Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Drayne, Johann Peter
Master of Science - MSc
University of British Columbia
The Kondo effect, first discovered in impure bulk metals during the 1930s and explained in the 1960s, has gained significant interest within the field of quantum devices. These devices offer a high degree of tunability and control, enabling rigorous testing of theoretical predictions. Previous studies on the Kondo effect have measured conductance through a quantum dot and observed a zero-bias peak between Coulomb peaks. This effect requires strong coupling between the quantum dot and leads. This work studies a relatively weak coupling, where the characteristic zero-bias peak between Coulomb peaks is not observed. However, as charge degeneracy of the quantum dot is approached, the Kondo temperature increases. This results in a small conductance enhancement at the shoulder of the Coulomb peak. We show that a simultaneous measurement of the quantum dot’s occupation can unveil the small enhancement of conductance due to the Kondo effect. To compare with Numerical Renormalisation Group (NRG) theory, conductance and occupation are measured across a range of temperatures to determine fitting parameters. Good agreement is found at a range of coupling strengths and charge sensor current setpoints. However, a discrepancy is found when the tunnel barriers to the quantum dot are asymmetrically tuned. Strong asymmetric coupling approaches a regime where the quantum dot is coupled to a single lead. A recent measurement of the entropy of a quantum dot coupled to a single lead where Kondo correlations were expected also observed a discrepancy between data and NRG. Interestingly, this work shows that conductance data displays greater Kondo enhancement than NRG, whereas previously measured entropy showed less Kondo enhancement than NRG. A direct comparison of conductance and entropy measured in the same device under similar settings holds promise for illuminating this discrepancy.
Occupation Resolved Conductance ofa Few Electron Quantum Dot:A Test for Kondo Correlations in an IntermediateCoupling RegimebyJohann Peter DrayneB.Sc., The University of British Columbia, 2022A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2024© Johann Peter Drayne 2024The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Occupation Resolved Conductance of a Few Electron QuantumDot: A Test for Kondo Correlations in an Intermediate CouplingRegimesubmitted by Johann Peter Drayne in partial fulfillment of the require-ments for the degree of Master of Science in Physics.Examining Committee:Joshua Folk, Professor, Physics and Astronomy, UBCSupervisorAlannah Hallas, Assistant Professor, Physics and Astronomy, UBCSupervisory Committee MemberiiAbstractThe Kondo effect, first discovered in impure bulk metals during the 1930sand explained in the 1960s, has gained significant interest within the fieldof quantum devices. These devices offer a high degree of tunability andcontrol, enabling rigorous testing of theoretical predictions. Previous studieson the Kondo effect have measured conductance through a quantum dotand observed a zero-bias peak between Coulomb peaks. This effect requiresstrong coupling between the quantum dot and leads.This work studies a relatively weak coupling, where the characteristiczero-bias peak between Coulomb peaks is not observed. However, as chargedegeneracy of the quantum dot is approached, the Kondo temperature in-creases. This results in a small conductance enhancement at the shoulder ofthe Coulomb peak. We show that a simultaneous measurement of the quan-tum dot’s occupation can unveil the small enhancement of conductance dueto the Kondo effect.To compare with Numerical Renormalisation Group (NRG) theory, con-ductance and occupation are measured across a range of temperatures todetermine fitting parameters. Good agreement is found at a range of cou-pling strengths and charge sensor current setpoints. However, a discrepancyis found when the tunnel barriers to the quantum dot are asymmetricallytuned.Strong asymmetric coupling approaches a regime where the quantumdot is coupled to a single lead. A recent measurement of the entropy of aquantum dot coupled to a