993). A. Konkar, “Unstrained and strained semiconductor nanostructure fabrication via molecular beam epitaxical growth on non-planar patterned GaAs (001) substrates,” Ph.D. dissertation, Physics, University of Southern California, 1999. A. Konkar, A. Madhukar, and P. Chen, “Creating three-dimensionally confined nanoscale strained structures via substrate encoded size-reducing epitaxy and the enhancement of critical thickness for island formation,” MRS Symposium Proc. 380, 17 (1995). A. Konkar, K. C. Rajkumar, Q. Xie, P. Chen, A. Madhukar, H. T. Lin and D. H. Rich, “In-situ fabrication of three dimensionally confined GaAs and InAs volumes via growth on non-planar patterned GaAs (001) substrates”, J. Cryst. Growth. 150, 311 (1995). A. V. Koudinov, I. A. Akimov, Y. G. Kusrayev and F. Henneberger, “ Optical and magnetic anisotropies of the hole states in stranski-krastanov quantum dots”, Phys. Rev. B 70, 241305 (2004). D. N. Krizhanovskii, A. Ebbens, A. I. Tartakovskii, F. Pulizzi, T. Wright, M. S. Skolnick and M. Hopkinson, “Individual neutral and charged In x Ga 1- xAs –GaAs quantum dots with strong in-plane optical anisotropy”, Physical Review B 72,161312 (2005). A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single photon source for distributed quantum networking”, Phys. Rev. Lett., 89, 067901(2002). C. Kurtsiefer, P. Zarda, S. Mayer and H. Weinfurter, “The breakdown flash of silicon avalanche photodiode- backdoor for eavesdropper attacks?” J. Mod. Opt. 48, 2039 (2001). L. Landin, M. S. Miller, M. –E. Pistol, C. E. Pryor and L. Samuelson, “Optical studies of individual InAs quantum dots in GaAs: few-particle effects”, Science, 280, 262 (1998). 331 A. Laucht, S. Pütz, T. Günthner, N. Hauke, R. Saive, S. Frédérick, M. Bichler, M. -C. Amann, A. W. Holleitner, M. Kaniber and J. J. Finley, “A waveguide-coupled on- chip single photon source”, Phys. Rev. X. 2, 011014 (2012). S. Laurent, S. Varoutsis, L. L. Gratiet, A. Lemaître, I. Sagnes, F. Raineri, A. Levenson, I. Robert-Philip and I. Abram, “Indistinguishable single photons from a single-quantum dot in a two-dimensional photonic crystal cavity”, Appl. Phys. Lett. 86, 163107 (2005) E. C. Le Ru, J. Fack and R. Murray, “Temperature and excitation density dependence of the photoluminescence from annealed InAs/GaAs quantum dots”, Phys. Rev. B 67, 245318 (2003). T. C. Lee, M. Y. Yen, P. Chen and A. Madhukar, “Kinetic processes in molecular beam epitaxy of GaAs (100) and AlAs (100) examined via static and dynamic behavior of reflection high-energy electron-diffraction intensities”, J. Vac. Sci. Tech. B 4, 884 (1986). Y. Léger, L. Besombes, L. Maingault and H. Mariette, “ Valence-band mixing in neutral, charged, and Mn-doped self-assembled quantum dots”, Phys. Rev. B 76,045331 (2007). X. Liu, H. Kumano, H. Nakajima, S. Odashima, T. Asano, T. Kuroda and I. Suemune, “Two-photon interference and coherent control of single InAs quantum dot emission in an Ag-embedded structure”, J. Appl. Phys. 116, 043103 (2014). P. Lodahl, S. Mahmoodian and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures”, Rev. Mod. Phys. 87, 347 (2015). R. Loudon, The quantum theory of light, Oxford Science, 2000. B. Lounis and W. E. Moerner, “Single photon on demand from s single molecule at room temperature”, Nature, 407, 491(2000). S. Lu, “Some studies of nanocrystal quantum dots on chemically functionalized substrates (semiconductors) for novel biological sensing”, PhD. Disseratation, University of Southern California, 2006. J. M. Luttinger, “Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory”, Phys. Rev. 102, 1030 (1956). J. M. Luttinger, W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields”, Phys. Rev. 97, 869 (1955). 332 A. Lyasota, S. Borghaardt, C. Jarlov, B. Dwir, P. Gallo, A. Rudra and E. Kapon, “Integration of multiple site-controlled pyramidal quantum dot systems with photonic-crystal membrane cavities”, J. Cryst. Growth. 414, 192 (2015). O. Madelung, Seminconductors: Data Handbook, Springer-Verlag Berlin Heidelberg, 2004. A. Madhukar, “A unified atomistic and kinetic framework for growth front morphology evolution and defect initiation in strained epitaxy”, J. Cryst. Growth. 163, 149 (1996). A. Madhukar, “Growth of semiconductor heterostructures on patterned substrates: defect reduction and nanostructures”, Thin Solid Films. 231, 8 (1993). A. Madhukar, “Stress engineered quantum dots: Nature’s Way”, Chapter 2 in Nano- Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. A. Madhukar, P. Chen, F. Voillot, M. Thomsen, J. Y. Kim, W. C. Tang and S. V. Ghaisas, “A combined computer simulation, RHEED intensity dynamics and photoluminescence study of the surface kinetics controlled interface formation in MBE grown GaAs/Al x Ga 1-x As (100) quantum well structures”, J. Cryst. Growth. 81, 26 (1987). A. Madhukar and S. V. Ghaisas, “The nature of molecular beam epitaxial growth examined via computer simulations,” CRC Critical Rev. Solid State Mater. Sci., 14, 1 (1988). K. H. Madsen, S. Ates, J. Liu, A. Javadi, S. M. Albrecht, I. Yeo, S. Stobbe and P. Lodahl, “Efficient out-coupling of high purity single photons from a coherent quantum dot in a photonic-crystal cavity”, Phys. Rev. B. 90, 155303 (2014). M. Mannoh, T. Yuasa, S. Naritsuka, K. Shinozaki and M. Ishii, “Pair-groove- substrate GaAs/AlGaAs multiquantum well lasers by molecular beam epitaxy”, Appl. Phys. Lett. 47, 728 (1985). J. Y. Marzin, J. M. Gérard, A. Izraël, D. Barrier and G. Bastard, “Photoluminescence of Single InAs Quantum Dots Obtained by Self-Organized Growth on GaAs,” Phys. Rev. Lett., 73, 716 (1994). A. E. McKinnon, A. G. Szabo, D. R. Miller, “Deconvolution of photoluminescence data”, J. Phys. Chem. 81, 1564 (1977). A. Messiah, Quantum mechanics, Dover, New York ,1999. P. Michler, Single semiconductor quantum dots, Springer-Verlag Berlin Heidelberg , 2009. 333 I. Mukhametzhanov, “Growth Control, Structural Characterization, and Electronic Structure of Stranski-Krastanov InAs/GaAs (001) Quantum Dots,” Ph.D. dissertation, Physics, University of Southern California, 2002. M. Müller, S. Bounouar, K. D. Jöns, M. Glässl and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs”, Nature Photon. 8, 224 (2014). J. L. O’Brien, A. Furusawa and J. Vučković, “Photonic quantum technologies”, Nature Photon. 3, 687 (2009). D. V. O’Conner, W. R. Ware, J. C. Andre, “Deconvolution of fluorescence decay curves – Critical comparison of techniques”, J. Phys. Chem. 83, 1333 (1979). S. B. Ogale, A. Madhukar, F. Voillot, M. Thomsen, W. C. Tang, T. C. Lee, J. Y. Kim and P. Chen, “Atomistic nature of heterointerfaces in III-V semiconductor- based quantum-well structures and its consequences for photoluminescence behavior”, Phys. Rev. B. 36, 1662 (1987). S. B. Ogale, M. Thomsen and A. Madhukar, “Surface kinetic processes and the morphology of equilibrium GaAs(100) surface: A Monte Carlo study,” Appl. Phys. Lett. 52, 723 (1988). M. Ohtsuka and A. Suzuki, “Simulation of epitaxial growth over patterned substrates”, J. Cryst. Growth. 95, 55 (1989). J. I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1971 R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. Ritchie and A. J. Shields, “Two-photon interference of the emission from electrically tunable remote quantum dots”, Nature Photon. 4, 632 (2010). S. Paul, J. B. Roy and P. K. Basu, “Empirical expressions for the alloy composition and temperature dependence of the band gap and intrinsic carrier density in Ga x In 1- x As”, J. Appl. Phys. 69, 827 (1991). M. Pelton, C. Santori, G. S. Solomon, O. Benson and Y. Yamamoto, “Triggered single photons and entangled photons from a quantum dot microcavity”, Eur. Phys. J. D 18, 179 (2002). E. Pelucchi, V. Dimastrodonato, A. Rudra, K. Leifer, E. Kapon, L. Bethke, P. A. Zestanakis, and D. Vvedensky, “Decomposition, diffusion, and growth rate anisotropies in self-limited profiles during metalorganic vapor-phase epitaxy of seeded nanostructures,” Phys. Rev. B. 83, 205409 (2011). G. E. Pikus and G. L. Bir, Sov. Phys.-Solid State 1, 136 (1959). 334 P. J. Poole, D. Dalacu, J. Lefebvre and R. L. Williams, “Selective epitaxy of semiconductor nanopyramids for nanophotonics”, Nanotechnology. 21, 295302 (2010). D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel and Y. Yamamoto, “Photon antibunching from a Single Quantum-Dot- Microcavity System in the Strong Coupling Regime”,Phys. Rev. Lett. 98, 117402 (2007). E. M. Purcell, H. C. Torrey and R. V. Pound, “Resonance absorption by nuclear magnetic moments in a solid”, Phys. Rev. 69, 37 (1946). K. C. Rajkumar, “Molecular beam epitaxical growth and characterization of III-V compounds on planar and patterned GaAs (111)B substrates towards the realization of quantum boxes,” Ph.D. dissertation, Physics, University of Southern California, 1994. K. C. Rajkumar, A. Madhukar, P. Chen, A. Konkar, L. Chen, K. Rammohan and D. H. Rich, “Realization of three-dimensionally confined structures via one-step in situ molecular beam epitaxy on appropriately patterned GaAs (111)B and GaAs (001)”, J. Vac. Sci. Tech. B 12, 1071 (1994). T. R. Ramachandran, R. Heitz, N. P. Kobayashi, A. Kalburge, W. Yu, P. Chen and A. Madhukar, “Re-entrant behavior of 2D to 3D morphology change and 3D island lateral size equalization via mass exchange in Stranski--Krastanow growth: InAs on GaAs(001)”, J. Cryst. Growth. 175, 216 (1997). M. E. Reimer, G. Bulgarini, N. Akopian, M. Hovevar, M. B. Bavinck, M. A. Verheijen, E. P.A.M. Bakkers, L. P. Kouwenhoven and V. Zwiller, “Bright single- photon sources in bottom-up tailored nanowires”, Nature Commun. 3, 737 (2012). M. E. Reimer, G. Bulgarini, A. Fognini, R. W. Heeres, B. J. Witek, M. A. M. Versteegh, A. Rubino, T. Braun, M. Kamp, S. Höfling, D. Dalacu, J. Lapointe, P. J. Poole and V. Zwiller, “Overcoming power broadening of the quantum dot emission in a pure wurtzite nanowire”, Phys. Rev. B. 93, 195316 (2016). M. Reischle, G. J. Beirne, W. M. Schulz, M. Eichfelder, R. Rosbach, M. Jetter, and P. Michler, “Electrically pumped single-photon emission in the visible spectral range up to 80K”, Opt. Express, 16, 12771 (2008). B. Rigal, C. Jarlov, P. Gallo, B. Dwir, A. Rudra, M. Calic, and E. Kapon, “Site- controlled quantum dots coupled to a photonic crystal molecule,” Appl. Phys. Lett. 107, 141103 (2015). 335 G. Sallen, A. Tribu, T. Aichele, R. André, L. Besombes, C. Bougerol, S. Tatarenko, K. Kheng and J. Ph. Poizat, “Excition dynamics of a single quantum dot embedded in a nanowire”, Phys. Rev. B. 80, 085310 (2009). C. M. Santori, “Generation of nonclassical light using semiconductor quantum dots”, Ph.D. dissertation, Applied Physics, Stanford University, 2003. C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from InAs quantum dots”, Phys. Rev. B. 69, 205324 (2004). C. Santori, D. Fattal and Y. Yamamoto, Single-Photon Device and Application, Weinheim: Wiley-VCH , 2010. C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot”, Phys. Rev. Lett. 86, 1502 (2001). C. Santori, G. S. Solomon, M. Pelton, and Y. Yamamoto, “Time-resolved spectroscopy of multiexcitonic decay in an InAs quantum dot”, Phys. Rev. B. 65, 073310 (2002). K. H. Schmidt, G. Medeiros-Ribeiro, M. Oestreich, P. M. Petroff and G. H. Dohler, “Carrier relaxation and electronic structure in InAs self-assembled quantum dots”, Phys. Rev. B. 54, 11346 (1996). W. -M. Schulz, R. Roβbach, M. Reischle, G. J. Beirne, M. Bommer, M. Jetter, and P. Michler, “Optical and structural properties of InP quantum dots embedded in (AlxGa1-x) 0.51 In 0.49 P,” Phys. Rev. B. 79, 035329 (2009). A. Schwagmann, S. Kalliakos, I. Farrer, J. P. Griffiths, G. A. C. Johes, D. Ritchie and A. J. Shields, “On-chip single photon emission from an integrated semiconductor quantum dot into a photonic crystal waveguide”, Appl. Phys. Lett. 99, 261108 (2011). M. O. Scully and M. S. Zubairy, Quantum optics, Cambridge University Press, 1997. R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl and D. Bimberg, “Size- dependent fine-structure splitting in self-organized InAs/GaAs quantum dots”, Phys. Rev. Lett. 95, 257402 (2005). G. Shan, Z. Yin, C. H. Shek and W. Huang, “Single photon sources with single semiconductor quantum dots”, Front. Phys. 9, 170 (2014). V. A. Shchukin and D. Bimberg, “Spontaneous ordering of nanostructures on crystal surfaces”, Rev. Mod. Phys. 71, 1125 (1999). 336 V. A. Shchukin, N. N. Ledentsov and D. Bimberg, Epitaxy of Nanostructures , Springer Verlag: Berlin, 2004. A. J. Shields, “Semiconductor quantum light sources”, Nat. Photonics, 1, 215 (2007). P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer”, SIAM J. Comput. 26, 1484 (1997). B. W. Shore and P. L. Knight. “The Jaynes-Cummings Model”, J. Mod. Opt. 40, 1195 (1993). J. Shumway, A. Franceschetti and A. Zunger, “Correlation versus mean-field contribution to excitons, multiexcitons, and charging energies in semiconductor quantum dots”, Phys. Rev. B 63, 155316 (2001). J. Singh, Physics of semiconductors and their heterostructures, McGraw-Hill, Inc, 1993. J. Skiba-Szymanska, A. Jamil, I. Farrer, M. Ward, C. Nicoll, D. Ellis, J. Griffiths, D. Anderson, G. Jones, D. Ritchie, and A. Shields, “Narrow emission linewidths of positioned InAs quantum dots grown on pre-patterned GaAs(100) substrates”, Nanotechnology. 22, 065302 (2011). J. S. Smith, P. L. Derry, S. Margalit and A. Yariv, “High quality molecular beam epitaxial growth on patterned GaAs substrates”, Appl. Phys. Lett. 47, 712 (1985). N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gónez, I. Sagnes, N. D. Lanzillotti- Kimura, A. Lemaítre, A. Auffeves, A. G.White, L. Lanco and P. Senellart, “Near- optimal single-photon sources in the solid state”, Nature Photon. 10, 340 (2016). D. J. Srolovitz, “On the stability of surfaces of stressed solids”, Acta metal. 37, 621 (1989). O. Stier, M. Grundmann and D. Bimberg, “Electronic and optical properties of strained quantum dots modeled by 8-band p k    theory”, Phys. Rev. B 59, 5688 (1999). O. Stier, A. Schliwa, R. Heitz, M. Grundmann and D. Bimberg, “Stability of biexcitons in pyramidal InAs/GaAs quantum dots”, Phys. Stat. Sol. (b) 224, 115 (2001). A. Strittmatter, A. Holzbecher, A. Schliwa, J. Schulze, D. Quandt, T. D. Germann, A. Dreismann, O. Hitzemann, E. Stock, I. A. Ostapenko, S. Rodt, W. Unrau, U. W. Pohl. 337 A. Hoffmann, D. Bimberg and V. Haisler, “Site-controlled quantum dot growth on buried oxide stressor layers”, Phys. Status Solidi A. 209, 2411 (2011). X. Su, R. Kalia, A. Nakano, P. Vashishta and A. Madhukar, “Million-atom molecular dynamics simulation of flat InAs overlayers with self-limiting thickness on GaAs square nanomesas”, Appl. Phys. Lett. 78, 3717 (2001). J. Suffczyński, T. Kazimierczuk, M. Goryca, B. Piechal, A. Trajnerowicz, K. Kowalik, P. Kissacki, A. Golnik, K. P. Korona, M. Nawrocki and J. A. Gaj, “Excitation mechnisms of individual CdTe/ZnTe quantum dots studies by photon correlation spectroscopy”, Phys. Rev. B 74, 085319 (2006). R. M. Thompson, R. M. Stevenson, A. J. Shields, I. Farrer, C. J. Lobo, D. A. Ritchie and M. L. Leadbeater, “Single-photon emission from exciton complexes in individual quantum dots”, Phys. Rev. B. 64, 201302 (R) (2001). C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots”, Phys. Rev. B. 85, 155303 (2012). W. T. Tsang and A. Y. Cho, “Growth of GaAs-Ga 1-x Al x As over preferentially etched channels by molecular beam epitaxy: A technique for two-dimensional thin-film definition”, Appl. Phys. Lett. 30, 293 (1977). Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors”, Physica. 34, 149 (1967). J. T. Verdeyen, Laser Electronics, Prentice Hall (1995). M.A. Versteegh, M.E. Reimer, K.D. Jöns, D. Dalacu, P.J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nature Commun. 5, 5298 (2014). F. Voillot, A. Madhukar, J. Y. Kim, P. Chen, N. M. Cho, W. C. Tang and P. G. Newman, “Observation of kinietcally controlled monolayer step height distribution at normal and inverted interfaces with ultrathin GaAs/Al x Ga 1-x As quantum wells”, App. Phys. Lett. 48, 1009 (1986). E. Waks, C. Santori and Y. Yamamoto, “Security aspect of quantum key distribution with sub-Poisson light”, Phys. Rev. A. 66, 042315 (2002). L. W. Wang, J. Kim and A. Zunger, “Electronic structures of [110]-faceted self- assembled pyramidal InAs/GaAs quantum dots”, Phys. Rev. B. 59, 5678 (1999). 338 M. B. Ward, T. Farrow, P. See, Z. L. Yuan, O. Z. Karimov, P. Atkinson, K. Cooper, and D. A. Ritchie, “Electrically driven telecommunication wavelength single-photon source”, Appl. Phys. Lett. 90, 063512 (2007). M. B. Ward, O. Z. Karimov, D. C. Unitt, Z. L. Yuan, P. See, D. G. Gevaux, A. J. Shields, P. Atkinson, and D. A. Ritchie, “On-demand single-photon source for 1.3 μm telecom fiber”,Appl. Phys. Lett. 86, 201111 (2005). M. E. Ware, E. A. Stinaff, D. Gammon, M. F. Doty, A. S. Bracker, D. Gershoni, V. L. Korenev, S. C. Bădescu, Y. Lyanda-Geller and T. L. Reinecke, “Polarized fine structure in the Photluminescence excitation spectrum of a negatively charged quantum dot”, Phys. Rev. Lett. 95, 177403 (2005). A. J. Williamson, L. W. Wang and A. Zunger, “Theoretical interpretation of the experimental electronic structure of lens-shaped self-assembled InAs/GaAs quantum dots”, Phys. Rev. B. 62, 12963 (2000). R. Winik, D. Cogan, Y. Don, I. Schwartz, L. Gantz, E. R. Schmidgall, N. Livneh, R. Rapaport, E. Buks and D. Gershoni, “On-demand source of maximally entangled photon-pairs using the biexciton-exciton radiative cascade”, arXiv:1703.04380v1. Y. H. Wu, M. Werner, K. L. Chen and S. Wang, “Single-longitudinal-mode GaAs/GaAlAs channeled-substrate lasers grown by molecular beam epitaxy”, Appl. Phys. Lett. 44, 834 (1984). Q. Xie, P. Chen and A. Madhukar, “InAs island-induced-strain driven adatom migration during GaAs overlayer growth”, Appl. Phys. Lett. 65, 2051 (1994). P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photonics Rev. 4, 499 (2010). L. Yang, S. G. Carter, A. S. Bracker, M. K. Yakes, M. Kim, C. S. Kim, P. M. Vora and D. Gammon, “Optical spectroscopy of site-controlled quantum dot in a Schottky diode”, Appl. Phys. Lett. 108, 233102 (2016). T. Yuasa, M. Mannoh, T. Yamada, S. Naritsuka, K. Shinozaki and M. Ishii, “Characteristics of molecular-beam epitaxially grown pair-groove-substrate GaAs/AlGaAs multiquantum-well lasers”, J. Appl. Phys. 62, 764 (1987). J. Zhang, S. Chattaraj, S. Lu and A. Madhukar, “Mesa-top quantum dot single photon emitter arrays: Growth, optical characteristics, and the simulated optical response of integrated dielectric nanoantenna-waveguide systems”, J. Appl. Phys. 120, 243103 (2016). 339 J. Zhang, Z. Lingley, S. Lu, and A. Madhukar, “Nanotemplate-directed InGaAs/GaAs single quantum dots: Toward addressable single photon emitter arrays,” J. Vac. Sci. Technol. B 32, 02C106 (2014). 