Quantum Circuit based on Electron Spins in Semiconductor Quantum Dots Chang-Yu Hsieh Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics1 Department of Physics Faculty of Science University of Ottawa c Chang-Yu Hsieh, Ottawa, Canada, 2012 1The Ph.D. program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute Of Physics Thesis advisor Author Dr. Pawel Hawrylak Chang-Yu Hsieh Quantum Circuit based on Electron Spins in Semiconductor Quantum Dots Abstract In this thesis, I present a microscopic theory of quantum circuits based on inter- acting electron spins in quantum dot molecules. We use the Linear Combination of Harmonic Orbitals-Configuration Interaction (LCHO-CI) formalism for microscopic calculations. We then derive effective Hubbard, t-J, and Heisenberg models. These models are used to predict the electronic, spin and transport properties of a triple quantum dot molecule (TQDM) as a function of topology, gate configuration, bias and magnetic field. With these theoretical tools and fully characterized TQDMs, we propose the fol- lowing applications: 1. Voltage tunable qubit encoded in the chiral states of a half-filled TQDM. We show how to perform single qubit operations by pulsing voltages. We propose the "chirality-to-charge" conversion as the measurement scheme and demon- strate the robustness of the chirality-encoded qubit due to charge fluctuations. We derive an effective qubit-qubit Hamiltonian and demonstrate the two-qubit gate. This provides all the necessary operations for a quantum computer built with chirality-encoded qubits. 2. Berry's phase. We explore the prospect of geometric quantum computing with chirality-encoded qubit. We construct a Herzberg circuit in the voltage space ii and show the accumulation of Berry's phase. 3. Macroscopic quantum states on a semiconductor chip. We consider a linear chain of TQDMs, each with 4 electrons, obtained by nanostructuring a metallic gate in a field effect transistor. We theoretically show that the low energy spectrum of the chain maps onto that of a spin-1 chain. Hence, we show that macroscopic quantum states, protected by a Haldane gap from the continuum, emerge. In order to minimize decoherence of electron spin qubits, we consider using electron spins in the p orbitals of the valence band (valence holes) as qubits. We develop a theory of valence hole qubit within the 4-band k.p model. We show that static magnetic fields can be used to perform single qubit operations. We also show that the qubit-qubit interactions are sensitive to the geometry of a quantum dot network. For vertical qubit arrays, we predict that there exists an optimal qubit separation suitable for the voltage control of qubit-qubit interactions. iii Contents Title Page . i Abstract . ii Table of Contents . iv List of Figures . vii Statement of Originality . ix Acknowledgments . xi 1 Introduction 1 1.1 Overview of Quantum Computation . 2 1.1.1 DiVincenzo's Criteria: Search for a Physical Qubit . 4 1.1.2 Semiconductor Qubits . 6 1.2 Quantum Dots . 9 1.2.1 Fabrication of Lateral Gated Quantum Dots . 11 1.2.2 Spectroscopic Tools for Lateral Gated Quantum Dots . 12 1.2.3 Fabrication of Self-Assembled Quantum Dots . 19 1.2.4 Photoluminescence Spectroscopy for Self-Assembled Quantum Dots . 21 1.3 Quantum Algorithm with Electron Spin-Based Qubits . 22 1.3.1 Single Qubit Operations . 23 1.3.2 Double Qubit Operations . 24 1.3.3 Quantum Teleportation in a Linear Triple Quantum Dot . 27 2 Electronic and Spin Properties of a Triple Quantum Dot 30 2.1 Models . 30 2.1.1 Linear Combination of Harmonic Orbitals - Configuration In- teractions . 31 2.1.2 Hubbard Model . 39 2.1.3 Effective Models . 43 2.2 Electronic and Spin Properties as a Function of Number of Electrons, Topology and Biasing of Central Dot . 48 2.2.1 One-Electron and One-Hole Case . 49 iv 2.2.2 2-Electrons and 2-Holes Case . 52 2.2.3 Three-Electron Case . 57 2.2.4 Stability Diagrams . 62 2.3 Aharonov-Bohm Oscillations in a Triple Quantum Dot . 63 2.4 Conclusion . 69 3 Chirality-based Coded Qubits in Triple Quantum Dot Molecules 72 3.1 Introduction . 72 3.2 Models of Chirality-based Coded Qubits . 74 3.2.1 LCHO-CI Microscopic Model . 77 3.2.2 Effective Models . 81 3.2.3 Chiral States . 83 3.3 Quantum Information Processing with Coded Qubits . 85 3.3.1 Single coded qubit: preparation and initialization . 85 3.3.2 Coherent Operations: single qubit gates . 