single lead where Kondo correlations were ex-pected also observed a discrepancy between data and NRG. Interestingly,this work shows that conductance data displays greater Kondo enhance-ment than NRG, whereas previously measured entropy showed less KondoiiiAbstractenhancement than NRG. A direct comparison of conductance and entropymeasured in the same device under similar settings holds promise for illu-minating this discrepancy.ivLay SummaryThe Kondo effect, a phenomenon originally observed in impure metalsdecades ago, has been studied more recently in quantum devices. Thesedevices offer precise control to test theoretical predictions. In these devices,a single electron on an isolated island is connected to a bath of electrons oneach side.Previous studies have measured the conductance through the island as asingle electron is added and observed a large increase in conductance as thesystem’s temperature was lowered. These experiments used a small barrierbetween the island and surrounding electrons.In our experiment, we used a large barrier, resulting in an increase inconductance that was too small to measure with previous methods. Tomeasure this small increase in conductance, we also measure the island’soccupation as a single electron is added. When the barriers to the islandare symmetric, we find agreement with theory, and surprising disagreementwhen asymmetric.vPrefaceThis thesis encompasses the research I conducted during my Master’s studiesin the Quantum Devices group, led by Joshua Folk, at the University ofBritish Columbia.Numerous discussions with my supervisor Joshua Folk, and research as-sociate Silvia Lu¨scher have greatly influenced the experimental concept, de-vice design, analysis and interpretation.The GaAs/AlGaAs heterostructures were provided by Michael Manfra’sgroup at the University of Purdue. Christian Olsen fabricated the mesasand ohmic contacts at the University of Copenhagen.Numerical Renormalisation Group (NRG) simulations that we compareour measurements to in Chapter 4, were performed by Yaakov Kleeorin,under the supervision of Yigal Meir.I joined the Quantum Devices group as an undergraduate (September2021) and spent the first eight months shadowing Timothy Child throughoutthe design, fabrication, measurement, and analysis processes.The following year (first year of my Master’s studies, September 2022), Ispent working closely with Timothy on device design, fabrication, and mea-surement. Timothy and I fabricated all devices used in this thesis duringthis period. All data taken in Chapter 2 and Chapter 3 was taken by Tim-othy and I. In particular, at the end of this period, Timothy and I tookpreliminary measurements in the regime explored in Chapter 4. These mea-surements demonstrated a working proof of how to measure in this regime.In my final year (starting May 2023), Timothy worked on his PhD, and Ibecame the lead researcher responsible for data collection and analysis. Alldata taken in Chapter 4 was taken during this period.All figures in this work are original.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Device Background . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Two Dimensional Electron Gas (2DEG) . . . . . . . . . . . . 52.3 Quantum Point Contact (QPC) . . . . . . . . . . . . . . . . 72.4 Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.1 Conductance Through a Quantum Dot . . . . . . . . 112.4.2 Charge Sensing a Quantum Dot . . . . . . . . . . . . 132.5 Determining Electron Temperature . . . . . . . . . . . . . . 15viiTable of Contents2.6 Measurement Procedures . . . . . . . . . . . . . . . . . . . . 172.6.1 Cooldown Bias . . . . . . . . . . . . . . . . . . . . . . 172.6.2 Gate Divider . . . . . . . . . . . . . . . . . . . . . . . 182.6.3 Virtual Gate . . . . . . . . . . . . . . . . . . . . . . . 183 Conductance in the Kondo Regime . . . . . . . . . . . . . . 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Kondo Effect in Bulk Materials . . . . . . . . . . . . . . . . 213.3 Kondo Effect in Quantum Dots . . . . . . . . . . . . . . . . 234 Occupation Resolved Conductance of the Kondo Effect . 284.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Occupation Resolved Conductance Method . . . . . . . . . . 