340 Appendix A: Quantum Dot Basics and Self-Assembled Quantum Dots In this appendix we cover the basic concept of a quantum dot, different implementation pathways for the QDs, and capture the essential background of the class of epitaxial quantum dots-- dubbed self-assembled quantum dot (SAQD) - - that have been examined the most for their potential as single photon sources suitable for on-chip integration. It is worth restating that the mesa-top single quantum dot (MTSQD) spatially regular arrays with considerably improved spectral emission uniformity that this dissertation demonstrates for the first time are the natural and viable scalable alternative to SAQDs as they naturally overcome the two inherent deficiencies of SAQD - - random spatial location and unacceptable fluctuations in size, shape, and composition. This considerable advantage of the MTSQDs arises because their underlying growth mechanism is driven by surface curvature associated stress gradient driven directional adatom migration and not lattice mismatch strain induced preferential upward adatom migration resulting in the spontaneous formation of 3D island like morphology leading to SAQDs. §A.1 Quantum dots: zero-dimensional electronic structures QDs are sub de Broglie wavelength-sized three-dimensional volumes of semiconductor A embedded in a higher bandgap semiconductor B in such a manner that the crystal potential of the latter provides a three-dimensional potential barrier for a certain range of otherwise band energies of the former, thus inducing three- 341 dimensionally confined electrons and holes in semiconductor A. Indeed when the size of the semiconductor confined by another semiconductor is reduced to smaller than the electron de Broglie wavelength in only one or two directions, the motion of the electrons is quantum confined only in the corresponding direction(s) and the resulting quantum confined structures are called quantum wells (or 2D systems) and quantum wires (or 1D systems), respectively. With reducing dimensionality of the system, the density of states (DOS) changes from DOS ~ E form for 3-D bulk crystals (Fig. A.1 (a)) to DOS ~ constant for 2-D systems (Fig. A.1 (b)), to DOS ~ 1/ E for 1-D systems (Fig. A.1 (c)), and finally to form DOS ~ () i E  for 0-D (Fig.A.1 (d)) where {E i } are the discrete energies of the bound electron states. Effectively, the simple descriptive relation 2 ~ 2 k E m with all k-vectors from the Brillouin zone is being transformed into description of a system with reduced space of k-vectors, where the disappearing k- points get packed closer to the selected energy states (defined by the confinement). The ultimate 0-D structure (i.e. QD) is the structure with no k-space at all, only selected energies with immensely packed DOS for every such level (δ-like DOS). Details of the QD electronic structure are discussed in Chapter 3. For any processes that matches the available energy states in QDs, all the carriers can participate without any waste of k-space in contrast to bulk and other reduced dimensional structures where only a fraction of the k-space is utilized due to k-conservation law. The price for increased performance is that if one is interested in a process with 342 characteristic energy that does not match the energies of QD system, there are no available carriers to participate in such a process. Consequently, QDs are thought to be attractive specifically for energy-selective applications, such as lasers, selected wavelength photodetectors, light modulators, and recently, single photon sources. Figure A.1 Schematic of density of states, and energy diagram of 3-D, 2-D, 1-D, and 0-D structures. So far four different major approaches have been pursued to realize three- dimensionally confined electronic structures. These result in quite different forms of QDs. The four approaches are: (1) nanocrystal quantum dots (NCQDs) synthesized by liquid colloidal chemistry [A.1], (2) lithographically patterned QDs realized by etching of a quantum wells to reduce its lateral dimensions [A.2], and (3) spontaneous formation of defect-free three-dimensional islands (dubbed SAQDs) on a substrate driven by lattice misfit strain between the island material and the substrate [A.3, A.4]. (4) nanowire quantum dot grown using metal nanoparticle 343 seeded vapor-liquid-solid (VLS) growth mechanism [A.5]. The nanowire diameter is controlled by the seed particle diameter and in turn the QD lateral size is controlled by the nanowire diameter. Thus for sufficiently small diameters, even for highly lattice mismatched combinations such as GaAs/InAs or InP/InAsP the shape of the QD is not island-like (i.e. not SAQD like) [A.5]. Among these different types of QDs, the SAQDs have been examined the most extensively for their single photon emission properties and high-performance optoelectronic device applications [A.6], including single photon emission behavior. In the following subsection we therefore briefly recall their formation process as it is also driven by stress but one that arises from lattice-mismatch strain between the quantum dot and barrier materials, unlike the MTSQDs that do not require lattice mismatch strain as discussed in Chapter 2. §A.2 Self-assembled quantum dots: Growth, Optical Behavior and Applications The well-known class of epitaxial self-assembled QDs is rooted in the lattice mismatch strain-driven controlled formation of defect-free 3D islands, first demonstrated (Fig. A.2 (a)) by the our group in 1990 using the GaAs(001)/In 0.5 Ga 0.5 As alloy system [A.7]. Figure A.2 (b) shows an atomic force microscope (AFM) image of binary InAs 3D islands on GaAs (001). Burying such defect-free three-dimensional islands coherently with higher bandgap materials would confine the electrons three-dimensionally in the islands, thus creating QDs [A.8,A.9]. 344 Figure A.2 (a) Cross-sectional TEM image showing the existence of coherent InGaAs Island on GaAs(001) [A.7]; (b) AFM image of InAs islands showing the inherent variation in size and random spatial positions. Shown also is the resulting broad photoluminescence and extremely sharp emission lines from individual islands when capped with a protective layer. §A.2.1 Self-assembled quantum dots: Formation Mechanism The SAQDs forms during the growth of material A on material B that have larger lattice constant compared to material B. For the growth of lattice mismatched layers on a planar substrate, the dominant stress is that associated with the lattice mismatch, defined as ( ) / L S S a a a a    , where a is the lattice constant and L and S denote the grown layer and substrate, respectively. Historically, such system has been examined in terms of the ground state energy of a fix number of particles, constrained to remain in a planar surface morphology, calculated for atomic arrangements without and with an array of interface edge dislocations as a function of film thickness [A.10]. It was found that for a given Δa there is a thickness, called the critical thickness for misfit dislocation formation. For low lattice mismatch (typically < ~ 2 % for tetrahedrally-bonded semiconductors) strained heteroepitaxial structures such a situation is usually realized, film growth proceeds as a layer-by- layer growth, referred to as the Frank and van der Merwe (FM) growth mode [A.11]. 345 At high lattice mismatch, stain relief can alternatively occur through the development of undulating surface morphology, often resembling 3-dimensional island-like features [A.12]. Two type of growth mode exists for the 3-dimensional island-like features: (1) Volmer-Weber (VW) growth mode [A.13] where 3-D islands are favored from the beginning of the heterogeneous growth and (2) Stranski- Krastanov (SK) growth mode [A.14] where 3D islands forms after initial thin layer- layer growth. The VW growth mode occurs when the atomic bonding between layer and substrate atom is weaker than bonds between atoms in the substrate. The gain in energy by island formation exceeds the cost of surface energy accompanying the increased surface area associated with the 3D islands compared to the flat film. The SK mode is an intermediate situation between FM and VK growth mode where at the initial stages the film morphology is flat because of the gain in energy to form A-B bond but, in the later stage of growth, B-B bonding energy is dominant compared to A-B bond energy and the film forms 3D islands. Following the reports of the observation of coherent 3D islands for “high” lattice mismatch of ~ 2 % to ~ 7 % for the tetrahedrally-bonded semiconductors (Fig. A.2), much continuum solid and thermodynamic ground state description based literature evolved describing the change in the morphology (initial 2D growth, or island growth in the SK mode, or ripening of islands) of a strained heteroepitaxial film as a function of the deposition amount and the lattice mismatch as decided by balancing the reduction of strain energy through elastic relaxation in 3D islands 346 against the cost of increased surface energy schematically shown in Fig. A.3 [A.15- A.19]. Figure A.3. Schematics of strain relaxation for lattice-mismatched strained heteroepitaxial growth with 3D island formation and elastically expansion of lattices inside the islands. The above noted thermodynamic ground state arguments capture the essence of the driving force for 3D island formation and act as a very good first cut at answering the question of “what controls island formation”. Experimental findings indicate that the real strained heteroepitaxial growth in SK mode is much more complex and rich in its physics. With the advance of in technologies to examine the island formation at increasingly microscopic level, thermodynamics base description is no longer adequate in advancing the understanding necessary for control the formation of nanostructures of size similar to 3D islands. The onset of the 3D island growth mode is highly dependent on the growth conditions and in many cases can be delayed via proper control of growth dynamics. Kapre et al. [A.20, A.21] has demonstrated that the 3D island formation can be suppressed with much smoother top interface and less islands between InGaAs and GaAs during the growth of In 0.5 Ga 0.5 As film on GaAs (001) substrate by reducing the growth temperature from 475°C to 420°C. Additionally, the growth rate of islands depends on its size as first 347 shown by Guha et al. [A.7]. As the 3-D islands become larger, extended defects form first at the island edges where the stress gradient is the steepest. The plastically relaxed islands are less strained than other coherent islands and become preferred adatom incorporation sites with continued deposition. Thus, the (large) dislocated islands grow faster than the (smaller) coherent islands. All these reveal the critical significance of the growth kinetics in affecting strained growth. Therefore, an atomistic and kinetic view of strained epitaxy is necessary to understand the growth of strained structures and account for the totality of experimental observations. Madhukar et al. in the early 1980s pursued the alternative conceptual approach to examining the nature of thin films by following their growth and viewing the problem as the dynamic evolution of the spatial configuration of a changing number of particles in an open system [A.22-A.25]. Understanding the evolution of a system of changing number of interacting particles requires examining the dynamics of growth evolution at an atomistic level with full consideration of the kinetics of the various processes involved. An atomistic configuration-dependent- reactive-incorporation (CDRI) model of epitaxial growth from vapor phase has been developed. We capture below the important kinetic processes involved in CDRI model for lattice mismatched heteroepitaxy: 1. Sticking coefficients for group III atoms and absorption coefficients for group V molecular (diatomic) species 2. Inter-planar migration jump rates of group III atoms that depend upon the local bonding configuration 348 3. Inter-planar migration rates, including asymmetry between up and down jump rates 4. Group III local number and configuration dependent reaction rates for the dissociative molecular reaction of physisorbed group V molecules and associative reaction of the chemisorbed group V atoms to covert back to physisorbed molecular group V species and its subsequent desorption. Basic kinetic process involved in CDRI model and the atomic view of GaAs homoepitaxy and lattice matched heteroepitaxy are captured in Appendix B. The local strain induced by lattice mismatch is included in the CDRI model through the modification of (1) intra- and inter-layer migration rate and (2) group V molecular dissociative reaction rates at cluster edges where the strain energy cost is lower than that in the central terraces of the clusters [A.23]. Compressive and tensile strain are taken to change the kinetic rates in opposite directions at different rates. From the Monte-Carlo simulations of the strained growth based upon the generalized CDRI model that incorporates strain effects, the types of strain effect on the atomic level kinetic processes significant to understanding strained epitaxy are [A.26]: 1. Enhanced intra-planar jump rates for compressive overlayer stress 2. Enhanced inter-planar up-jumps over down-jumps for compressive overlayer stress 3. Local strain and hence cluster size dependent activated incorporation behavior for atom incorporation at the cluster edge and at cluster coalescence boundaries 349 4. Enhancement in the incorporation rates of physisorbed molecular anions at steps and cluster edges compared to that on the flat terraces or central region of the cluster Figure A.4 shows the schematic representation of the generalized CDRI model with strain effect incorporated. At the earlies stage of the growth when 2D clusters form, the atomic relaxation with respect to the substrate lattice varies in space (Fig. A.4 (a)). With increasing 2D cluster size, incorporation of atoms at periphery will become increasingly energy-cost-ineffective. The strain dependent atomic interplaner migration is shown in Fig.A.4 (b). The upward migration is favored under strain owing to increasing strain relaxation from away from the interface. The net upward migration is characterized by activation barriers that depend upon both the volume strain relief to be gained and the cost in surface energy. In the growth of film under compressive strain, incorporation of arriving adatoms to the edge of 2D or 3D islands encounters a high potential barrier for incorporation owing to the need for a coordinated movement of several atoms surrounding the incorporation site. Additionally, at the edges of the formed islands exist the steepest stress gradients that provide a driving force for upward migration of adatoms that do arrive at the edges [A.27]. Formation of 3D islands emerges from an initial formation of 2D clusters as a function of increasing surface coverage (Fig. A.4(b)). Consequently, the formation of defect-free (coherent) 3D islands (Fig. A.2 (a)) for a certain lattice mismatch regime is the natural course of growth evolution. 350 Figure A.4 (a) Schematic of the strain in affecting the atomic kinetic processes and the initial 2D cluster growth. (b) Schematic of local, cluster edge, stress-driven enhanced tendency for up-ward interplanar adatom jump and formation of 3D islands. Given the above mentioned kinetic process, the 3D islands should have self- limiting base size feature which was confirmed experimentally in the work of Mukhametzhanov et al. [A.28]. The growth evolution of 3D island formation was studied by Ramachandran et al [4.29, 4.30] through systematic structural and optical study of InAs grown on GaAs as a function of InAs deposition from 0.87 ML to 1.61 ML. Various structural features were observed in-situ STM/AFM imaging [4.29] and were classified as 2D (i.e. 1ML high) small (< 20nm) and large, quasi-3D (Q3D) clusters small and large, 3D islands (i.e. >5ML high). Figure A.5 below shows the density of various InAs features as a function of InAs delivery amount extracted from in-situ UHV STM studies. 2D clusters form first on the sample, followed by the formation of Q3D clusters before the onset of 3D islands. The small Q3D clusters disappear with 1.35ML deposition and reappear with 1.45ML deposition and 351 eventually disappear with 1.75ML deposition. Randomly formed 3D islands density rises with 1.57ML deposition and rises rapidly with 2ML deposition. It clearly reveals the nature of the evolution of 3D islands from 2D clusters and also indicated the critical role of Q3D clusters as precursors providing kinetic pathway for the evolution of 3D islands. These observations reveal that the kinetics of island formation is more relevant to the strained epitaxy. Figure A.5. Density of (a) 2D clusters, (b) Q3D clusters and (c) 3D islands as a function of InAs deposition amount extracted from in-situ STM studies from Ref. A.29. 3D island formation can be controlled and manipulated based on the understanding of lattice misfit strain affected growth kinetics. Size uniformity of 3D islands can be controlled by controlling the deposition amount. 