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Imaging Transport Resonances in the Quantum Hall Effect Gary Alexander Steele Imaging Transport Resonances in the Quantum Hall Effect by Gary Alexander Steele Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2006 °c Gary Alexander Steele, MMVI. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Physics January 19, 2006 Certified by. Raymond C. Ashoori Professor Thesis Supervisor Accepted by . Thomas J. Greytak Professor, Associate Department Head for Education Imaging Transport Resonances in the Quantum Hall Effect by Gary Alexander Steele Submitted to the Department of Physics on January 19, 2006, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We image charge transport in the quantum Hall effect using a scanning charge ac- cumulation microscope. Applying a DC bias voltage to the tip induces a highly resistive ring-shaped incompressible strip (IS) in a very high mobility 2D electron system (2DES). The IS moves with the tip as it is scanned, and acts as a barrier that prevents charging of the region under the tip. At certain tip positions, short-range disorder in the 2DES creates a quantum dot island inside the IS that enables breach- ing of the IS barrier by means of resonant tunneling through the island. Striking ring shapes appear in the images that directly reflect the shape of the IS created in the 2DES by the tip. Through the measurements of leakage across the IS, we extract information about energy gaps in the quantum Hall system. Varying the magnetic field, the tunneling resistance of the IS varies significantly, and takes on drastically different values at different filling factors. Measuring this tunneling resistance provides a unique micro- scopic probe of energy gaps in the quantum Hall system. Simulations of the interaction of the tip with the quantum Hall liquid show that native disorder from remote ionized donors can create the islands. The simulations predict the shape of the IS created in the 2DES in the presence of disorder, and comparison of the images with simulation results provides a direct and quantitative view of the disorder potential of a very high mobility 2DES. We also draw a connection to bulk transport. At quantum Hall plateaus, electrons in the bulk are localized by a network of ISs. We have observed that the conductance across one IS is drastically enhanced by resonant tunneling through quantum dot islands. Similarly, this resonant tunneling process will dramatically enhance the con- ductance of certain hopping paths in the localized bulk and could play an important role in dissipative transport at quantum Hall plateaus. Thesis Supervisor: Raymond C. Ashoori Title: Professor Acknowledgments First and foremost, I would like to thank my advisor Ray. In my graduate career, I have learned a great deal from Ray: in particular, his openness, his remarkable physical intuition, his enthusiasm for physics, and his creativity are all things that I aspire to. It is these qualities that make him an excellent physicist and teacher. Success in this project would not have been possible without the drive and excitement he brought to the experiment. I would also like to thank my thesis committee, in particular Professor Leonid Levitov for the helpful and insightful discussions we had. In experimental physics, nothing is accomplished alone. Whether it is borrowing a vacuum fitting, discussing physics on a white board in the hallway, or fixing the liquefier, interacting with a community of people around you is a key to a successful experiment. For me, that community was the people of the second floor of Building 13, including the all the members of our research group as well as the Kastner and Greytak groups. I would like to thank Oliver Dial in particular for his endless help with writing software and building electronics, and even more so for the stimulating physics and non-physics discussions we have had in our time here. I would like to thank all the Ashoori group members who have helped me over the years, in particular Paul Glicofridis who taught me to run the microscope and Nemanja Spasojevic who was instrumental in our simulation work. Finally, I would like to thank my family for their constant support and encour- agement, and Susan Dunne, who was always there for me. 6 Contents 1 Introduction 19 1.1 The 2D Electron System and the Quantum Hall Effect . 19 1.2 Conductivity in the Quantum Hall Effect . 23 1.3 Our work ................................. 27 2 Scanning Charge Accumulation Imaging 31 2.1 Scanning Probe Microscope Design ................... 31 2.2 Capacitance Bridge Charge Sensor ................... 35 2.3 What does SCA measure? ........................ 