304.2.1 Numerical Renormalisation Group (NRG) . . . . . . 324.2.2 Conductance Global Fit to NRG . . . . . . . . . . . . 334.2.3 Charge Transition Fit to NRG . . . . . . . . . . . . . 334.2.4 Determining Occupation . . . . . . . . . . . . . . . . 344.2.5 Conductance versus Occupation . . . . . . . . . . . . 364.3 Kondo Effect with Varying Coupling Strength . . . . . . . . 364.4 Kondo Effect with Varying Charge Sensor Current . . . . . . 384.4.1 Varying Charge Transition . . . . . . . . . . . . . . . 384.4.2 Conductance versus Occupation . . . . . . . . . . . . 394.5 Kondo Effect with Varying Coupling Symmetry . . . . . . . 414.5.1 Symmetric to Asymmetric Coupling . . . . . . . . . . 414.5.2 Conductance versus Occupation . . . . . . . . . . . . 445 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47AppendicesA Quantum Device Fabrication . . . . . . . . . . . . . . . . . . 60A.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60A.2 Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A.2.1 Gallium Removal . . . . . . . . . . . . . . . . . . . . 61viiiTable of ContentsA.2.2 Mesa Etch . . . . . . . . . . . . . . . . . . . . . . . . 62A.2.3 Ohmic Contact . . . . . . . . . . . . . . . . . . . . . 62A.2.4 Gating . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2.5 Wire Bonding . . . . . . . . . . . . . . . . . . . . . . 66B Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68B.1 Open-Source Thesis . . . . . . . . . . . . . . . . . . . . . . . 68B.2 Data Acquisition and Analysis . . . . . . . . . . . . . . . . . 68B.3 Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . 68ixList of Tables4.1 Γ/kBT From Conductance and Charge Transition Global Fits 43xList of Figures1.1 Fridge to Quantum Dot Scale Illustration . . . . . . . . . . . 22.1 GaAs/AlGaAs Heterostructure . . . . . . . . . . . . . . . . . 62.2 Quantum Point Contact . . . . . . . . . . . . . . . . . . . . . 82.3 Quantum Dot Energy Levels . . . . . . . . . . . . . . . . . . 102.4 Conductance Through a Quantum Dot . . . . . . . . . . . . . 122.5 Charge Sensing a Quantum Dot . . . . . . . . . . . . . . . . . 142.6 Determining Electron Temperature . . . . . . . . . . . . . . . 152.7 Effects of a Virtual Gate . . . . . . . . . . . . . . . . . . . . . 193.1 Kondo Effect in Bulk Materials . . . . . . . . . . . . . . . . . 223.2 Kondo Effect Illustration : Bulk Material . . . . . . . . . . . 233.3 Kondo Effect Illustration : Quantum Dot . . . . . . . . . . . 243.4 Kondo Effect in a Strongly Coupled Quantum Dot . . . . . . 264.1 Fittin Conductance and Charge Transition to NRG . . . . . . 344.2 Conductance versus Occupation : Varying Temperature . . . 354.3 Conductance versus Occupation : Varying Coupling Strength 374.4 Varying Charge Sensor Current Setpoint . . . . . . . . . . . . 394.5 Conductance versus Occupation : Varying Charge SensorCurrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.6 Symmetric to Asymmetric Coupling . . . . . . . . . . . . . . 424.7 Conductance versus Occupation : Varying Coupling Symmetry 45A.1 Gate Fabrication Illustration . . . . . . . . . . . . . . . . . . 65A.2 Wirebonding a Chip . . . . . . . . . . . . . . . . . . . . . . . 67xiList of FiguresB.1 Loading a DXF Design File . . . . . . . . . . . . . . . . . . . 69B.2 Electric Potential Calculation . . . . . . . . . . . . . . . . . . 70B.3 kwant : Tight Binding Model and Band Structure . . . . . . 70B.4 kwant : Transport Simulation . . . . . . . . . . . . . . . . . 71xiiAcknowledgementsAs I write this final section of my thesis on a rainy Vancouver evening witha cup of ‘Belfast Brew’ in hand, I can’t help but reminisce on this chapter oflife as it comes to an end. I have spent the last seven years at the Universityof British Columbia, where the last three in the Quantum Devices Grouphave been a particular highlight. I must start by thanking all current andformer members of the group. Much work is required to study quantumdots, and none of my work would have been possible without all the effortthat has been put into creating an educational and thought-provoking labenvironment.Of course, my first personal thanks goes to my supervisor, ProfessorJoshua Folk. Over the last three years, his ability to explain tricky concepts,patience to ensure my