3D islands formed at the early stage of island formation are of small size with large size variation (FWHM in interband PL ~ 100 meV). At the later stage, the formed 3-D islands cause tensile 352 strain field in the substrate which directs the adatom migration preferentially toward small islands resulting in more uniform 3D islands on the substrate [A.31] with improved size variation (FWHM in interband PL ~50meV). Mukhametzhanov et al. took advantage of such kinetics to form uniform island distribution by introducing the punctuated island growth (PIG) approach wherein the growth is punctuated after an initial deposition amount (say, of ~ 2 ML InAs) to allow the existing In adatoms to organize. The PIG technique gives extremely uniform distribution of SAQDs as manifested in the narrowest FWHM of ~ 23 meV in the interband PL peak [A.28]. In general, 3D island size, uniformity and density can be controlled through the control of growth kinetics by choosing right growth parameters and procedures [A.31]. To convert the 3D islands into quantum dots, the islands need to be buried (capped) without inducing defects. The islands induced stress affects the growth of capping layer. The 3D InGaAs islands formed on the GaAs substrate have increasing lattice relaxation as a function of the distance from the interface [A.7]. The capping GaAs layer grown on top of islands shows a depression that tends to decrease with increasing capping layer thickness due to the stress between InGaAs islands and the GaAs capping layer as shown in Fig. A. 6 (a). Ga adatoms are driven away by the tensile stress at the island surface to form GaAs preferentially in regions between islands [A.32]. For capping layers thin enough (~ 10 nm for InAs/GaAs system), the strain field propagates to the surface of the capping layer and lateral gradients in the surface stress are created as the region of the capping layer material just above the 353 island apex is in tensile strain to minimize strain energy while the regions away are in compression [A.32]. Xie et al. estimated that the range of the island induced stress in the capping layer to be from 10 nm to 40 nm from the island surface. It was argued that such strain gradient will induce a preferential migration of adatoms deposited for the formation of the subsequent island layer to the most tensile strain region above the buried island apex to minimize the local strain energy [A.32]. As a result, the 3D island QDs can be vertically aligned and the degree of vertical self- organization can be controlled through control on the capping layer composition and thickness. Figure A.6 (b) shows the schematic capturing the essential nature of atomistic phenomenon of stress-gradient directed surface migration of adatoms to realize vertical alignment of 3D island QDs. The first modeling of the dynamics of such vertically self-organized 3D island QDs formation as well as experimental demonstration was done by Xie et al [A.33]. Figure A.6 (c) shows the cross-section TEM image of a stack of five island QD layers of InAs separated by 36ML GaAs spacer layers. The tensile strain at the growth front increases the total amount of In delivery needed for 3D island formation [A.33, A.34] and increases the size of the typical islands in the upper layers. An important implication of vertical self- organization is that the size uniformity of the 3D island QDs within the layer is improved with deposition of subsequent layers, as subsequently demonstrated in references [A.35, A.36]. 354 Figure A.6. (a) Cross-section TEM image of MBE growth of GaAs cap layer interspersed with AlGaAs 3ML marker layers from Ref. [A.32] (b) Schematic of inhomogeneous stress distribution at the GaAs spacer layer surface responsible for directed indium migration leading to vertically self-organized growth of 3D islands from Ref. [A.33] (c) Cross-section TEM image of a stack of fine InAs 1.74ML deposit islands separated by 36 ML GaAs spacer grown at 400°C by migration enhanced epitaxy from Ref. [A.33] To summarize, the lattice mismatch induced strain affects the atomic scale kinetic processes such as atom migration and incorporation that control the formation and evolution of the 3D islands. Controlling the local strain through indium concentration and deposition amount, the synthesized 3D islands size, uniformity and density can be manipulated. §A.2.2 Self-assembled quantum dots: Optical properties and Applications Shortly after the demonstration of coherent 3D island formation, these island SAQDs were demonstrated to be efficient light emitters when buried under a protective overlayer (without inducing defects) [A.8, A.9] through study of their ensemble photoluminescence (PL). A lot of studies have been carried out on the electronic structure and the optical transitions of the SAQD quantum confined states 355 [A.6, A.37, A.38]. By engineering the growth process, SAQD shape can be controlled to have pyramidal shape with shallow or deep sidewalls [A.39] or to have shell like shape [A.40]. The change of shape modifies the ensemble average transition energy, energy separation between QD ground and excited states, as well as the oscillator strengths of the transitions [A.39-A.41]. Additionally, the size uniformity of the 3D island QDs within the layer can be improved using the vertical self-organization of the SAQDs [A.35, A.36]. The advances achieved in the synthesis and the controlled realization of electronic structure/ response of the ensemble behavior of these SAQDs make SAQDs suited for various advanced optoelectronic device applications such as lasers [A.42, A.43], amplifiers [A.44] and detectors [A.45, A.46] etc. With development of nanolithography, emission from a single SAQD could be probed and studied [A.47, A.48]. A single SAQD has been found to have homogeneous linewidth of <100 μeV [A.49]. The fine structure of neutral exciton [A.50], the multiexctionic states (trions and biexciton) [A.51] emission and its dynamics [A.52] have been studied. The extensive knowledge of the optical properties of individual SAQDs demonstrated that single SAQDs can act as single photon source (SPS) showing anti-bunched emission statistics at 4K [A.49] and entangled photon sources using biexciton-exciton cascade process [A.53]. It opens a new application for SAQDs in the quantum information science. Details of the literature status of SAQDs acting as SPSs for quantum information processing and 356 their inherent limitations owing to their inhomogeneity in size, shape, composition as well as the random locations in space are captured in §2.1. Appendix A References: [A.1] A.P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science, 271, 933 (1996). [A.2] A. Forchel, H. Leier, B. Maile and R. Germann, "Fabrication and optical spectroscopy of ultra small III–V compound semiconductor structures," in Festkörperprobleme 28, 99 (1988). [A.3] V. A. Shchukin, N. N. Ledentsov and D. Bimberg, Epitaxy of Nanostructures , Springer Verlag: Berlin, 2004. [A.4] A. Madhukar, “Stress engineered quantum dots: Nature’s Way”, Chapter 2 in Nano-Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. [A.5] M. T. Bjŏrk, B. J. Ohlsson, T. Sass, A. I. Persson, C. Thelander, M. H. Magnusson, K. Deppert, L. R. Wallenberg and L. Samuelson, “One-dimensional steeplechase for Electrons realized”, Nano Lett. 2, 87 (2002). [A.6] M. Grundmann, Nano-Optoelectronics: Concepts, physics and devices, Springer Verlag: Berlin, 2002. [A.7] S. Guha, A. Madhukar and K. C. Rajkumar, “Onset of incoherency and defect introduction in the initial stages of molecular beam epitaxical growth of highly strained InxGa1-xAs on GaAs(100)”, Appl. Phys. Lett. 57, 2110 (1990). [A.8] Q. Xie, P. Chen, A. Kalburge, T. R. Ramachandran, A. Nayfonov, A. Konkar and A. Madhukar, “Realization of optically active strained InAs island quantum boxes on GaAs(100) via molecular beam epitaxy and the role of island induced strain fields,” J. Cryst. Growth, 150, 357 (1995). [A.9] D. Leonard, M. Krishnamurthy, C. M. Reaves, S. P. Denbaars and P. M. Petroff, “Direct formation of quantum-sized dots from uniform coherent islands of InGaAs on GaAs surfaces,” Appl. Phys. Lett. 63, 3203 (1993). [A.10] S. C. Jain, H. E. Maes and K. Pinardi, “Stresses in strained GeSi stripes and quantum structures: calculation using the finite element method and determination using micro-Raman and other measurements”, Thin. Solid Films 292, 218 (1997). 357 [A.11] F. C. Frank and J. H. van der Merwe, “One-Dimensional Dislocations. I. Static Theory,” Proc. Roy. Soc. London A, 198, 205 (1949). [A.12] S. Guha, “MBE Growth of InGaAs and AlGaAs on patterned and nonpatterned GaAs(100): A study of inter-facet migration, morphology and defect formation”, Ph.D. dissertation, Materials Science, University of Southern California, 1991. [A.13] M. Volmer and A. Weber, Z. Physik. Chem. 119, 277 (1926). [A.14]I. N. Stranski and L. V. Krastanow, Sitzungsberichte der Akademie der Wissenschaften in Wien, Mathemetisch-Naturwissenschaftliche Klasse 146, 797 (1939). [A.15] I. Daruka and A.-L. Barabási, “Dislocation-Free Island Formation in Heteroepitaxial Growth: A Study at Equilibrium”, Phys. Rev. Lett. 79, 3708 (1997). [A.16] V. A. Shchukin, N. N. Ledentsov, P. S. Kop'ev and D. Bimberg, “Spontaneous Ordering of Arrays of Coherent Strained Islands”, Phys. Rev. Lett. 75, 2968 (1995). [A.17] C. Ratsch and A. Zangwill, “Equilibrium theory of the Stranski-Krastanov epitaxial morphology”, Surf. Sci. 293, 123 (1993). [A.18] J. Tersoff and F. K. LeGoues, “Competing relaxation mechanisms in strained layers”, Phys. Rev. Lett. 72, 3570 (1994). [A.19] B. J. Spencer and J. Tersoff, “Equilibrium Shapes and Properties of Epitaxially Strained Islands”, Phys. Rev. Lett. 79, 4858 (1997). [A.20] R. M. Kapre, A. Madhukar and S. Guha, “Highly srained GaAs/InGaAs/AlAs resonant tunneling diodes with simultaneously high peak current densities and peak- to-valley ratios at room temperature”, Appl. Phys. Lett. 58, 2255 (1991). [A.21] R. M. Kapre, A. Madhukar and S. Guha, “Highle strained pseudomorphic In x Ga 1-x As/AlAs based resonant tunneling diodes grown on patterned and non- patterned GaAs (100) substrates”, J. Cryst. Growth. 111, 1110 (1991). [A.22] A. Madhukar, “Far from equilibrium vapour phase growth of lattice matched III-V compound semiconductor interfaces: Some basic concepts and monte-carlo computer simulations”, Surf. Sci. 132, 344 (1983). [A.23] J. Singh and A. Madhukar, “Surface orientation dependent surface kinetics and interface roughening in molecular beam epitaxial growth of III--V semiconductors: A Monte Carlo study”, J. Vac. Sci. Technol. B 1, 305 (1983). 358 [A.24] S. V. Ghaisas and A. Madhukar, “Role of Surface Molecular Reactions in Influencing the Growth Mechanism and the Nature of Nonequilibrium Surfaces: A Monte Carlo Study of Molecular-Beam Epitaxy”, Phys. Rev. Lett. 56, 1066 (1986). [A.25] A. Madhukar and S. V. Ghaisas, “The nature of molecular beam epitaxial growth examined via computer simulations”, CRC Critical Rev. Solid State Mater. Sci. 14, 1 (1988). [A.26] S. V. Ghaisas and A. Madhukar, “Influence of compressive and tensile strain on growth mode during epitaxical growth: A computer simulation study”, Appl. Phys. Lett. 53, 1599 (1988). [A.27] S. V. Ghaisas and A. Madhukar, “Kinetic aspects of growth front surface morphology and defect formation during molecular-beam epitaxy growth of strained thin films”, J. Vac. Sci. Technol. B 7, 264 (1989). [A.28] I. Mukhametzhanov, Z. Wei, R. Heitz and A. Madhukar, “Punctuated island growth: An approach to examination and control of quantum dot density, size, and shape evolution”, Appl. Phys. Lett. 75, 85 (1999). [A.29] T. R. Ramachandran, R. Heitz, N. P. Kobayashi, A. Kalburge, W. Yu, P. Chen and A. Madhukar, “Re-entrant behavior of 2D to 3D morphology change and 3D island lateral size equalization via mass exchange in Stranski--Krastanow growth: InAs on GaAs(001)”, J. Cryst. Growth 175, 216 (1997). [A.30] R. Heitz, T. R. Ramachandran, A. Kalburge, Q. Xie, I. Mukhametzhanov, P. Chen and A. Madhukar, “Observation of Reentrant 2D to 3D Morphology Transition in Highly Strained Epitaxy: InAs on GaAs”, Phys. Rev. Lett. 78, 4071 (1997). [A.31] I. Mukhametzhanov, “Growth Control, Structural Characterization, and Electronic Structure of Stranski-Krastanov InAs/GaAs (001) Quantum Dots,” Ph.D. dissertation, Physics, University of Southern California, 2002. [A.32] Q. Xie, P. Chen and A. Madhukar, “InAs island-induced-strain driven adatom migration during GaAs overlayer growth”, Appl. Phys. Lett. 65, 2051 (1994). [A.33] Q. Xie, A. Madhukar, P. Chen and N. P. Kobayashi, “Vertically Self- Organized InAs Quantum Box Islands on GaAs(100)”, Phys. Rev. Lett. 75, 2542 (1995). [A.34] O. G. Schmidt, O. Kienzle, Y. Hao, K. Eberl and F. Ernst, “Modified Stranski--Krastanov growth in stacked layers of self-assembled islands”, Appl. Phys. Lett. 74, 1272 (1999). 359 [A.35] C. Teichert, M. G. Lagally, L. J. Peticolas, J. C. Bean and J. Tersoff, “Stress- induced self-organization of nanoscale structures in SiGe/Si multilayer films”, Phys. Rev. B 53, 16334 (1996). [A.36] G. S. Solomon, J. A. Trezza, A. F. Marshall and J. J. S. Harris, “Vertically Aligned and Electronically Coupled Growth Induced InAs Islands in GaAs”, Phys. Rev. Lett. 76, 952 (1996). [A.37] R. Heitz, O. Stier, I. Mukhametzhanov, A. Madhukar and D. Bimberg, “Quantum size effect in self-organized InAs/GaAs quantum dots,” Phys. Rev. B, 62, 11017 (2000). [A.38] R. Heitz, I. Mukhametzhanov, A. Madhukar, A. Hoffmann and D. Bimberg, “Temperature dependent optical properties of self-organized InAs/GaAs Quantum dots”, J. Electron Mater, 28, 520 (1999). [A.39] I. Mukhametzhanov, “Growth Control, Structural Characterization, and Electronic Structure of Stranski-Krastanov InAs/GaAs (001) Quantum Dots,” Ph.D. dissertation, Physics, University of Southern California, 2002. [A.40] R. Heitz, F. Guffarth, K. Pötschke, A. Schliwa and D. Bimberg, “Shell-like formation of self-organized InAs/GaAs quantum dots”, Phys. Rev. B 71, 045325 (2005). [A.41] F. Guffarth, R. Heitz, A. Schliwa, O. Stier, N.N. Ledentsov, A. R. Kovsh, V. M. Ustinov and D. Bimberg, “Strain engineering of self-organized InAs quantum dots”, Phys. Rev. B. 64, 085305 (2001). [A.42] M. Asada, Y. Miyamoto and Y. Suematsu, "Gain and the threshold of three- dimensional quantum-box lasers," IEEE J. Quantum Electron., 22, 1915 (1986). [A.43] Q. Xie, N. P. Kobayashi, T. R. Ramachandran, A. Kalburge, P. Chen and A. Madhukar, “Strained coherent InAs quantum box islands on GaAs(100): size equalization, vertical self-organization, and optical properties”, J. Vac. Sci. Technol. B 14, 2203 (1996). [A.44] M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B, 69, 235332 (2004). [A.45] V. Ryzhii, “The theory of quantum-dot infrared phototransistors,” Semicond. Sci. Technol. 11, 759 (1996). [A.46] T. Asano, C. Hu, Y. Zhang, M. Liu, J. C. Campbell and A. Madhukar, “Design consideration and demonstration of resonant-cavity-enhanced quantum dot 360 infrared photodetectors in Mid-infrared wavelength regime (3-5μm)”, J. Quantum Electronics. 46, 1484 (2010). [A.47] J. Y. Marzin, J. M. Gérard, A. Izraël, D. Barrier and G. Bastard, “Photoluminescence of Single InAs Quantum Dots Obtained by Self-Organized Growth on GaAs,” Phys. Rev. Lett. 73, 716 (1994). [A.48] M. Grundmann, J. Christen, N. N. Ledentsov, J. Böhrer, D. Bimberg, S. S. Ruvimov, P. Werner, U. Richter, U. Gösele, J. Heydenreich, V. M. Ustinov, A. Yu, A. E. Zhukov, P. S. Kop’ev and Z. I. Alferov, “Untranarrow luminescence lines from single quantum dots”, Phys. Rev. Lett. 74, 4043 (1995). [A.49] C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot”, Phys. Rev. Lett. 86, 1502 (2001). [A.50] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke and S. N. Walck, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots”, Phys. Rev. B 65, 195315 (2002). [A.51] L. Landin, M. S. Miller, M. –E. Pistol, C. E. Pryor and L. Samuelson, “Optical studies of individual InAs quantum dots in GaAs: few-particle effects”, Science, 280, 262 (1998). [A.52] R. M. Thompson, R. M. Stevenson, A. J. Shields, I. Farrer, C. J. Lobo, D. A. Ritchie and M. L. Leadbeater, “Single-photon emission from exciton complexes in individual quantum dots”, Phys. Rev. B. 64, 201302 (R) (2001). [A.53] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs”, Nature Photon. 8, 224 (2014). 361 Appendix B: Equipment and experimental techniques In this Appendix, we discuss four major experimental techniques and setups used extensively in the present thesis research work for the research objective of studying the potential use of mesa-top single quantum dot (MTSQD) synthesized through substrate-encoded size reducing epitaxy (SESRE) as single photon sources (SPSs) for quantum information processing: (1) molecular beam epitaxy (MBE) system in which all the samples were grown; (2) scanning electron microscope (SEM) and (3) atomic force microscopy (AFM) which are utilized to measure size information on nanomesas and study the growth evolution on nanomesas; (4) photoluminescence (PL) , PL excitation (PLE) and time-resolved PL (TRPL) spectroscopy. The PL/PLE/TRPL instrumentation captured here are the pre-existing large area PL/PLE/TRPL setup in the group. Detailed discussion of the instrumentation of our home-build micro-PL/PLE/TRPL setup for studying optical property of the synthesized MTSQDs is captured in detail in Appendix C. §B.1 Molecular beam epitaxy Molecular beam epitaxy (MBE) is a vapor phase crystal growth technique and is currently the most advanced technique to study crystal growth phenomena as well as to grow semiconductors heterostructures and oxides. In MBE growth, materials are delivered to substrates from sources as vapor under ultra-high vacuum (UHV, ~10 -9 torr) condition with long mean free path (> 100cm). The impinging flux thus reaches the substrate ballistically without collision between the atoms and/or molecules unlike MOCVD (metal organic chemical vapor deposition). MBE 362 offers several advantages as epitaxial techniques in comparison with others such as: (1) UHV environment with utilization of pure sources (typically 7N or better) minimizes impurity incorporation (typically, < 10 14 /cm 3 background impurity); (2) MBE allows integration with surface analyzer tools (such as RHEED, reflection high energy electron diffraction, requires UHV environment) to monitor in-situ the surface condition and its dynamic evolution (i.e. monolayer growth) during growth; (3) growth thickness can be calibrated using RHEED and controlled to obtain accuracies of one atomic layer or even down to sub-atomic layer. The sub-monolayer (ML) control on the deposition amounts originates from the Ballistic nature of the imping flux which allows sharp interruptions through the control of the K-cell shutter shut off time to obtain ~ 0.01 – 0.1 ML deposition accuracy; (4) as the substrate temperature can be independent of the temperature of source materials, growth can be performed at relatively low temperature compared to other traditional epitaxial growth techniques such as liquid phase epitaxy (LPE) or standard vapor phase epitaxy (VPE), which minimizes inter-diffusion of materials at interfaces; and thus, the abrupt junctions of heterostructures can be created. These features make MBE the most suitable technique to grow quantum nanostructures for high performance optoelectronic devices. §B.1.1 Conceptual picture of MBE MBE growth process can be simply visualized as impingement of a beam of atoms/molecules on a single crystal substrate from the vapor phase as schematically 363 shown in Fig. B.1. The atomic/molecular vapor beams are generated through Figure B.1 Schematic picture of molecular beam epitaxy. material evaporation by heating crucibles containing elemental sources using a filament wrapped around. Upon impingement these atoms/molecules react with the substrate surface at growth temperature controlled by the substrate heater and are incorporated into the single crystal substrate epitaxially. The advance in experimental RHEED data obtained during crystal growth and theoretical computer simulation revealed the critical role of cation and anion kinetics at growth front in understanding crystal growth [B.1]. Comprehensive ideas about MBE growth mechanism and detailed description of the MBE growth process are captured in a review by Madhukar and Ghaisas [B.2]. The