43 2.4 Understanding the influence of the measurement on the 2D electron system ................................... 46 2.5 Sample Design ............................... 52 2.6 Characterizing the sample using Magnetocapacitance . 55 2.7 The shape of the tip ........................... 64 2.7.1 Analysis of the capacitance tip approach curve . 65 2.7.2 Analysis of the SCA gate images . 69 2.8 Finding the surface of the sample .................... 73 2.9 Producing blunt tips with a controlled geometry . 77 3 Imaging Transport Resonances in the Quantum Hall Effect 81 3.1 Background ................................ 81 3.2 Imaging the resistance of an Incompressible Strip . 87 3.3 What are the hotspots? .......................... 90 7 3.4 Imaging the resistance of a ν = 1 IR at lower magnetic fields: Creating a hole-bubble ............................... 95 3.5 Frequency Dependence .......................... 98 3.6 Behavior of the imaged arcs with bias voltage and magnetic field . 100 3.7 Measuring the signal with the tip at a fixed position . 104 3.8 Why Partial Rings? ............................ 108 3.9 Testing the quantum dot island model: Imaging with a large tip- induced density gradient . 110 3.10 IRs at other filling factors . 113 3.11 Features of the uncharging bubble . 117 3.12 Excitation Dependence . 119 3.13 Hotspots are not “defects” . 124 3.14 Summary ................................. 125 4 Measuring the Capacitance of the 2DES to a Fixed Metal Pad 129 4.1 Motivation ................................. 129 4.2 Experimental Setup ............................ 130 4.3 Measuring the Charging Signal . 132 4.4 Measuring Oscillations of the Density of States . 135 4.5 Density inhomogeneity at the edge of the gate . 140 4.6 Sample Density .............................. 144 4.7 Conclusions ................................ 145 5 Simulating the Interaction of a Metallic Scanning Probe with the Quantum Hall Liquid 149 5.1 Motivation ................................. 149 5.2 Numerical Methods ............................ 150 5.3 The Non-Linear Poisson Equation . 154 5.4 Boundary Conditions and Non-Uniform Grids . 157 5.5 Results from 2D Simulations Using Cylindrical Symmetry . 159 5.5.1 Tip Approach Curves . 159 8 5.5.2 Capacitance Change Seen in Gate Images . 161 5.5.3 Simulating the Incompressible Ring . 164 5.5.4 Calculating the Lever Arm . 165 5.5.5 Incompressible Strip Widths in the Magnetocapacitance Exper- iment ............................... 167 5.5.6 Jellium Approximation for the Tip Perturbation . 168 5.6 3D simulations .............................. 170 5.6.1 Simulating the IR in the Presence of Disorder . 170 5.6.2 Effects of Donor Correlations . 173 5.6.3 Moving the Tip: Creating a Simulated SCA Image . 175 5.7 Simulating the incompressible bulk . 178 5.8 Conclusions ................................ 179 6 Future Research Directions 181 A Measuring the Input Noise of the Capacitance Bridge 185 B Transconductance Calculations and Noise Considerations for the HEMT Amplifier 189 B.1 Properties of a FET . 189 B.2 Modelling the FET ............................ 191 B.3 Amplifier Design ............................. 194 B.4 Charge Transconductance . 195 B.5 Noise .................................... 197 C Images Taken with a Smashed Tip 199 D Depleting the Sample with a Global Backgate 205 E Distributed RC Network 209 9 10 List of Figures 1-1 The Quantized Hall Resistance ..................... 20 1-2 The GaAs 2DES ............................. 22 1-3 Conductivity in the quantum Hall effect . 23 1-4 Landau Levels ............................... 24 1-5 Deviation from activated transport at low T . 26 1-6 Probing σxx microscopically ....................... 28 2-1 The scanning probe microscope ..................... 32 2-2 Besocke coarse approach motor ..................... 34 2-3 Schematic of a capacitance bridge .................... 36 2-4 Pictures of the SCA charge sensor .................... 38 2-5 Schematic of amplifier .......................... 38 2-6 Noise spectrum of lock-in input ..................... 40 2-7 Distributed RC charging model ..................... 44 2-8 Simple RC charging model ........................ 45 2-9 Effect of AC excitation on fully charging 2DES . 48 2-10 Capacitive lever arm ........................... 49 2-11 Images of the sample with topgate .................... 53 2-12 AFM images of the sample ........................ 54 2-13 Magnetocapacitance with no gate .................... 55 2-14 Global capacitance contribution from tip . 56 2-15 Magnetocapacitance with a top gate . 58 2-16 Charging the 2D layer through the gate capacitance . 59 11 2-17 Hysteresis
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