90 3.3.3 Coherent Operations: double qubit gates . 90 3.3.4 Measurement of a single coded qubit . 96 3.4 Decoherence due to Fluctuating Charged Impurities . 97 3.4.1 Triple Quantum Dot Molecule with Charged Impurity . 98 3.4.2 Effects of Impurity on the One-Electron Energy Spectrum . 100 3.4.3 Effects of an Impurity on the Coded Qubit Levels . 103 3.4.4 Mapping of Charge Fluctuation Induced Decoherence into Spin in a Random Magnetic Field Problem . 108 3.4.5 Conclusion . 112 4 Macroscopic Quantum States in a Chain of Linear TQDs 115 4.1 Introduction . 116 4.2 Theory and methods . 117 4.3 Results and discussion . 123 4.4 Conclusions . 128 5 Herzberg Circuit and Berry's Phase in a Chirality-based Coded Qubit in a Triangular Triple Quantum Dot 129 5.1 Introduction . 130 5.2 The model . 132 5.3 Generation and Detection of Berry's phase . 135 5.4 Conclusion . 144 6 Sequential Tunneling and Spin Blockade in a Linear Triple Quantum Dot 145 6.1 Introduction . 146 6.2 Model . 149 v 6.3 Theory of Sequential Tunneling . 154 6.4 Transport and Spin Blockade . 157 6.4.1 Quantum Interference-based Spin Blockade . 159 6.4.2 Symmetrical and Asymmetrical Spin Blockade . 169 6.5 Conclusion . 176 7 Valence Hole Qubit in Self-Assembled Quantum Dots 177 7.1 Motivation . 178 7.2 Model . 181 7.3 Hole states in single quantum dot . 184 7.3.1 Hole under Bz field . 184 7.3.2 Hole under Bx field . 187 7.4 Hole states in coupled quantum dots . 190 7.4.1 Hole states in vertically coupled quantum dots . 192 7.4.2 Hole states in laterally coupled quantum dots . 193 7.5 Conclusion . 194 8 Conclusion 195 A Linear Combination of Harmonic Oscillator - Configuration Interac- tion 207 A.1 Single Particle in a Parabolic Dot with Magnetic Field . 208 A.1.1 Fock-Darwin Orbitals . 209 A.1.2 Fock-Darwin Orbitals in terms of Harmonic Oscillator Orbitals 211 A.2 Single Particle in a Multi-Dot Device . 214 A.3 Many Electrons in a Multi-Dot Device . 223 B L¨owdinPerturbation Theory 226 B.1 Division of Hamiltonian and Hilbert Space . 226 B.2 Analytical Transformations . 227 B.2.1 First Order . 229 B.2.2 Second Order . 231 B.3 Iterative Approach to Numerical Transformation . 232 C Cotunneling Through a Triple Quantum Dot 236 vi List of Figures 1.1 Fabrication of lateral gated quantum dots . 11 1.2 Coulomb blockade in lateral gated quantum dots . 13 1.3 Quantum point contact and charge sensing . 15 1.4 Spin blockade in a linear triple quantum dot . 16 1.5 Characterization of a triple quantum dot . 18 1.6 Fabrication of self-assembled quantum dots . 20 1.7 quantum circuit diagram for quantum teleportation . 27 2.1 Model of triple quantum dots . 32 2.2 Single particle energy spectra of a triangular triple quantum dot as a function of complete shells of FD orbitals included in the basis for exact diagonalization. 36 2.3 single-particle spectrum calculated with LCHO . 37 2.4 Spatial representation of localized Hubbard orbitals. 41 2.5 2-electron energy spectra calculated from LCHO-CI, Hubbard, and t-J models . 47 2.6 Two-hole triplet-singlet transition . 58 2.7 Three-electron energy spectra calculated from LCHO-CI, Hubbard, and Heisenberg models . 60 2.8 Stability diagram of triple quantum dots . 63 2.9 AB oscillations in triple quantum dots . 65 2.10 AB oscillations in triple quantum dots 2 . 68 2.11 Stability diagram of a triple quantum dot in the presence of magnetic field . 70 3.1 Quasi-momentum-resolved subspaces of a half-filled triangular triple quantum dot . 76 3.2 Architectures of triple quantum dot network . 80 3.3 Preparation of a coded qubit . 86 3.4 Preparation of coded qubit 2 . 89 3.5 Energy spectrum of two coupled triple quantum dots . 95 vii 3.6 Effects of a charged impurity on single-particle states . 102 3.7 Effects of a charged impurity on single-particle energy spectrum . 104 3.8 Effects of a charged impurity on three-electron energy spectrum . 105 3.9 Simulation of random magnetic fields . 111 3.10 Charged impurity-induced decoherence of coded qubit . 113 4.1 Model of quantum dot network on a semiconductor chip . 118 4.2 Models of effective spin chain . 121 4.3 Comparison of Energy spectra of spin-1/2 chain and TQD chain . 124 4.4 Comparison of energy spectra of spin-1/2 and spin-1 chains . 126 5.1 Herzberg circuit . 136 5.2 Deviation from Heisenberg model . 139 5.3 Herzberg circuit 2 . 140 5.4 accumulation of Berry's phase . ..