elementary kinetic processes that lead Electron gun K-cell Shutter Moly block Heater RHEED screen Electron beam Substrate Molecular/atomic flux 364 up to the incorporation of the impinging species can be succinctly summarized as follows: The rates for these kinetic processes are controlled by the substrate temperature and the local configuration around the atomic or molecular species. For the growth of III-V compound semiconductors, cation atoms (Ga or Al) and anion molecules (As 4 or As 2 ) are impinged on the compound semiconductor substrate surface. The sticking coefficient of group V anions on anion-stabilized surface is essentially zero at any reasonable growth temperature while the sticking coefficient of group III cation atoms is unity regardless of the types of surface atoms. Thus MBE growth is generally performed under high overpressure of group V element (flux V to flux III ratio is typically 10 to 50), so that arriving cation species always have available anion bonds for incorporation into the crystal and the sticking of arriving group V element adjusts to the III-group element coverage to keep group V element incorporation as required for the one-to-one stoichiometry of the compound. The growth rate and the evolution of the growth front are hence mainly Random Impingement Physisorption (for molecular species) Surface Migration Chemisorption (dissociation of molecules) Surface Migration Epitaxial Incorporation Desorption Desorption 365 controlled by cation impinging rate and kinetics of cation adatom migration, respectively. Impinging cations migrate on the substrate surface and get incorporated at various incorporation sites on the substrate surface. Incorporation of cations at sites at low energy such as kinks/terrace edges results in two-dimensional (2D) layer- by-layer growth. To obtain a smooth layer by layer growth with best quality of the grown planar films, the cation migration length should be maximized to enhance the probability of adatom reaching and being incorporated into terrace edges for enhanced average terrace width, hence improved structural quality of the film. For a given cation flux, two parameters are important in controlling cation migration length: (1) substrate temperature and (2) anion flux. Since the cation diffusion is a thermally activated process, to realize the longest possible adatom diffusion length for the best quality of materials, sufficient thermal energy needs to be provided to the adatoms via substrate temperature. However, very high substrate temperatures may cause significant desorption of group V anions and in turn cause shortage of incorporation sites for III-group cations which leads to the formation of metallic clusters on growth surface that degrade crystal quality. The deficiency caused by very high substrate temperature is equivalent to that caused by low group V flux. On the other hand, very high anion flux (pressure) can lead to the reduction of cation migration length due to the reduction of free migration time of cation before being pinned down by anion incorporation. Therefore, for any given growth rate (group III controlled) there is an optimum combination of substrate temperature and group V flux to achieve the smoothest growth. In our MBE machine, for GaAs homoepitaxy 366 we generally employ growth rate of 0.2-1ML/sec, substrate temperature T s ~580- 600 0 C, and As beam equivalent pressure ~2-6E-6 Torr. Definitions of these terms and their calibration will be addressed later in section §B.1.2 and §B.1.3. Besides the control of group III and V flux and substrate temperature in the standard continuous MBE growth mentioned above, the time sequence of the supply of group III and V flux can be modulated to enhance cation migration length for the growth of highest quality structures, especially heterostructures [B.3]. Growth interruption has been introduces based on the understanding of kinetics of MBE growth [B.4] and used to heal and reduce surface step density for sharp interface between heterostructures with high quality layer growth, i.e. GaAs/AlGaAs. In the growth of heterostructures, i.e InAs/GaAs, the two material of heterostructure require two different growth temperatures (~350-500°C for InAs, ~580 – 600°C for GaAs) controlled by the different kinetic rate and activation energy for chemical reaction of these two cations. To enhance the migration length of III-group adatoms even at the temperature significantly lower (~ 350 – 500°C for GaAs) than the optimum growth temperature for standard MBE growth (~ 580 – 600°C for GaAs) for high quality material growth with sharp interface, V-group flux can be reduced to enhance diffusion time of III-group adatoms and hence migration length. In the extreme case, V-group flux can be completely terminated during III-group flux delivery. Due to the absence of V-group flux during III-group flux impingement, the diffusion length of III-group adatoms is significantly enhanced at low temperature. The growth of III-V compound is done with alternating supply of III-group and V- 367 group flux. After the delivery of 1-ML worth amount of III-group, V-group flux will be delivered without III-group flux to complete the formation of the III-V compound. Such growth technique was, first, demonstrated by Yen et al. [B.3], and later studied in detail by Horikoshi [B.5] and was named as migration enhanced epitaxy, MEE for short. Growth of GaAs at low temperature (T < 400°C) using MEE approach to cap InAs SAQDs has been demonstrated to significantly lower point defect density in the low temperature MEE grown GaAs cap, thus enhancing capture rate of the excitons generated in the GaAs by the quantum dots and provide photoluminescence efficiency than standard MBE GaAs (400 – 480°C)-capped InAs SAQDs [B.6, B.7]. During this dissertation works, the MEE growth condition was employed to grow GaAs capping after the growth of InGaAs on mesa top. The growth condition for MEE was set to be 4 sec/ML of Ga flux and As pressure ~ 1 × 10 -6 torr with As#2 cell. After the Ga cell shutters was opened for the 1-ML growth time, the Ga cell shutters were closed, and simultaneously the As#2 cell shutter was opened. At the end of the Ga cell shutter opening, the RHEED pattern momentarily (for less than ~ 0.5 sec) showed Ga (4 × 2), Ga rich surface. Then, arsenic is supplied for 6 second to form As-stabilized surface. The growth condition was tested with RHEED to ensure high RHEED specular spot intensity for more than 10ML MEE growth. Details of calibration of supply time for arsenic and corresponding RHEED oscillation is discussed in §B.1.3. §B.1.2 Description of RIBER 32P MBE System 368 The experimental results used in this dissertation were obtained from samples grown in the RIBER 3200P MBE machines. The RIBER MBE machine at USC is part of a six UHV chamber interconnected Growth-Processing-Characterization system as shown in Fig. B.2 (as shown as Fig. 2.9 in Chapter 2, recaptured here for discussion purpose) which contains (a) molecular beam epitaxial (MBE) growth chamber, (b) UHV chemical vapor deposition (UHV-CVD) chamber with electron cyclotron resonance (ECR) plasma source, (c) an RF-plasma enhanced chemical vapor deposition (PECVD) chamber, (d) a focused ion beam (FIB) assisted direct – write patterning chamber, (e) a metallization/H-cleaning chamber, and (f) a UHV STM/AFM system. Here we discuss only the MBE part of the system. Figure B.2. Inter-connected UHV semiconductor growth, processing and characterization system at USC. The MBE system consists of a sample introduction chamber (module B in Fig. B.2), a growth chamber (GC), and sample transfer system (Modutrack TM ) inside FIB Chamber H-Cleaning/ e-beam metal deposition Chamber A B PECVD Chamber E C D F G H III-V MBE Growth Chamber Dry Glove Box ECR plasma-assisted UHV-CVD/etching Chamber 1 UHV-STM/AFM chamber UHV sample transfer modules Heating station FIB Chamber H-Cleaning/ e-beam metal deposition Chamber A B PECVD Chamber E C D F G H III-V MBE Growth Chamber III-V MBE Growth Chamber Dry Glove Box ECR plasma-assisted UHV-CVD/etching Chamber 1 UHV-STM/AFM chamber 1 UHV-STM/AFM chamber UHV sample transfer modules Heating station 369 modules for transferring the sample between these two chambers. Substrate loading is through an environment controlled glove box attached to the loading chamber of the system (module A in Fig. B.2). The substrate is transferred to sample introduction chamber and degassed on the heating station in module C by heating the sample holder resistively to above 250°C to drive off moisture and other adsorbed impurities before it is moved to the growth chamber. The introduction chamber is pumped by a turbomolecular pump and isolated from the growth chamber by a gate valve. The introduction chamber acts as a buffer between the atmosphere and the growth chamber. Breakage of vacuum in sample loading process does not compromise the vacuum level in GC. This ensures a very clean environment in the growth chamber. The sample is transferred from the introduction chamber to the growth chamber with the aid of a magnetically coupled transfer rod. The sample holder is mounted on a manipulator inside the growth chamber. The GC is then isolated from the introduction chamber by closing the gate valve after withdrawing the transfer rod. The RIBER MBE growth chamber mainly consists of (1) a pumping well with three different pumps (ion pump, cryopump, and titanium sublimation pump (TSP)) and one ion gauge, (2) a source flange equipped with effusion cells containing III and V group charges (Knudsen cell, or K-cell), shutters, and a cryopanel, (3) a sample manipulator with a substrate heater and an ion gauge to measure source beam flux. The MBE chamber is also equipped with in-situ monitoring and characterization tools such as Residual Gas Analyzer (RGA) and 370 RHEED to enable real time in-situ chamber condition and growth monitor. We will address the detailed description and characteristics of each components, its role and calibration for growth below. (a) The Pumping system The RIBER MBE growth chamber contains three UHV pumps: ion pump, cryopump, and titanium sublimation pump (TSP) to minimize impurity species in the chamber and their incorporation into grown samples. Each pump is effective for certain type of molecule species and the three combined maintains the chamber pressure to be <5E-9 torr with low background doping of <5E14/cm 3 induced by impurity incorporation in the grown sample. The ion pump consists of a flat titanium cathode, a cylindrical anode and an axial magnetic field. Gas is ionized by energetic free electrons and the positive ions bombard the cathode, sputtering titanium to form getter films on the anode and the opposite cathode. The titanium reacts with all active gases forming stable compounds, and also a considerable number of bombarding gas molecules are buried in the cathode. Therefore, the ion pump is most effective to pump reactive gases with reasonable ionized efficiency such as H 2 , O 2 , N 2 , H 2 O, CO 2 etc. The pumping speed of our ion pump is ~ 400 l/sec. Cryopump captures gases by condensing gases at low temperature (~ 10 K) into porous materials (activated charcoals, in our system) and retaining it there. The pumping speed is proportional to the surface area refrigerated and is usually very fast owing to its high surface area, and that of our cryopump (Oxford, CT-8) is ~ 2,000 371 l/sec. The cooling of the cryopanel is carried out in the same principle as household refrigerators but with highly compressed (~ 200 to 250 psig) ultra-high purity (99.999%) He used as a coolant. Cryopump is effective to pump high vapor pressure gases, particularly water. The cryopump has to be regenerated at intervals depending upon the gas load pumped, and when the cryo surfaces are fully 'loaded' with condensate. TSP is a gettering pump utilizing high reactivity of Ti, and most effective for pumping reactive species such as O 2 , CO and water but is very ineffective at pumping inert components such as the noble gasses. High electric current ( ~ 47 A) is passed through Ti filament periodically (typically once every few hours to 24 hours, depending on the GC chamber background pressure) to sublimate Ti to deposit Ti on the inner walls of the pumping well. The Ti film getters gas molecules impinging on it. Since firing TSP outgases impurities and temporarily raises background pressure, TSP is turned off during growth. These three pumps combined can pump both inert gases and reactive gases well and can hold the GC chamber at pressure <5E-9 torr at room temperature. Chamber total pressure is constantly monitored by Bayard Alpert ionization pressure gauge. In the Bayard Alpert ionization gauge, thermally emitted electrons are accelerated under high bias (140 V) and the accelerated electrons collide with gas molecules to ionize. Then, the ionized molecules are collected as ion current which is proportional to the pressure. The pressure measurement by ionization gauge varies 372 with ionization efficiency of gas species, and our gauge is calibrated with ultrahigh pure (99.999%) nitrogen gas. In addition to the monitoring of the total pressure of the chamber by the ion gauge, our MBE chamber is also equipped with residual gas analyzser (RGA) to find the partial pressure of contributing molecule species in the chamber. The total pressure and RGA spectrum of partial pressure of each individual molecule tells us the overall condition (cleanness) of chamber and the three pumps. The main component of RGA is quadropole mass spectrometer that can filter the ions by its mass-to-the-charge ratio. The working principle of RGA except for the quadropole part is essentially the same way as Bayard Alpert ionization gauge, namely ionized molecules by accelerated electrons are collected and ion current is measured for each mass-to-the-charge ratio. Thus, to accurately obtain the partial pressure of each mass-to-the-charge ratio, the measured ion current needs to be weighted with ionization efficiency of each species. Given that the ionization efficiency of the species of our interest is typically within factor of two to three with respect to nitrogen [B.8], the partial pressure readings of most of the species are correct within factor of two to three in our RGA (MKS, Microvision Plus) which is calibrated with ultrahigh pure (99.999%) nitrogen gas. (b) Effusion cells The RIBER MBE growth chamber contains a source flange which is equipped with seven Knudsen effusion cells (K-cells) (shown in Fig. B.3) for source evaporation, shutters to turn on/off the source fluxes, and a cryopanel. Each K-cell 373 contains a crucible (39 cc) made of pyrolyte boron nitride (PBN), a tantalum filament based heating assembly, tantalum foil for shielding the heat and a type C thermocouple (5%Rh-W/25% Rh-W) to monitor cell temperature. Three K-cells and crucibles are used for group III (Ga, Al, and In), two for dopants (Si and Be) and two for group V material (As). Cells shutters are mounted at the mouse of K-cell inside GC and used to start/terminate the flux to the substrate. It is controlled by a stepper- motor with time accuracy of 0.1sec. A master shutter is placed in front of all the cells and is used for avoiding initial transient overshooting of the fluxes upon opening of the cell shutters. The cell temperatures are automatically controlled by proportional- integral-derivative (PID) controllers (Eurotherm model 818, 820, 2408, or 2404) that give PID feedback signal to the power supplies and control the output power to the cells to control the cell temperatures within < ± 0.5 °C with the models 818 and 820, and < ± 0.1 °C with the models 2408 and 2404. All the cells are usually kept at stand-by temperatures as shown in table II.1. In order to avoid thermal interference between cells and the outgasing of impurities from the vicinity of heated cells, liquid nitrogen is passed through the cryoshroud during growth. 374 Figure B.3. Schematic drawing of K-cell and substrate geometry for Riber 3200 MBE system at USC. Table B.1 List of cell stand by temperatures for Riber 3200 MBE system Cell As 1 As 2 Ga In Al Si Be Temperature 100° 100° 600° 500° 700° 600° 250°  Uniformity of flux The uniformity of composition and thickness of the structures grown by MBE depend on the uniformity of the beam flux and hence are related to the geometry of source position and the substrate. The fluctuation in composition and thickness of structure can result in inhomogeneous broadening of emission line. It is known that for InGaAs SAQDs on GaAs (001), for an example, its emission energy can vary over 50meV with 10% variation of the size (<1-2nm) and its size distribution is very sensitive to even 0.05ML deposition. Similarly, it has been established that a variation of QD height of 1ML can shift QD emission energy by ~10-25meV for Substrate Normal ϕ γ=7° σ ψ Horizontal Plane Substrate γ=20° α=33° α=11° 5° Si As#2 In Al Ga Pyro As#1 Be Substrate Normal ϕ γ=7° σ ψ Horizontal Plane Substrate γ=20° α=33° α=11° 5° Si As#2 In Al Ga Pyro As#1 Be 375 truncated pyramidal QD, same shape as our MTSQDs [B.9-B.11]. Maintaining high flux uniformity for the synthesis of uniform MTSQDs array is an important requirement. The flux distribution in our growth chamber (GC) was calculated in detail by Hu [B.12] and Xie [B.13] in their thesis. Below we reproduce the most important parts of their results of flux uniformity with and without rotating substrate. Figure B.4 captures the substrate-source relation in a Cartesian coordinate with notification defined below. Point O: Center of the substrate and the origin of the axis Point S: The orifice of the cell Point P: An arbitrary point on the substrate that can be represent as (x,y,0) in the Cartesian coordinate z: The normal of the substrate x: The projection of OS to the substrate plane along z direction y: The direction perpendicular to x and z direction θ: The angle between OS and PS δ: The angle between PS and z axis φ: The angle between OS and z axis Assuming the source-cell is an ideal Knudsen-effusion-cell, the flux at any point P (j P ) with respect to the flux at point O (j o ) shown in Fig. B.4 can be expressed as [B.14] 376 2 o P o cos cos cos r j jd        (Eq. B.1) where r o and d is the distance between OS and PS respectively. Using coordinates at point P (x,y,0) and point S (r 0 sin ϕ, 0, r 0 cos ϕ), Eq. B.1 can be written as 3 00 P 2 2 2 2 O 0 0 ( sin ) ( 2 sin ) r x r j j r x y xr        (Eq. B.2) Considering the geometry of the chamber, the distance r 0 is 127 to 152 mm and value for x and y is ±25.4mm due to the 2” size of moly-blocks used. The term ( 𝑥 𝑟 0 ) 3 and ( 𝑦 𝑟 0 ) 3 is less than 8E-3 and can be ignored. Therefore, Eq.2.2 can be simplified as below 2 2 2 2 P 2 O 0 0 3 sin 2( 2 sin ) 1 j x x y x j r r      (Eq.B.3) From Eq.B.3, the flux non-uniformity in the case without substrate rotation is dominated by the linearly x-dependent term. When substrate rotation is used during the growth, the flux may be averaged over the eccentric circles if the number of rotation is large enough during the growth of the relevant layer. The averaged flux can be written as: 2 0 2 2 2 0 O P ) sin 1 ( 2 1 2 1 r r d j j j j O P          (Eq.B.4) where Θ is the angle between OP and x axis and 22 r x y  . Flux distribution now has a central symmetry with the linearly x-dependent term being averaged out with substrate rotation. 