understanding and attention to detail has made me abetter physicist. One of many skills that I will take with me is understandingwhen I do and do not understand.Dr. Silvia Lu¨scher: Much have I learnt from Silvia on mesoscopic physicsand the art of fabrication. In meetings, Silvia’s insights offered alternatepaths to a solution. These insights were a reminder to explore all optionsbefore settling on an answer. Also, her Swiss chocolate recommendation(‘Ragusa’) is excellent.Dr. Timothy Child: A senior PhD student who was my mentor duringmy first year in the group. We worked together closely in my second year.Many long days were spent together being serenaded by the pulse tube. Itdidn’t take long for us to become friends. Joined by a genuine passion forlearning and adventure, our many discussions and deliberations were notonly of great academic but also personal interest. I look forward to stayingin touch and perhaps trying caving in Slovenia.xiiiAcknowledgementsVahid Mohaved: A PhD student who joined in my final year here. Ourlove for coffee brought us together on many Tim Horton’s trips where wediscussed physics and more.Elena Cornick: A Ph.D. student who joined me in my final half-yearhere. Although we only had a short cross-over, we worked efficiently, andElena’s theoretical insights were a welcome challenge to my ‘tennis ball’model of electrons.Lily Watt: An undergraduate student who worked on perfecting ourohmic recipe. I always enjoyed the insightful questions and wish her thebest as president of UBCQ.Raveel Tejani: An undergraduate who tackled the brave task of devel-oping ‘master-slave’ functionality for our measuring instruments. Our lunchwalks for pizza slices and discussions on climbing were enjoyable breaks fromweeks of confusing measurements.Dr. Yigal Meir, Dr. Eran Sela, and Dr. Andrew Mitchell: Our theorycollaborators for their expertise and understanding of how to translate theirtheoretical frameworks onto our experiments.Our ‘Quantropy’ collaborators at ETH Zurich (Klauss Ensslin and ThomasIhn’s group), and Universite´ Paris-Saclay (Frederic Pierre’s group), thankyou for organising the Quantropy conferences over the past three years. Thewide range of disciplines and shared goals fostered fruitful conversations.Witnessing the birth and debate of new experimental ideas was remarkable.Diego, Mason, Pearce and Luke: My best friends from University andback home in Ireland. We have all moved to different cities, but our friend-ships remain strong. I always look forward to our calls for fresh perspectivesand annual, sometimes bi-annual, meetups for a beer or three. Whether weare discussing the details of heat pumps or walking 60 km for a fish andchips, our adventures are always a pleasure.Cˇenˇka: My sister, we have grown up greatly over the past few years,although it seems that distance has brought us closer. I love all that we dotogether, from our debates to our dance-offs; I look forward to being closerto home in the future.Jana and Peter: My parents, it goes without saying that you influencexivAcknowledgementsmuch of who I am. I always look forward to our weekly catchups, whetherthey are long or short; they keep me rooted. They say that home is whereyou are known the most and appreciated the least. But not in this case.You were my first educators and now my best friends.Olivia: My fiance´e and future wife. Little did I know that seven yearsago, when I decided to cross the Atlantic, I would find a life partner. Weare there for each other, for the big and small, the highs and lows. Fromour protein bar assembly line to our 14 hour hikes and evening tea time,who knows what our next chapter will hold. I look forward to it with eageranticipation.xvDedicationBe Bright,Be Brief,Be Off.