377 Figure B.4. Geometry of cell and substrate for the calculation of flux distribution over substrate surface. Given the cell geometry of the MBE chamber as shown in Fig.B.3, the distribution of flux coming out of cells has been calculated. Figure B.5 shows the calculated flux distribution of flux coming out of Ga or In cell which have same φ and r 0 value. The relative flux can vary from 0.92 to 1.06 over a ±10mm region without substrate rotation as shown in Fig. B.5(a). The flux variation within ±10mm region can be improved to be varying from 0.99 to 1 (Fig. B.5(b)). For all our MTSQD samples, the patterned nanomesa region are designed to be with in ±10mm region with respect to the center of the substrate and the subsequent growth on patterned substrate for the formation of MTSQDs is done with substrate rotation employed. d O r 0 P (x,y,0) r  Substrate surface ϕ δ D B S θ Source x y d d O r 0 P (x,y,0) r  Substrate surface ϕ δ D B S θ Source x y dz 378 Figure B.5. Panel (a) and (b) shows the calculated Ga or In flux distribution in the case without substrate rotation and with substrate rotation. (c) The Sample Manipulator The sample manipulator of the growth chamber, hosting a substrate heater and a Bayert-Apert type ion gauge, has the capability of 5-axis motion (x-, y-, z- transverse axes, orbital rotation and spinning rotation). 379 The moly-block with substrate mounted is transferred from Mod B into growth chamber and positioned on the substrate heater by a clamp mechanism. The substrate can be rotated about the axis perpendicular to its surface at a rotation speed up to 60 rmp. The rotation degree of freedom about the mounting axis of the manipulator on the chamber allows the manipulator to be adjusted to different orientation, e.g. the growth position (substrate heater facing the cells), sample transfer position (substrate heater facing Mod B) and beam flux measurement (BFM) position (ion gauge facing the cells). The lateral (parallel to the substrate surface) and angular position of the growth position is calibrated by aligning the center of the substrate heater with respect to the center of the central view-port of source flange. The vertical (normal to substrate surface) position of the sample is adjusted with respect to RHEED axis so that the diffraction condition of electron beam satisfies the off-Bragg condition for the one monolayer step height of GaAs to maximize the sensitivity of RHEED intensity to the step density on the sample surface [B.15]. Sample is radiatively heated by tantalum filament behind the moly-block on the substrate heater. The temperature of the substrate is sensed by a type C thermocouple hosted in the central hole on the backside of the moly-block. A Eurothern 820 temperature control unit takes the output from the thermocouple and sends a PID feedback to the corresponding power supply unit to control the substrate temperature with accuracy of <±0.5°C. However, the temperature measured by this thermocouple behind the Moly-block does not represent the actual sample surface temperature. A pyrometer that can directly measure the sample surface temperature 380 by sensing the infrared intensity of the heated object is used to determine the sample surface temperature. The principle and calibration method of the pyrometer is to be discussed in a later subsection §B.1.3. A Bayard-Alpert ionization pressure gauge on the manipulator is used to measure molecular beam flux equivalent pressure. Pressure of gases, p, can be, in general, converted to flux, j, using the following relationship under ideal gas approximation [B.16], T Mk p j B  2  (Eq.B.5) , where M is the molecular mass, k B is Boltzmann constant, and T is temperature of the molecules. The pressure measured by the ion gauge provides a guidance on the flux impinging on the substrate but not representing directly the real absolute value of impinging flux due to the difference of ionization energy of different atoms/ molecules [B.17] and the different geometry of each cell with respect to the pressure gauge. Therefore, absolute values of fluxes need to be determined by another in-situ method such as RHEED oscillation discussed in detail in §B.1.3. (d) Reflection High-Energy Electron Diffraction (RHEED) Reflection high energy electron diffraction (RHEED) is a very powerful technique and widely used in charactering the crystal surface structure and the most critical tool for in-situ control of MBE growth. In this section, the principle and surface characterization using RHEED is briefly reviewed.  Basic concept of RHEED 381 The basic concept of RHEED is captured in Fig.B.6 (shown in Fig.2.10 in Chapter 2, recaptured here for discussion purpose). Electrons thermally emitted from a tungsten filament are accelerated to anode that is biased at high potential of 10- 20kV. These accelerated electrons have the wave characteristic with wavelength predicated by the equation below due to wave-particle duality:           2 0 0 e 2 1 2 c m E E m h  (Eq.B.6) where h is Planck’s constant, m 0 is electron rest mass, E is kinetic energy of electron, and c is speed of light. The term E/2m 0 c 2 in the denominator is derived from relativistic effect. For energy of 10kV, the wavelength of electron is ~ 0.12 Å, which is ~ 10 times shorter than atomic distance of a typical crystal, and thus, is suited to characterize atomic arrangement of solids. The accelerated electron beam is then incident on a substrate at a glancing angle less than ~ 1 degree. Because of the use of glazing incident angle, the momentum component along the normal of the surface is small. The electrons, hence, can only penetrate into the crystal for at most several monolayers and is very sensitive to the crystal surface. The electrons can be diffracted by the surface lattice following the general Bragg condition and the diffraction pattern that can be constructed from the Ewald sphere is observed on a phosphor screen (shown in Fig. B.6). Given the small interaction depth between electron beam and crystal, the reciprocal lattice points of this interacting volume is elongated in the direction perpendicular to the surface. The length of these elongated rods is inversely 382 proportional to the thickness of the interacting volume. Because of this elongation, the overlap between Ewald sphere and the reciprocal rods are also elongated in a perpendicular direction, giving rise to streaky RHEED pattern from the crystal surface on the projected screen. Figure B.6. Schematic picture of RHEED with geometry of sample, diffracted beams and RHEED screen. The incident beam is at glancing angle with respect to the substrate surface. The diffracted electron beam is projected on fluorescent screen. The relative phase of the diffracted beams from two consecutive planes parallel to the surface plane with separation d in between can be varied by adjusting the electron beam incident angle θ. The intensity of diffracted beam from two consecutive planes can reach maximum when a constructive interference occurs with incident angle following the Bragg condition below: mλ=2dsinθ, m=0,1,2…. (Eq.B.7) 383 or the intensity can reach its minimum when a destructive interference occurs when the incident angle is set at the off-Bragg condition below: (m+1/2)λ=2dsinθ, m=0,1,2…. (Eq.B.8)  Surface reconstruction and surface phase diagram determination The surface structure is usually different from the termination of bulk lattice due to the reconstruction of dangling bonds. In case of GaAs (001) surface, in both cases for Ga-stabilized surface or As-stabilized surface, the surface atoms have two lone pairs of sp 3 atomic orbital along [110] (for Ga-stabilized surface) or [11 __ 0] direction (for As-stabilized surface). These dangling orbitals cause attractive interactions between adjacent atoms and these adjacent atoms along the direction of dangling orbitals are attracted to each other and paired, which reconstruct the surface crystal structure different from the simple termination of bulk crystal structure. GaAs (001) surface exhibits many reconstructions as a function of the chosen Ga partial pressure, As pressure and substrate temperature. At a given temperature, it can range from a C(4×4) structure at heavily As-stabilized condition corresponding to As coverage (θ As ) no less than 0.8 to As-stabilized C(2×8) or (2×4) reconstructions for 0.5<θ As <0.8 to intermediate (3×1) reconstruction to Ga-stabilized C(8×2) or (4×2) reconstructions for θ As <0.4 as atom evaporation rate increases with substrate temperature [B.18]. Thus, at a given As pressure, with the increase of substrate temperature from 400°C to 650°C, GaAs (001) surface will go from Ga-stabilized C(8×2) or (4×2) to (3×1) reconstruction to As-stabilized C(2×8) or (2×4) to C(4×4) surface reconstruction. The surface reconstruction can be directly determined from 384 the RHEED pattern. We use RHEED to determine the surface crystal structure of GaAs with different combination of substrate temperature, Ga flux, and As flux, and created surface phase diagram of GaAs (001) as shown in Fig. B.7. Since phase diagram is an intrinsic material property of the GaAs substrate, we routinely use this phase diagram to maintain the reproducibility of experimental conditions such as As flux and substrate temperature to compensate the possible errors caused in the reading of growth parameter from MBE machine. Figure B.7. Measured surface phase diagram of GaAs (001) surface with and without Ga flux using As#1 cell from Ref. B.19.  RHEED intensity and intensity oscillation —growth kinetics control Morphology of growth front and its evolution during MBE growth can be analyzed by monitoring RHEED intensity, particularly specular beam intensity. The specular beam oscillation during MBE growth was first observed by several groups 105 110 115 120 125 130 10 -6 10 -5 c(4 × 4) (2 × 4) (3 × 1) T S ( o C) (2×4) - (3×1) Without Ga flux With Ga flux at  GaAs ~4 sec/ML With Ga flux at  GaAs ~2 sec/ML P As#1 Cell (torr) 10 5 / T S (10 5 /K) (2×4) - C(4×4) Without Ga flux 660 630 600 570 540 510 480 385 [B.20]. The intensity oscillation was later explained as due to monolayer (ML) -high- step density oscillation during the growth of crystals. The simple picture of RHEED oscillation (Fig. B.8(a)) can be summarized below: consider a flat starting surface with a certain step density and specular beam spot intensity I 0 (Fig. B.8(b)); As growth proceeds, the steps form and reach maximum density near half a monolayer of coverage (Fig. B.8 (c)). The RHEED specular spot intensity drops to a minimum with beam incident angle set at off-Bragg condition shown in Eq.B.8. The diffracted beams from two adjacent layers of the surface are exactly out of phase and interfere destructively. With continued growth and increase of surface coverage, the incoming adatoms are preferentially incorporated into edges of the existing clusters, and the clusters are laterally grown. Eventually, the clusters start to be merged, the step density is reduced and the RHEED intensity increases again and reaches it maximum upon a near complete layer deposition as shown in Fig. B.8(d). As growth continues, this process on the surface will be repeated, and RHEED specular beam intensity oscillates accordingly at the period of 1 ML growth time as shown in Fig.B.8(a). The intensity of the specular spot represents the surface step density. Since the oscillation in the specular beam intensity are consistent with 1ML deposition, periodic RHEED intensity oscillation can be used to measure the growth rates. Details of growth rate calibration is discussed in section §B.1.3. 386 Figure B.8. (a) Schematic picture of RHEED specular spot intensity oscillation during MBE growth. (b)-(c) The correspondence of specular beam intensity to surface monolayer step density during the first 1ML growth. Since the specular spot intensity of the RHEED is inversely related to the surface step density, the specular spot intensity can be used to tell us the condition of the surface morphologic condition to growth structures with highest surface smoothness. GaAs surface reconstruction can remain in As (2×4) for a wide range of substrate temperatures under a given As pressure as seen in Fig.B.7. While the surface reconstruction remains in As (2×4), the surface morphology, i.e. step density, changes as the substrate temperature changes. To ensure that growth is conducted 387 under the condition with high surface smoothness and reproducibility, we track the intensity of specular spot as a function of sample temperature under a fixed As pressure and conduct cap measurement as shown in Fig.B.9. The optimal growth condition, i.e. growth temperature, is chosen to be at the point where the intensity just dropped after the reach of the flat maximum region. Figure B.9. Result of a cap measurement. It shows the measured specular spot intensity as a function of substrate temperature with As pressure of 1.8E-6 torr. §B.1.3 System calibration For MBE growth, growth condition is controlled by the group III and V flux, growth temperature and growth time (shutter opening time). Equipment related to the control of the above mentioned factor related to growth condition in our Riber 3200 system need to be calibrated to obtain reproducible growth from run to run. In this 500 520 540 560 580 600 620 640 0.4 0.6 0.8 Intesity (A.U) T Pyro [e=0.22] (C) As(2x4) As(3x1) C(4x4) P As =1.8E -6 Torr Optimal growth condition 388 subsection, the calibration methods we employed for temperature and beam flux measurement and control are discussed. (a) Pyrometer calibration The pyrometer (IRCON, model V12C05) measures the surface temperature by detecting the light emitted (0.91 μm-0.97 μm wavelength) from the surface of substrate due to thermal radiation and converting the measured light intensity to temperature reading. The radiation intensity of infrared emission from a heated object can be represented as I (λ) = Φ (λ, T) ∙ ε λ (T), where I is the emission intensity, Φ (λ, T) is the blackbody radiation, and ε λ (T) is spectral emissivity of the object that is determined by material properties and surface morphology. Therefore, the measured infrared radiation can be converted to the temperature of the surface with known emissivity. The emissivity of any object satisfies the following relationship: ε λ = 1- R λ - T λ (Eq.B.9) where T λ , and R λ is transmittance and reflectance of the object at wavelength λ. For the wavelength shorter than the cut-off wavelength which is determined by the bandgap of the material, transmittance of semiconductor material is essentially zero with thick enough material. The emissivity is only related to the reflectance of the object and can be represented as ε λ ~ 1 – R λ . In addition, in this wavelength regime, the refractive index of semiconductor material is relatively insensitive to temperature, and thus, the temperature dependence of emissivity can be ignored. Thus, pyrometer can measure temperature of the object that with a correct known emissivity. For the 389 wavelength longer than the cut-off wavelength of the material, the material is transparent and emissivity is close to zero. Pyrometer becomes ineffective in this wavelength region. Given the 0.91 μm-0.97 μm detection wavelength region, the pyrometer can be used for semiconductors with bandgap small enough compared to 1.28 eV. For GaAs, the pyrometer can measure its temperature in the region of 450°C to 1200°C. Although emissivity is a known material property for the material with mirror-like surface, the measured infrared intensity is disturbed by extrinsic factors such as a transmission of quartz view-port in front of the pyrometer. For accurate temperature measurement, we need to calibrate the “emissivity parameter” which is an input parameter used for the pyrometer to convert the detected light intensity to substrate temperature. We calibrated this “emissivity parameter” of GaAs substrate for the pyrometer in our chamber by observing Al-Si eutectic phase transition occurring at ~577°C with 20 % of Al composition. We use a Si substrate coated with 200 nm Al mounted beside GaAs substrate. The Al-Si eutectic transition can be visible by eye as the reflectance changes on the surface. Thus, with this Al-Si sample mounted beside GaAs wafer and monitoring its eutectic transition, we can adjust the “emissivity parameter” for GaAs at 577 °C. With calibrated “emissivity parameter”, the pyrometer can measure surface temperature with accuracy of <10°C. The typical errors come from (1) the change in the effective emissivity due to the change of transmission of the quartz view port with gradual arsenic-coating during growth; (2) stray light from the system; (3) the radiation from the heater on the backside of the 390 substrate. To prevent the arsenic-coating on the pyrometer viewport, the viewport is always kept in a moderately heated condition. (b) Equivalent beam flux calibration The fluxes coming out of effusion cells are controlled by the temperature of the charges. Given the flux coming out of cells through evaporation, the flux and temperature of charges satisfy the following relationship:           T k E j B evap a, exp (Eq.B.10) where E a, evap is activation energy of evaporation and k B is the Boltzmann constant. We measure cell flux induced partial pressure (P BFM ) using the BFM gauge on the sample manipulator. Fig. B.10 shows the measured P BFM ~1/T curve for As#2 cell. Given the proportional relationship between partial pressure and beam flux and Eq.B.10, the slop of the curve represents the E a, evap of As 4 . The intercept of the P BFM ~1/T curve with vertical axis provide information on the amount of charges used up in the cells. For cells with newly-loaded charges, the intercept moves upward due to the increased surface area resulting from the initial break up of charges rods into small fragments. As the charges getting depleted inside the cell, the intercept moves down wards. Therefore, this type of curves of cells can not only help guide to choose the desired fluxes but also to track the charge consumption in the cells over a long term. The temperature controller has accuracy of < ± 0.5 °C and can provide flux with fluctuation within ~ 1 % arose from the limitation in the accuracy of the cell temperature. 