—My Irish granny, Eileen DraynexviChapter 1Introduction“In mesoscopic physics, you really need to build up intuition, becauseit is not the world you know.”Carlo Beenakker [1]1.1 ContextMesoscopic physics is in the domain of condensed matter physics with sys-tem scale sizes between nanometre and micrometre. Mesoscopic systemsunveil a rich zoo of quantum phenomena and material properties. One ofthe initial measurements of a mesoscopic system studied the conductancethrough a quantum wire [2], [3]. Classically, the conductance is expected tobe proportional to the width of the wire. However, as the width varies, itwas found that a quantum wire exhibits steps at integer conductance val-ues. Each step in conductance is the result of an additional 1D-sub-bandthat contributes to the transport. Another milestone in the field was themeasurement of Coulomb blockade oscillations in the conductance measuredacross a quantum dot [4]. In this experiment, a small isolated island of elec-trons was connected to two baths of electrons through tunnel barriers. Theisland’s small size results in charge quantisation and significant spacing be-tween orbital energy levels, forming an ‘artificial atom’. The benefits of thissystem come from a high degree of tunability and control. The relevantscales that allow human control over this ‘artificial atom’ are illustrated inFig. 1.1. The scope of mesoscopic physics is continually expanding, withongoing developments in new materials [5], measurement methods [6], anddevice architectures [7], [8].11.1. Contexta) b) c)d)e)f)80 cm 30 cm 5mm200 nm 10 μm 1mmFigure 1.1: Illustrating the relationships from macroscopic to nanoscale todemonstrate the tunability of quantum devices. (a) Photograph of the Au-plated cold plates within a dilution refrigerator. The lowest plate, known asthe mixing chamber, can reach temperatures of 8mK. (b) Photograph of theSi chip carrier, onto which the heterostructure is attached. (c) An opticalmicroscope image shows the heterostructure attached to the chip carrier.Thin Al wire bonds connect the quantum device to the fridge wires. (d)Optical microscope image of a single mesa on the heterostructure. A mesais an isolated area of the heterostructure where new designs are fabricated.The black lines around the outside are wire bonds and the bright Au arethick (100 nm) ‘outer gates’ which connect to thin (10 nm) ‘inner gates’. (e)Scanning electron micrograph (SEM) of the inner gates. The mean free pathof the electrons is of order 3 µm. (f) SEM image of the quantum dot. Anisolated puddle of electrons with a total occupation between 0 and 10 isengineered by carefully tuning the voltages on the inner gates.Theorists and experimentalists enjoy a symbiotic relationship in meso-scopic physics. Experiments validate established theories, while theorists are21.1. Contexttasked with modelling surprising and repeatable phenomena discovered inexperiments. Notable examples of successful experiments validating theoryinclude the investigation of Wigner crystals [9], isolation of graphene [10],and measurement of the charge of quasiparticles in the fractional quantumHall effect [11]. Conversely, instances of unexpected experimental findingsthat prompted theoretical inquiry include integer quantum Hall effect [12],anomalous Landau quantisation [13], and superconductivity in twisted dou-ble bilayer graphene [8].This thesis focuses on a phenomenon initially discovered through exper-imental observation, prompting theorists to develop a new model capable ofpredicting the striking behaviour observed in measurements. The ‘Kondoeffect’, first discovered in impure bulk metals in the 1930s [14] and explainedlater in the 1960s [15], has seen significant interest in the field of quantumdevices. These devices offer high tunability and control, allowing for testingtheoretical predictions [16]. The Kondo effect in quantum devices involves anet spin (odd number of electrons) in a quantum dot that is strongly coupledto two electron reservoirs (leads). Virtual tunnel events through the quan-tum dot, with possible spin flips, lead to a correlated state called the Kondosinglet. The Kondo effect can be seen in the enhanced conductance betweenCoulomb peaks with decreasing temperature. This conductance approachesthe unitary limit, where the transmission probability through the quantumdot is one [17]–[19]. This is a remarkable effect as the tunnel barriers intothe quantum dot each have a transmission probability less than one. Morerecent experiments have explored exotic regimes of the Kondo effect. Forexample, Kondo effect in graphene [20], Kondo lattices which are used tostudy quantum criticality [21], multi-channel Kondo which could lead tonon-Fermi liquid