391 110 120 130 140 150 160 170 180 190 200 1E-8 1E-7 1E-6 1E-5 P As#2 (Torr) T As#2 (C) As#2 P BFM vs As#2 Cell Temperature Activation energy of 1.1eV Figure B.10. Plot of measured As 4 flux as a function of As#2 cell temperature. More accurate calibration of beam flux is done by measuring the growth rate via RHEED oscillation. The oscillation period corresponds to 1 ML-worth delivery amount (details refer to the principle of RHEED and its intensity variation discussed in §B.1.2). Because the ML-worth sheet density of atoms is exactly fixed by the crystal structure of the substrate (6.27 × 10 14 /cm 2 for GaAs and 5.46 × 10 14 /cm 2 for InAs), the oscillation period represents exactly the flux of materials. The stability of the flux over the typical growth time (typically ~ 3 – 10 hours) was checked time-to- time, and the variation of flux is typically less than ~ 2 %. (c) Arsenic Controlled Oscillations: For MEE growth MEE growth technique as discussed in section §B.1.2 is used to enhance the migration length of III-group adatoms at temperatures significantly lower (~ 300 – 400 °C for GaAs) than the optimum growth temperature for standard MBE growth (~ 392 580 – 600 °C for GaAs) for high quality material growth with sharp interface by alternating the supply of group III and group V flux during growth. With reduced group V flux, group III adatoms diffusion time is enhanced and hence migration length. For the growth of GaAs using MEE technique, the GaAs surface reconstruction turns from As (2×4) to Ga (4×2) after 1ML deposition of Ga onto the sample surface without As. With the opening of As cell, delivered Ga incorporates As and turns into GaAs. The time it takes for 1ML of Ga to be fully turned into GaAs and recover the As (2×4) surface reconstruction is calibrated with RHEED by observing the Arsenic-induced oscillation. Fig. B.11 below shows a measured Arsenic-induced RHEED oscillation for 2ML of GaAs growth. After opening Ga shutter, the surface becomes Ga rich. The specular spot intensity increased. Ga atoms impinging on the surface migrate more freely without As pressure and start forming gallium droplets which roughs the surface and blocks electron beam and reduces the specular spot intensity. As soon as the As cell shutter is opened, the specular spot intensity starts to oscillate because of the incorporation of As and the formation of GaAs. The surface returns back to As (2×4). The time interval period of RHEED oscillation induced by the incorporation of As determines the needed As cell open time for complete formation of 1ML GaAs using MEE. In this dissertation, the calibrated As cell open time for 1ML GaAs MEE growth is 6 sec with As pressure ~ 1E-6 torr. This time used in the MEE growth of GaAs. 393 0 20 40 60 80 100 0.25 0.30 0.35 0.40 0.45 0.50 2ML GaAs MEE growth As(2×4) Intensity (a.u.) Time(sec) openGa close As open As As(2×4) Ga(4×2) As induced oscillation Figure B.11. Plot of measured RHEED oscillation of 2ML MEE growth with growth rate of 4sec/ML and As pressure of 1.8E-6 torr. §B.1.4 Substrate preparation and Mounting The preparation of substrate for MBE growth contains three parts: (1) preparation of substrate, (2) preparation of moly-block (sample holder) and (3) mounting of substrate on moly-block. In this dissertation work, both planar substrate and patterned substrate with etched array of sub-micro sized mesas are used. Planar substrates are mainly used for calibration purpose while the patterned substrates with nanomesas are used for MTSQD growth. Detailed preparation steps for these two types of substrates are discussed in this section. The preparation method for moly- block and the mounting of substrate on top are independent of the type of substrate used and is discussed first below followed by the discussion of substrate preparation. 394 (a) Moly-block preparation and Indium bounding A 2” molybdenum block (moly-block) is used as the sample holder for MBE growth in our system. The moly-block preparation contains three major steps: (1) moly-block polishing, (2) moly-block degreasing and (3) moly-block etching. For the moly-block polishing, we polish it with SiC abrasive paper with average grain size of 30 μm (Grit# 400) followed by the one with 15 μm grain size (Grit # 600) and one with ~6um gran size (Grit # 1200) to remove material deposited on it in the previous MBE growth. The polished moly-block is then degreased by trichloroethylene (TCE), acetone, and methanol in ultrasonic bath, in the sequence from most hydrophobic solvent to the most hydrophilic solvent to get rid of any possible grease and organic junks. After rinsing with deionized (DI) water, the moly- block is etched with NH 4 OH:H 2 O 2 :H 2 O = 1:1:10 for 10 sec to remove residual inorganic impurities followed by DI water rinse. The water is then blown off with high pressure ultrahigh purity (UHP) nitrogen gas. Occasionally, the Moly-block is thermally cleaned by heating it up to 750 °C in a preparation module under UHV for ~ 8 hours. Moly-block that has been cleaned and dried following above mentioned recipes is ready for sample mounting using Indium. For sample mounting, the moly- block is heated to ~200°C on a clean hot plate. A small piece of Indium is placed on top and spread uniformity (upon melting of Indium) over the moly-block. Then, the prepared substrate is affixed on the moly-block and moved around by a clean stainless steel forceps to allowing good wetting between the GaAs substrate and 395 moly-block till substrate stick onto moly-block. A good uniform wetting between GaAs substrate and moly-block is essential for achieving good thermo-conductivity and uniform heating of substrate in the chamber. (b) Substrate preparation GaAs wafer used in this dissertation work are epi-ready 2” GaAs (001) ± 0.1° substrates with etch pit density less than 3,000 /cm 2 (AXT, Inc.). Substrate preparation contains three major steps: (1) wafer cleaving, (2) substrate degreasing and (3) substrate etching. The 2” GaAs wafer is cleaved into pieces of substrates of size ~20mm × 20mm by cleaving along the <110> cleavage direction. The cleaved substrate is degreased in TCE, acetone, methanol, and DI water in ultrasonic bath, same as degreasing moly-blocks. For planar substrate, it is etched with H 2 SO 4 :H 2 O 2 :H 2 O = 8:1:1 at 40 – 50 °C for 1 minute to remove native oxide on GaAs surface and etches GaAs substrate to remove ~1 μm thick GaAs to clean the surface. The etched substrate is then rinsed with DI water and kept in DI water for 5 mins for the formation of a fresh and uniform native oxide layer (~50A) on top of the clean surface before drying the substrate with UHP N 2 gas. For the patterned substrate, the cleaved substrate goes through the following preparation steps: 1. Degrease the substrate with TCE, acetone, methanol, and DI water in ultrasonic bath, same as planar substrate 396 2. Spin coat the electron beam resist on the substrate. The substrate is heated at 120°C for 5minutes on a hot plate to drive off adsorbed moisture. The resist is HSQ from Ellsworth Adhesives and is spin coated on the substrate in the Solitec spinner with a uniform layer of thickness ~75nm. Substrate with resist on is baked at 120°C on hot plate for 2 minutes to drive off all the volatiles in the resist and also improves the adhesion between the resist and the sample. 3. Patterning the resist with electron beam lithography. The substrate is patterned using Raith electron beam lithography system. It contains patterned region and an L shape unpatterned region for RHEED characterization. In the patterned region, there are arrays of mesas with mesa edges oriented along <100> direction. Details of the pattern design for MTSQD growth is discussed in section §2.3.2. 4. Resist development. The substrate is immersed into a solution of 4 wt% NaCl and 1 wt% NaOH for approximately ~ 40 s. During development the regions of the resist exposed to the electron beam stays due to the formation of SiO 2 in the exposure process and the non-exposes region dissolve. The developed substrate is then rinsed in DI water for cleaning. 5. Etch mesas into the sample. The substrate is dipped into the etchant made of NH 4 OH:H 2 O 2 :H 2 O = 4:1:20 which gives etch rate of ~30nm/sec for an appropriate amount of time to realize mesas with the required depth and mesa top size. The substrate is rinsed in DI water after etching. 397 6. Strip resist. The etched substrate is dipped in 50% HF solution for 30sec to remove the resist on the etched mesas. The substrate is then rinsed in DI water after removal of resist. The prepared substrate (planar or patterned) mounted on a moly-block using Indium discussed before is loaded into a sample loading module (Mod A) in a glove box. After pumping down the sample loading module by turbo molecular pump (TMP) to ~10 -7 – 10 -6 torr, the moly-block with substrate mounted is transferred to the sample preparation module (Mod C) where it is heated up to ~ 350 °C for 2 – 3 hours under UHV to drive-off moistures and other impurities. After degassing the substrate in the Mod C, the moly-block is loaded into MBE chamber through Mod B. The substrate is heated up to ~ 580 °C under arsenic pressure, and around ~ 580 °C surface oxide desorbs. This oxide desorption is clearly observed by RHEED pattern changing from a completely diffused pattern (indicating amorphous surface) to a periodic diffraction pattern (indicating crystalline GaAs substrate). Subsequent growth can be conducted on the substrate under calibrated growth condition discussed in section §B.1.2 and §B.1.3. §B.2 Structural Characterization Electron microscope and atomic force microscope have been used extensively as structural characterization tools in the studies presented in this dissertation. Scanning electron microscopy (SEM) has been used for characterizing the as-patterned mesa profiles, quick inspection of the as-patterned and post-growth mesas, and to determine gross features in mesa profile evolution with growth. 398 Atomic force microscopy (AFM) has been used largely for the characterization of InGaAs SAQDs size distribution and density that act as reference for the understanding of atom migration in patterned structures and MTSQDs’ optical properties. In this section, we capture here a brief review of the working principle of these two types of structural characterization tools. §B.2.1 Scanning Electron Microscope Scanning electron microscopy (SEM) produces images of a sample by scanning the surface with a focused beam of electrons. The electrons interact with atoms in the sample, producing various signals that contain information about the sample's surface topography and composition. SEM is widely used to study the surface of the sample. Figure B.12 shows the schematic diagram of SEM. The electron gun produces the electron beam using thermionic emission from a heated filament or field emission from a metal in high electric field using the effect of electron tunneling. For the thermionic emission gun, the electrons emitted from the heated filament are accelerated rapidly towards the anode to the energy of 1 keV to 100 keV. The brightness of the beam, defined as the beam current density per unit solid angle, of a thermionic gun is represented as the following [B.21]: 5 2 10 exp( / ) B TV kT     (Eq.B.11) where T is the temperature of the filament and  is the thermionic work function of the filament. Brightness is a measure of how many electrons per second can be directed at a given area of the sample. Thermionic gun with tungsten filament usually provide brightness of 10 9 Am -2 sr -1 limited by its melting temperature (3653K) 399 and work function (4.5eV). For even higher brightness, desirable for high resolution imaging, the electron guns used are field emission guns to overcome the brightness limitation of thermionic guns. Figure. B.12 Schematic diagram of major component of a standard SEM. In the field emission guns, electrons is extracted from a fine tip of tungsten subject to an extremely high electric field (>10 9 V/m) in ultra-high vacuum environment (P<10 -7 torr) and accelerated to energy of 1 keV to 100 keV. Electrons 400 can tunnel out of the surface without giving the energy of work function of the metal when metal is subjecting to high electric field. The field emission current is related to the applied electric field as shown below [B.21]: 0.5 2 9 1.5 6 ( / ) 6.8 10 6.2 10 exp( ) f f EF j EF          (Eq.B.12) where E f is the fermi energy and F is the applied electric field. For field higher than 5×10 9 V/m, the current from field emission exceeds that from thermionic emission. The electrons emitted from field emission gun has energy spread less than 0.5 eV better than that from thermionic guns which is around 1-2 eV. Therefore, field emission gun is used for generation of high bright “monochromatic” beam for the high resolution imaging and study of cathodoluminescence from sample. The emitted electron beam from the electron gun is collected by the condenser lens and gets focused down on the sample through objective lens with a beam spot size of ~2-10 nm. Deflection coils are used to control the beam spot position on the sample and scan the beam spot on sample surface. When the electron beam hits the thick specimen, the energy of the incident electron is dissipated and results in various secondary emissions from the specimen as shown in Fig. B. 13. Each signal is a result of a particular interaction between the beam and sample and can provide different information of the sample. X-rays are used for chemical analysis and cathodoluminescence are used for band structure analysis. Backscattered electrons and secondary electrons are usually detected and used as detection signal in SEM for imaging and studying the surface of sample. The 401 backscattered electron has broad energy spread. Those of the highest energy are scattered only a few times and originated near the incident beam. These electrons are capable of giving surface information at high resolution. Those backscattered electrons of low energy that have undergone multiple scatterings come from a larger area and provide a worse spatial resolution. The detected secondary electron, on the other side, are mainly from a region just little larger than the incident beam. Therefore, secondary electron having small sampling volume is better suited for high spatial resolution imaging. Fig. B.13 Schematic diagram showing the type of signals from electron beam interacting with specimen that can be used in SEM for signal processing. In imaging the sample surface, the image is formed by scanning the electron beam with the detector counting either backscattered electrons or secondary electrons giving off from each point on the sample surface and forming a scanned image on the screen with brightness of pixel representing the signal from detector. The resolution of SEM is therefore related to the ratio of the pixel size of the image 402 and the scanning pixel on sample. To achieve the best resolution, the spot size of the electron beam on the sample should be the same as the beam scanning pixel. In this dissertation, all the SEM works were done using the Hitachi S4800 field emission SEM with extraction voltage up to 6.5 kV and accelerating voltage up to 30 kV. It can provide resolution of 1 nm with 15 kV accelerating voltage and 4mm working distance. We conduct our SEM work under the condition of 1 kV accelerating voltage and 1.5mm working distance which gives 2nm resolution, enough for characterizing patterned mesa and growth evolution of MTSQDs. §B.2.2 Atomic Force Microscope Atomic force microscopy (AFM) measures the sample surface morphology based on the sensing the force between sample surface and the scanning tip. AFM has two different operating modes: contact mode (contact-AFM) and tapping mode (noncontact-AFM). The detailed description of its operating principle is discussed in Ref. [B.22, B.23]. In this section, we briefly review the principle of AFM and capture the key important issues related to AFM. When the tip is approaching the sample surface, it senses the van atomic interaction force (van der Waals force, shown in Fig. B.14) between the tip and the sample surface. Contact-AFM is usually operated in the repulsive force region (Fig. B.14, region A). A thin pointing tip (typical tip radius ~ 10 nm) at the edge of a cantilever is brought in contact with the sample of interest. The deflection of the cantilever in the vertical direction due to the repulsive force acting on it from the sample is measured using position sensitive photodiode (PSPD) and the tip is 403 scanned across the surface keeping deflection a constant as shown in Fig B.15 (a). The resulting image is a map of the constant force acting on the tip due to sample. Since the tip is in contact with sample, contact-AFM may cause physical damage to the sample. Fig. B.14 Schematic diagram of the force acting between the tip and sample separated by the distance z. The blue lines marked two operation regions A and B respectively for contact-AFM and noncontact-AFM. The technique of noncontact-AFM was invented based partly on the desire to measure long-range force between materials and to avoid the potentially destructive interactions between tip and sample observed in contact-AFM. Noncontact AFM uses the fundamental principle that the presence of an external force (usually attractive force, Fig. B.14 region B) on a vibrating cantilever modifies its fundamental vibrational properties. Instead of measuring the quasi-static deflection of the cantilever in contact-AFM, the cantilever is made to vibrate at or near its 404 resonance frequency, v 0 and changes in the resonance frequency (and/or the changes in vibration amplitude, A) due to tip-sample interaction are measured and used to represent surface morphology as schematically shown in Fig.B.15(b). The cantilever can be modeled as a spring with spring constant k. In the absence of external force, we assume the net force acting on the tip to be zero. The force on the tip when it is perturbed infinitesimally from its equilibrium position by an amount Δz with the presence of the force between tip and sample can be represented as following: ( ) ( ) s tip F F z k z z      (Eq.B.13) where F s represents the force between tip and sample. This is identical to the equation of motion of a harmonic oscillator with an effective spring constant. The corresponding resonant vibration frequency is 1 2 s eff F k z V m      (Eq.B.14) The noncontact AFM image that uses the vibration frequency as the detected signal can be simply interpreted as a map of constant force gradient of interaction between the sample and amplitude in the limit of small amplitude A. For ambient noncontact AFM, the amplitude of the oscillation of cantilever is allowed to be modified upon interaction with the surface while the frequency of the cantilever is maintained as a constant. The general equation describing the vibration of the cantilever is shown as:   0 22 0 / 1 ( ) o o vv A v A v v Q vv   (Eq.B.15) 405 where Q is the quality factor and A 0 is the amplitude of cantilever at resonant frequency. The change in amplitude of the cantilever is measured by the PSPD and sent to the feedback loop to control the position of tip in the scan on the sample surface. Fig. B.15 Schematic diagram of (a) contact-AFM and (b) noncontact-AFM. In the contact-AFM, PSPD sense the deflection of the tip by measuring the position of light spot on the detector. In noncontact-AFM, PSPD senses the cantilever vibration frequency (v) and amplitude (A). In this dissertation, we used the ambient DI Multiscan AFM with a Si- cantilever (typically, length × width × thickness ~ 125 μm × 35 μm × 4 μm, the 406 resonant frequency is ~ 300 kHz) under amplitude-modulated scanning mode (tapping mode) to measure patterned mesas as well as InGaAs SAQDs reference samples. §B.3 Optical Characterization Optical properties of QDs, i.e. emission uniformity, electronic structures, lifetime, and coherent time, etc., can be characterized by studying their photoluminescence. In this section, we briefly discuss the pre-existing photoluminescence instrumentation in the group. The pre-existing photoluminescence instrumentation has large excitation spot size on sample and hence probing only ensemble averaged large area PL/PLE/TRPL properties of QDs. The photoluminescence from individual QD is studied using a micro-PL system built under this dissertation work. Details of the instrumentation of the micro-PL setup is discussed in Appendix C. §B.3.1 Pre-existing time-integrated PL/PLE instrumentation In the pre-existing time-integrated PL/PLE setup shown in Fig.B.16, an Ar + laser (Innova 310, Coherent Inc.) with 10W peak power is used either as an excitation source for E=2.41eV energy in PL measurement or as a pump for the Ti- Sa tunable CW laser (Spectra Physics 3900) in PLE or near-resonant/ resonant PL measurement. The Ti-Sa laser is tuned by a motorized micrometer (Oriel 18240) controlled by Oriel 18011 micrometer controller through RS232 interface. The laser beam (Ar or Ti-Sa) is chopped at frequency ~78Hz, steered using mirrors (M) through neutral density (ND) filter (F1), and finally focused using a f=300mm lens 407 (L1) onto the sample. The excitation laser on the sample is ~100 μm for Ar + laser and ~ 200 μm for Ti-Sa laser. The luminescence is collected by a f=75mm lens (L2) and focused onto the entrance slit of the double path double grating f=0.85m spectrometer (model SPEX 1404). The dispersed photoluminescence after the spectrometer is detected using LN 2 cooled Ge detector (North Coast). The detected signal from the detector is then fed to a lock-in amplifier (model SR830), phase locked at the frequency at which the exciting laser is chopped, and then stored in the main computer. The details of this instrumentation can be found in the dissertation of former group members [B.24, B.25]. Figure B.16 Schematic diagram for our existing time-integrated PL/PLE set-up. The setup needs to be routinely calibrated for optical studies on QDs. The typical calibrations are listed below: (1) Spectrometer wavelength calibration using Ar + lines and their second harmonics. 408 (2) Spectral response calibration of Ge detector in the range of 800- 1500nm using a tungsten lamp which can be approximated by a blackbody of a color temperature specified by the manufacturer with known spectral response. §B.3.2 Pre-existing time-resolved PL instrumentation The pre-existing time-resolved PL (TRPL) setup was built around 2004 by a former Ph.D student, Siyuan Lu. Details of the instrumentation can be found in his dissertation [B.26]. We only capture briefly the instrumentation here in this section. In the TRPL setup shown in Fig. B.17, a cavity dumped fs/ps Ti:S laser consisting of a Coherent Mira 900D fs/ps laser with Mira PulseSwitch cavity dumper (Mira900D and PulseSwitch together are hereafter referred to as Mira) pumped by Ar + laser (Coherent Innova 310) is used as excitation light source. The laser pulse from the Mira output has 200fs or 3ps pulse width for fs and ps operation mode respectively. The repetition rate of the Mira900D pulses by itself is 76MHz. The laser pulse coming out of Mira goes through cavity dumper which employs an acousto-optic Brag cell to select desired pulses out of the pulse train of Mira900D for experiment use. The laser pulse is focused on the sample using lens with a spot size ~ 200 μm. The photoluminescence from sample is dispersed by a 1/4m subtractive monochromator (CVI DK242) with spectral resolution of 3.2nm at 1mm slit width and detected by a MCP-PMT (Hamamatsu R3809-U, 25ps T.T.S., 400-1200nm). 409 Figure B.17 Schematic diagram for our existing time-resolved PL set-up including processing time correlation single photon counting (TCSPC) electronics. The electronic signal from the detector and electronic signal from laser generated by a home-built Silicon diode is fed into the time-correlation single photon counting electronic system that contains pico-timing discriminators, delay unit, and pulse to height convertor. The signal from detector goes though discriminator and delay unit and then fed into pulse to height converter as “start” signal. The discriminated signal from home-built Silicon diode representing the timing sequence of laser pulse is fed into pulse to height converter as “stop” signal. The time difference between “start” and “stop” signal is measured and stored for the measurement of TRPL spectrum. 410 Appendix B References: [B.1] A. Madhukar, P. Chen, F. Voillot, M. Thomsen, J. Y. Kim, W. C. Tang and S. V. Ghaisas, “A combined computer simulation, RHEED intensity dynamics and photoluminescence study of the surface kinetics controlled interface formation in MBE grown GaAs/Al x Ga 1-x As(100) quantum well structures,” J. Cryst. Growth, 81, 26 (1987). [B.2] A. Madhukar and S. V. Ghaisas, “The nature of molecular beam epitaxial growth examined via computer simulations,” CRC Critical Rev. Solid State Mater. Sci., 14, 1 (1988). [B.3] M. Y. Yen, A. Madhukar, B. F. Lewis, R. Fernandez, L. Eng and F. J. Grunthaner, “Cross-Sectioal Transmission Electron Microscopy of GaAs / InAs (100) Strain Layer Modulated Structures Grown by Molecular Beam Epitaxy,” Surf. Sci., 174, 606 (1986). [B.4] A. Madhukar, T. C. Lee, M. Y. Yen, P. Chen, J. Y. Kim, S. V. Ghaisas and P. G. Newman, “Role of surface kinetics and interrupted growth during molecular beam epitaxial growth of normal and inverted GaAs/AlGaAs(100) interfaces: A reflection high-energy electron diffraction intensity dynamics study,” Appl. Phys. Lett., 46, 1148 (1985). [B.5] Y. Horikoshi, “Migration-enhanced epitaxy of GaAs and AlGaAs,” Semicond. Sci. Technol., 8, 1032 (1993). [B.6] Q. Xie, P. Chen, A. Kalburge, T. R. Ramachandran, A. Nayfonov, A. Konkar and A. Madhukar, “Realization of optically active strained InAs island quantum boxes on GaAs(100) via molecular beam epitaxy and the role of island induced strain fields,” J. Cryst. Growth, 150, 357 (1995). [B.7] T. Asano, Z. Fang, and A. Madhukar, “Deep levels in GaAs(001)/InAs/InGaAs/GaAs self-assembled quantum dot structures and their effect on quantum dot devices”, J. Appl. Phys. 107, 07311(2010). [B.8] T. A. Flaim and P. D. Ownby, “Observations on Bayard-Alpert Ion Gauge Sensitivities to Various Gases,” J. Vac. Sci. Technol. 8, 661 (1971). [B.9] O. Stier, M. Grundmann and D. Bimberg, “ Electronic and optical properties of strained quantum dots modeled by 8-band k·p theory”, Phys. Rev. B 59, 5688 (1999). [B.10] F. Guffarth, R. Heitz, A. Schliwa, O. Stier, N. N. Ledentsov. A. R. Kovsh, V. M. Ustinov and D. Bimberg, “Strain engineering of self-organized InAs quantum dots”, Phys. Rev. B 64, 085305 (2001) 411 [B.11] R. Heitz, F. Guffarth, K. Pötschke, A. Schliwa and D. Bimber, “ Shell-like formation of self-organized InAs/GaAs quantum dots”, Phys. Rev. B 71, 045325 (2005). [B.12] K. Hu, “Strained InGaAs/AlGaAs Multiple Quantum Well Based Asymmetric Fabry-Perot Light Modulators,” Ph.D. dissertation, Physics, University of Southern California, 1993. [B.13] Q. Xie, “Molecular Beam Epitaxical Growth of Strained InAs Island Quantum Boxes on GaAs (100): Growth Kinetics, Vertical Self-Organization, and Lasing Characteristics,” Ph.D. dissertation, Materials Science, University of Southern California, 1996. [B.14] M. A. Herman and H. Sitter, Molecular Beam Epitaxy: Fundamentals and Current Status, Springer-Verlag 1989. [B.15] B. F. Lewis, T. C. Lee, F. J. Grunthaner, A. Madhukar, R. Fernandez and J. Maserjian, “RHEED oscillation studies of MBE growth kinetics and lattice mismatch strain-induced effects during InGaAs growth on GaAs(100),” J. Vac. Sci. Technol. B, 2, 419 (1984). [B.16] A. Chambers, Modern Vacuum Physics, Boca Raon, FL, Chapman & Hall / CRC, 2005. [B.17] T. A. Flaim and P. D. Ownby, “Observations on Bayard-Alpert Ion Gauge Sensitivities to Various Gases,” J. Vac. Sci. Technol. 8, 661 (1971). [B.18] S. B. Ogale, M. Thomsen and A. Madhukar, “Surface kinetic processes and the morphology of equilibrium GaAs(100) surface: A Monte Carlo study,” Appl. Phys. Lett. 52, 723 (1988). [B.19] T. Asano, “Growth control and design principles of self-assembled quantum dot multiple layer structures for photodetector applications”, Ph.D. dissertation, Materials Science, University of Southern California, 2010. [B.20] J. H. Neave, B. A. Joyce, D. J. Dobson and N. Norton, “Dynamics of film growth of GaAs by MBE from RHEED observations”, Appl. Phys. A. 31, 1 (1983). [B.21] P. J. Goodhew, J. Humphreys and R. Beanland, Electron Microscopy and Analysis, Tayler & Francies 2001. [B.22] T. R. Ramachandran, “Scanning Probe Microscopy Studies of the Highly Strained Epitaxy of InAs on GaAs (001) and Scanning Probe Based Imaging and Manipulation of Nanoscale Three-Dimensional Objects,” Ph.D. dissertation, Materials Science, University of Southern California, 1998. 412 [B.23] T. Sukurai and Y. Watanable, Advances in Scanning Probe Microscopy, Springer Science & Business Media 2012. [B.24] I. Mukhametzhanov, “Growth control, structural characterization, and electronic structure of Stranski-Krastanow InAs/GaAs (001) quantum dots”, Ph.D. Disseration, University of Southern California (2002). [B.25] E. T. Kim, “Tuning InAs quantum dot electronic structure using (InAlGa)As capping layers and application to infrared photodetectors”, Ph.D. Dissertation, University of Southern California (2003). [B.26] S. Lu, “Some studies on nanocrystal quantum dots on chemically functionalized substrates (semiconductors) for novel biological sensing”, Ph.D. Dissertation, University of Southern California (2006). 413 Appendix C: Micro-PL Instrumentation This appendix discusses the detailed instrumentation of the micro-PL setup built for studying the excitonic emission, electronic structure, and single photon emission characteristics of the synthesized MTSQD array. §C.1 Micro-PL instrumentation: major components The micro-PL setup built in the group is capable of measuring the time- integrated PL/PLE/time-resolved PL behavior and photon correlation response from individual QDs for NIR wavelength regime (700-1000nm excitation, 700-950nm detection) and 10ps-1000ns time dynamic range. The setup contains three major parts as captured in Figure. C.1 : (1) laser system for optical excitation, (2) optical detection system for collecting and subsequent detection of photons emitted from MTSQDs and (3) signal processing system to record the electronic signal from spectrometer and detectors. 414 Figure C.1. Schematic of the home built micro-PL setup containing laser system, optical detection and signal processing system.  Laser system: The micro-PL setup contains three major light sources: (1) fiber coupled 640nm diode laser (PicoQuant model LDH-P-C-640B) with adjustable repetition rate of 80MHz, 40MHz, 20MHz, 10MHz, 5MHz and 2.5MHz with maximum output power of 20mW (for 80MHz repetition rate). (2) Ti-Sa mode-locked pulsed laser (Coherent Mira 900D) pumped by 8W 532nm solid state laser (Coherent Verdi G). The Mira 900D is a passive mode- locked Ti:S laser, capable of generating fs (~200fs) or ps (<3ps) pulsed light at 415 repetition rate 76MHz and maximum average power >1.5W (depending on wavelength (700-1000nm) and pumping power). The Mira900D exploits the Kerr lens effect of the Ti:S crystal which narrows the beam diameter of fs/ps high-energy pulsed lasing mode. A slit is strategically placed in the laser cavity to select the narrowed pulsed lasing mode and suppress the CW (continuous-wave) mode. (3) Ti-Sa CW laser (Spectraphysics 3900) pumped by 8W 532nm solid state laser (Coherent Verdi G). This Ti-Sa CW laser can provide maximum output power ~1W (depending on wavelength and pumping power). The wavelength of the Ti:Sa CW excitation laser is scanned by adjusting the position of a step motor that is attached to the prism inside cavity to adjust its position and thus tuning the wavelength. The motor position is computer controlled and can provide wavelength adjustment accuracy of 0.2A. For time-integrated PL measurement, all three lasers can be used as excitation light source covering 640-1000nm excitation wavelength range needed for non-resonant and resonant excitation of InGaAs QDs that has emission ~900- 1000nm. For time-integrated PLE measurement, the Ti-Sa CW laser (Spectraphysics 3900) pumped by 8W 532nm solid state laser (Coherent Verdi G). For time-resolved PL (TRPL) and photon correlation (g (2) (0) ) measurement, pulsed lasers are required. The fiber coupled 640nm diode laser (PicoQuant model LDH-P-C-640B) and Ti-Sa mode-locked pulsed laser (Coherent Mira 900D) pumped by 8W 532nm solid state laser (Coherent Verdi G) are used as excitation light source 416 covering wavelength of 640-1000nm needed for studying optical behavior of the InGaAs QDs that has emission ~900-1000nm using either non-resonant or resonant excitation scheme.  Optical detection system: The micro-PL setup is designed for NIR wavelength detection regime (700- 1600nm detection). It contains three major components in the detection system: (1) An f=0.3m single stage imaging spectrograph (Acton SP300i). Three gratings of 300g/mm blazing at 1000nm, 600g/mm blazing at 1000nm, 1200g/mm blazing at 750nm are installed in the spectrometer Acton SP300i. The spectrometer has a light dispersion of ~2.5nm per mm slit width and diffractive limited resolution of 0.2nm (at 1000nm using the 1200g/mm grating). Spectrally dispersed light can be steered to either of the two exit ports of the spectrograph : one port fitted with LN 2 cooled InGaAs photodiode array (1×512 linear array, Princeton Instruments Inc.) and silicon avalanche photodiode (APD, PicoQuant model τ-SPAD) in the Hanbury- Brown and Twiss setup (discussed in section §C.2) respectively. General consideration and caution of using an imaging spectrograph need to be followed. (2) The InGaAs array detector allows high throughput time-integrated PL measurements. The InGaAs array has spectral response from 0.8-1.6 μm. It is controlled by a PI (Princeton Instrument) ST-133 camera controller which is in turn connected to a PC with PI spectra interfacing card and Winspec software. Adjustment is needed to ensure that the dispersed light is focused sharply on the InGaAs array detector to avoid blurring of the spectrum. The InGaAs array needs to 417 be calibrated carefully to display wavelength correctly throughout the array using Xe calibration lamp with sharp atomic emission lines. In addition, the InGaAs array has a relatively high dark current (~15,000e/sec/pixel) which, as a background, needs to be subtracted from the measured spectrum. (3) The Silicon APD detector (Excelitas, model SPCM-NIR-14-FC and PicoQuant, model τ-SPAD) has spectral response of 0.8-1 μm (detection efficiency drops below 10% at 1 μm). Its signal is measured by a photon counter (EG&G, model 1109) which is connected to PC with Oriel interfacing card and a custom program written by Dr. Siyuan Lu. Silicon APD can only detect photon counts up to 1million c/sec (~200 fW photon energy with wavelength ~1 μm). Caution of not exposing APD to high intensity light need to be followed. Si APD has dark counts ~150 c/sec for APD from PicoQuant (model τ-SPAD) and ~50 c/sec for APD from Excelitas (model SPCM-NIR-14-FC) as background. The InGaAs array detector and Si APDs combined covers the detection range of 700 nm-1.6 μm detection range for time-integrated PL measurements. The Si APDs are used as detector for time-integrated PL /PLE /TRPL and photon correlation (g (2) (0) ) measurement with spectral range of 700-950nm.  Signal processing system: For the time-integrated PL/ PLE measurement, the signal from the spectrometer and the number of photons detected by Si APDs (or InGaAs array detector) read by the photon counter (EG&G, model 1109) or InGaAs array detector signal read by the PI spectra interfacing card are registered to PC. 418 For the time-resolved PL and g (2) (0) measurements, the timing interval of between the excitation photon and luminescence photon or between two luminescence photon need to be measured. In our micro-PL setup, we use time- correlated single photon counting (TCSPC) system to perform the above mentioned measurements. The TCSPC system contains the following major elements: (1) Constant Fraction Discriminator (CFD): In CFD, the input pulse is split into two pulses of equal amplitude. The first pulse is inverted and delayed for a preset period of time. The second pulse is attenuated and added to the first one, generating a bipolar pulse with zero-crossing that occurs at the same fraction of the input pulse independent of its amplitude. Based on the zero-crossing, CFD sends a NIM-standard logic pulse representing the precise timing of the APD output to the time-to-pulse-amplitude converter (TAC) to serve as the “start” or “stop”. The uncertainty in the pulse timing determined by the CFD for pulses of 1:10 amplitude range is less than 25ps for the CFD used (Ortec 9037) in use. (2) Time-to-Amplitude Conversion: The TAC (Ortec 457) measures the time interval between the “start” and “stop” NIM pulses, and generates an analogous output whose voltage is proportional to the time interval, namely the time difference between the excitation photon and PL photon (or between two PL photon). The TAC output is then fed into a multi-channel analyzer (MCA, Ortec Trump 8K). Upon a large number of such measurements the MCA accumulates a histogram of the TAC output voltages that present the probability distribution of photon emission in various 419 time windows after the excitation, namely the TRPL or photon correlation histogram spectrum. §C.2 Detailed design of the setup In the micro-PL setup, excitation light (blue shade and arrow, in Fig. C.2) from either of the laser mentioned in the previous section is coupled into a multimode optical fiber with core 50 μm and NA 0.2, filtered by a 900nm short pass and a 900nm dichromatic filters and focused down to ~ 1.25μm diameter through a 40× NA 0.65 objective lens on to sample mounted inside a continuous flow cryostat (Janis ST-500, 4K to 360K). A CCD (charge-coupled device) mounted on the microscope is used to form an image of the sample illuminated by the light (black arrow) from fiber coupled halogen lamp (Fiber-Lite, model 190) to adjust the focus of the objective. The light path from the excitation fiber and the illumination fiber to the sample is adjusted to be identical to ensure focus of the light spot of sample when the objective is in focus with sample. Detailed drawing of the interference filters in the microscope and the CCD is shown in Fig. C.2. The emitted photons from an individual QD is collected in the same vertical geometry using the same objective lens, focused by a lens tube with f=200mm, filtered by a 900nm dichromatic and a 900nm long pass filter. The focused and spectrally filtered light (cutting excitation background) is coupled to a FC-PC adjustable collimator (Thorlab, CFC-2X-B) and focused into multimode optical fiber with core 25 μm and NA 0.1. The positon of the collimator on the microscope is adjusted to have almost all light collected by the collimator into the detection fiber. 420 The detection fiber attached with the collimator is mounted on the x-y translation stage for the alignment of the collimator with respect to the light beam collected (labeled as red shade and arrow in Fig. C. 2). Figure C.2.Detailed schematic of the configuration of the microscope of micro-PL setup. The photoluminescence collected through this optical fiber is focused by a pair of NIR-coated achromatic 1” f=25mm lens into a spot of ~25-30 μm into the entrance slit of an f=0.3m single stage imaging spectrograph (Acton SP300i). The distance and focal length of the focusing lens in front of the spectrometer (Fig. C.1) is chosen to match the NA of the light from detection fiber to the f number of the spectrometer to get all light from fiber coupled into spectrometer. Spectrally dispersed light passing through spectral entrance and exit slit window set by the spectrometer can be either directed to the InGaAs array photodetector or collimated 421 by a NIR-coated achromatic 1” f=75mm lens and directed to a NIR 50/50 beam splitter of the Hanbury-Brown Twiss (HBT) setup. The light at both the transmitted and reflected direction of beam splitter is focused down to a spot with size <150 μm (the active area of APDs) on silicon APDs (Excelitas model SPCM-NIR-14-FC and PicoQuant model τ-SPAD) by NIR-coated achromatic 1” f=50mm lens. The 50/50 NIR beam splitter together with the two silicon APD detector is the standard realization of the HBT setup. The NIR 50/50 beam splitter is mounted on an x-y translation stage and can be moved out of light path for PL/PLE measurement using APD as detector. A 950nm OD 4 short pass filter is inserted in front of the APD at reflected arm of beam splitter to prevent detection of photons emitted by the APDs during the avalanche process and creating detector cross-talk. §C.3 Instrument detection efficiency The detection efficiency of the micro-PL setup (shown in Fig. C.1), det  , defined as the ratio of the photons detected by the APDs in the HBT setup and the photons impinging on the objective within the objective’s collection cone, is controlled by the three major components: (1) collection efficiency of photons into the detection fiber ( micro  ), (2) transmission efficiency of the spectrometer ( spec T ) and (3) detection efficiency of the Si APD ( APD  ). Guided by the design of the setup, the collection efficiency of the photon into the detection fiber is purely controlled by the transmission efficiency of the lenses and interference filters in the microscope. Therefore, the theoretical estimation of micro  gives 64% micro   . Similarly, the transmission efficiency of the spectrometer 422 is also controlled by the refection of mirrors and grating used due to the designed matching of the F number of the light entering the slit and the F number of the spectrometer discussed in §C.2. The estimated transmission efficiency of the spectrometer at 930nm (the MTSQD emission wavelength) is 52.4% spec T  and 39.6% spec T  respectively for the 600gr/mm and 1200gr/mm used in this dissertation work. Accounting for the know detection efficiency of APD at 930nm ( 25% APD   ), the theoretically estimated overall detection efficiency of the setup, det  , at 930nm with 600gr/mm used in the spectrometer can, therefore, be captured as 22 det 2.4% micro f l spec BS l APD T T T T T            (Eq. C.1) where f T , l T and BS T are the transmission efficiency of the fiber, lens and NIR beam splitter respectively. Similarly the detection efficiency of the setup at 930nm with 1200gr/mm used in the spectrometer is det 1.8%   . To verify and check the coupling efficiency of optical component in the micro-PL setup and the design of setup, the detection efficiency of the setup, det  , is measured by tracing the power loss of 930nm laser line along the optical path in the micro-PL setup. To assess the collection efficiency of the photon into the detection fiber, we reversed the light path and coupled the 930nm CW laser light from the Ti- Sa CW laser (Spectraphysics 3900) into the detection fiber of the micro-PL setup. The power of the 930nm laser line coming out of the detection fiber and that coming out of the microscope objective are measured. The measured micro  is 25.2%, 45% of the estimated value, indicating possible additional loss of the photons due to 423 coupling loss of the fiber and collimator and possible of abbreviation in the light path. The transmission efficiency of the spectrometer and detection efficiency of the APD are verified and measured by directly shooting the 930nm laser line through the detection fiber into spectrometer and tracing the power of the laser line in the light path. The transmission efficiency of the spectrometer with 600gr/mm is found to be 38% and the measured detection efficiency of the APD is ~25%. The measured ratio of the photons detected by the APD and the number of photons coming out of the detection fiber is ~4.48%. Therefore, we obtain that the measured detection efficiency of the setup is det 1.13%   at 930nm with 600gr/mm in the light path. The measured detection efficiency of setup is ~47% of the theoretical estimated detection efficiency in Eq. C.1. The discrepancy probably lies in the losses in the fiber/collimator connections, spherical abbreviation of lens and imperfection of mirrors etc. The closeness of the measured and estimated detection efficiency indicates that the optical design and optical alignment of the setup are in good condition. The measured detection efficiency of setup at 930nm with the 600gr/mm in the light path (as in g (2) (0) measurement discussed in Chapter 4) is used to estimate the number of single photons emitted from the QD into the collection cone angle of the objective. §C.4 Micro-PL instrumentation: timing electronics As briefly mentioned in section §C.1, we use time-correlated single photon counting (TCSPC) system to perform TRPL and g (2) (0) measurement. Here in this 424 section, we discuss in detail the timing process of the TCSPC in generating TRPL spectra and photon correlation histogram. In the TRPL measurement, the TTL (Transistor-Transistor Logic) electrical pulse signal with voltage ~3V recoding the PL detection event from MTSQD from the Si detector is inverted and attenuated to <200mV into the CFD to generate the NIM (nuclear instrumentation module) pulse representing the timing of the detection of photon from APD used as “start” signal into TAC. The laser NIM electronic pulse representing the optical excitation pulse from the laser itself is fed into the CFD to determine the timing of optical excitation event. Such signal is delayed and used as “stop” signal into the TAC. In TRPL measurement, the detection event of the APD detector is normally <1% of the repetition rate of the excitation laser itself. The use of laser signal as “stop” single ensures that there is always a corresponding “stop” for the “start” signal. The delay of the “stop” needs to be long enough to account for the time of light propagation, all instrument delay, and the desired time range of TRPL measurement. The timing configuration of the TCSPC for TRPL measurement is captured in Fig. C.3. The TRPL setup has a finite timing resolution limited by the APD time jitter and the jitter of the processing electronics with the FWHM (full width at half maximum) of the instrument response function (IRF) of our setup of ~150ps (Chapter 3). In the g (2) (0) measurement, however, the timing signal from the two APDs in the HBT setup is used as “start” and “stop” for the TAC. The “stop” signal is delayed before feeding into the TAC. No timing sequence is recorded for the excitation laser 425 but the timing sequence of photons detected by the two APDs. The timing configuration of the TCSPC for TRPL measurement is captured in Fig. C.3(b). The FWHM (full width at half maximum) of the instrument response function (IRF) of our setup for g (2) (0) measurement is ~750ps (Chapter 4) limited by the jitter of the two APDs. Figure C.3. The timing configuration of TC/SPC for (a) TRPL and (b) g (2) (0) shown in figure C.3. 426 Since the TC/SPC electronic works at as high as 1GHz frequency, all cautions in constructing a ultra-high frequency instrumentation need to be accounted, such as the proper choice and termination of cables, and the attenuation and distortion of signal as a function of cable length. 427 Appendix D: Quantization of the electromagnetic field This appendix follows essentially chapter 1.1 of ref [4.1]: M. O. Scully and M. S. Zubairy, Quantum optics, Cambridge University Press, 1997. The classical description of electromagnetic field is based on Maxwell’s equations shown below: , D H t      (Eq. D.1a) , B E t       (Eq. D.1b) 0, B  (Eq. D.1c) 0, D  (Eq. D.1d) These equations relate the electric and magnetic field. The magnetic field H and inductive vector B and the electric field E and displacement vector D satisfy the constitutive relations, , o BH   (Eq. D.2) , o DE   (Eq. D.3) where o  and o  are the free space permittivity and permeability, respectively, and 2 0 o c    , c is being the speed of light in vacuum. Given the constitutive relations in Eq. 4.2 and Eq. 4.3, the electric field ( , ) E r t satisfies the following wave equation: 2 2 22 1 0 E E ct      (Eq. D.4) The electric field can thus be represented as, 428   , ik r i t k k E r t E e     (Eq. D.5) where k is the wave vector with | | 2 / k   with  being the wavelength and  represents the frequency of the wave . Light can be understood as the wave-like undulation of the electric and magnetic fields propagating through space. However, the particle nature of light needs to be taken into account to understand certain effects, i.e. ultraviolet catastrophe and photoelectric effects. Dirac combines the wave- and particle-like aspects of the light to explain all interference effects of the radiation field. Following Dirac, each mode of the radiation filed is associated with a quantized simple harmonic oscillator which is an essence of the quantum theory of radiation. To understand the quantization of the continuum wave into harmonic oscillators, we capture here the quantization process of the electric field inside a one dimensional cavity to illustrate the concept of field quantization and then extend it to a three dimensional continuum space to capture the harmonic oscillator based description of the radiation field in free space. Now consider the electric field confined in a cavity resonator of length L as shown in Fig. D.1. We take the electric field to be linearly polarized in the x- direction and expand it in the normal modes of the cavity, 429 Figure 4.1. Schematic of electromagnetic field of frequency ω inside a cavity. The electric field is assumed to be transverse with polarization along the x-axis.   , sin( ), x j j j j E z t A q k z   (Eq. D.6)   0 , / cos( ), j y j j j j dq H z t A k k z dt    (Eq. D.7) where 1/2 2 0 2 jj j m A V        , V is the volume of cavity, q j (t) is the normal mode amplitude with the dimension of a length, k j =jπ/L, with j=1,2,3….. and w j = c k j . The classical Hamiltonian for the electromagnetic field is, 22 00 1 () 2 xy V d E H        (Eq. D.8a) with the integration over the volume of the cavity. Following Eq. D.6-D.8, the Hamiltonian is represented as 2 2 2 2 2 2 2 0 11 ( ( ) ) ( ) 22 jj j j j j j j jj j dq p H A q m q dt m         (Eq. D.8b) where j jj dq pm dt  is the canonical momentum of the j th mode. The Hamiltonian of this radiative field is expressed as a sum of the independent oscillators (Eq. D.8b). 430 Each mode of the field is dynamically equivalent to a mechanical harmonic oscillator. This dynamical problem can be quantized by identifying q j and p j as operators which obey the following commutation relations: '' [ , ] , j j jj q p i   (Eq. D.9a) '' [ , ] [ , ] 0 jj jj q q p p  (Eq. D.9b) We can make the following canonical transformation of q j and p j operators and obtain a j and a j + operators: 1 ( ), 2 it j j j j j jj a e m q ip m      (Eq. D.10a) 1 () 2 it j j j j j jj a e m q ip m      (Eq. D.10b) The Hamiltonian in Eq. D.8 can thus be written as 1 () 2 j j j j H a a     (Eq. D.11) The operator a j and a j + , referred to as annihilation and creation operator, satisfy the following commutation relations: '' [ , ] , j j jj aa    (Eq. D.12a) '' [ , ] [ , ] 0 jj jj a a a a   (Eq. D.12b) Correspondingly, the electric field and magnetic field (Eq.D.6 and Eq. D.7) in the cavity can be represented in terms of annihilation and creation operator (a j and a j + ):   , ( )sin( ), i t i t x j j j j j E z t a e a e k z      (Eq. D.13) 431   , ( )cos( ) i t i t y o j j j j j H z t i c a e a e k z        (Eq. D.14) where the quantity 1/2 j o V        has the dimension of an electric field. The radiation field in a finite one-dimensional cavity shown in Eq. D.13 and Eq. D.14 is quantized. The radiative field in the unbounded free space can be quantized in a similar way. We consider the field in a large but finite cubic cavity of size L. Cavity is treated as a region of space without any specific boundaries. We consider the running-wave solutions instead of standing waves considered above and impose periodic boundary conditions. The classical electric and magnetic fields can be expanded in terms of the plane waves as shown below:   ˆ,. k ik r i t k k k k E r t e c c        (Eq. D.15)   0 ˆ 1 ,. k ik r i t k kk k k k H r t e c c w         (Eq. D.16) where c.c stands for complex conjugate, ˆ k  is a unit polarization vector, k  is a dimensionless amplitude, 1/2 2 k k o V        and the summation is taken over an infinite discrete set of values of wave vector k . The boundary condition requires that, 2 2 2 , , , y x z x y z n n n k k k L L L       (Eq. D.17) where n x , n y , n z are integers. A set of numbers (n x , n y , n z ) defines a mode of the electromagnetic field. 432 The radiative field is quantized by identifying α k and α k * with harmonic oscillator operators a k and a k + , respectively, which satisfy the commutation relation [a k , a k + ]=1. The quantized electric and magnetic field then take the following form:   ˆ,. k ik r i t k k k k E r t a e H c      (Eq. D.18)   0 ˆ 1 ,. k ik r i t k kk k k k H r t a e H c w         (Eq. D.19) where H.c. represents Hermitian conjugate. The change from a discrete distribution of modes to a continuous distribution when L  can be made by replacing the sum in Eq. D.18 and D.19 with an integral: 3 3 2 k L dk        (Eq. D.20) Using the integration approach, the electric and magnetic field is found to satisfy the following commutation relations [ ( , ), ( , )] 0 jj E r t H r t  (j=x,y,z) (Eq.D.21) 2 (3) ' [ ( , ), ( , )] ( ) jk E r t H r t i c r r l       (Eq.D.22) where j, k and l form a cyclic permutation of x, y and z. The electric and magnetic field strength do not commute and they thus cannot be measured simultaneously. From the above discussion, the radiation field is described in the form of now harmonic oscillators that captures the wave- and particle-like nature in the quantum theory of light. Such harmonic oscillator representation is used to represents the state of photons discussed in Chapter 4.

Abstract (if available)

Abstract This dissertation contributes to the subject of realizing on-chip integrated nanophotonic systems comprising light source, light manipulating elements (LMEs such as cavity, waveguide, etc.), and detectors operating down to a single photon level for quantum information processing. We envision a scalable architecture for building such information processing circuits with ordered single quantum dot based single photon and entangled photon source integrated with either conventional 2D photonic crystal or, as we propose, dielectric building block (DBB) based multifunctional light manipulation unit (LMU). The dissertation covers several different issues that underlie the challenge of realizing co-designed scalable, ordered, and spectrally uniform single photon source (SPS) and entangled photon source (EPS) arrays integrable with LMEs. Specifically, it addresses: ❧ (1) The spatially selective synthesis of ordered and spectrally uniform on-chip integrable single quantum dot (SQD) arrays with controlled size and shape

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Creator Zhang, Jiefei (author)

Core Title Single photon emission characteristics of on-chip integrable ordered single quantum dots: towards scalable quantum optical circuits

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 11/20/2017

Defense Date 10/03/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,on-chip,ordered quantum dot,scalable quantum optical circuits,single photon emission

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Madhukar, Anupam (committee chair), Brun, Todd (committee member), Hashemi, Hossein (committee member), Kresin, Vitaly (committee member), Lu, Grace (committee member)

Creator Email jfz1130@gmail.com,jiefeizh@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-458924

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