behaviour [22]–[24], and the novel topological Kondo ef-fect which could demonstrate the non-local quantum dynamics of Majoranafermions [25]–[27].Nevertheless, there remains considerable potential for further insightsby extending the scope of the initial, seemingly ’simple’ measurements onthe Kondo effect into new parameter regimes. In this thesis, we developand test a method for measuring the Kondo effect with weaker coupling31.2. Outlinethan previously explored. This regime is rarely explored as the conductanceenhancement from Kondo is small, and cannot be reliably extracted witha conductance measurement only. Interestingly, a recent measurement ofentropy in this regime found a disagreement with theoretical predictions [28].Hence, this measurement of conductance taken in a similar regime couldilluminate potential reasons for the discrepancy.1.2 OutlineChapter 2 provides an overview of quantum devices. The chip material(GaAas/AlGaAs heterostructure) is introduced and how the metal gates ontop of the chip are used to engineer quantum structures such as quantumpoint contacts and quantum dots. Additionally, measurements of conduc-tance and occupation through a quantum dot are described.Chapter 3 introduces the Kondo effect in bulk materials and quantumdots. It includes a measurement demonstrating the large conductance en-hancement caused by the Kondo effect in the ‘Kondo regime’, highlightingthe differences from the regime discussed in the subsequent Chapter 4.Chapter 4 demonstrates a method to measure the conductance enhance-ment from the Kondo effect with weaker coupling. Various dot configura-tions are tested, with results compared to the Numerical RenormalisationGroup (NRG) theory.Chapter 5 consolidates the main findings from Chapter 4, and outlinesfuture directions for this research.4Chapter 2Device Background“If I have seen further it is by standing on the shoulders of Giants.”Isaac Newton [29]2.1 IntroductionBefore describing the measurements that are the focus of this thesis, it is im-portant to first introduce the devices used to engineer interesting states andthe tools used to measure them. As an overview, the chip is a GaAs/AlGaAsheterostructure; the precise layering of the semiconductors form a two-dimensional electron gas (2DEG) around 50 -100 nm below the surface of theheterostructure. Voltages applied to patterned metallic gates on top of theheterostructure tune the potential landscape in the 2DEG, controlling wherethe electrons can go. This tunability allows for forming quantum structuressuch as quantum point contacts (QPCs) and quantum dots. Ohmic contactsare used to contact the 2DEG to measure transport through such structures.2.2 Two Dimensional Electron Gas (2DEG)The quantum point contacts (QPCs) and quantum dots described later inthis thesis are engineered in a two-dimensional electron gas (2DEG). A2DEG can be thought of as a two-dimensional plane of electrons. The elec-trons can move freely along the x and y directions but are tightly confinedin the z-direction. The 2DEGs in this thesis are realised in a GaAs/AlGaAsheterostructure illustrated in Fig. 2.1. A semiconductor heterostructure52.2. Two Dimensional Electron Gas (2DEG)EnergyEF2DEG14.3 nm n − Al0.36Ga0.64As60.0 nm Al0.36Ga0.64As12.0 nm Al0.36Ga0.64As500 nm GaAs2mm Superlattice Buffer5.0 nm GaAs20.0 nm Al2O3363 nm Ohmic12 nm Au GateFigure 2.1: Illustration of a GaAs/AlGaAs heterostructure. At the bound-ary between GaAs and AlGaAs, a two-dimensional electron gas (2DEG) isformed. A 2DEG can be thought of as a two-dimensional plane of electronsthat can move freely along the x and y directions but are tightly confined inthe z-direction. Ohmic contacts are made by annealing Ni-Au-Ge; the metaldiffuses through the heterostructure, making contact with the 2DEG. Ohmiccontacts are used to measure conductance through the 2DEG. Voltages areapplied to gates to control the potential in the 2DEG, forming quantumpoint contacts (QPCs) or quantum dots. A negative (positive) voltage onthe gate repels (attracts) the electrons.refers to a material system composed of two or more semiconductor mate-rials with different bandgaps or lattice constants that are layered together.These layers are typically grown on top of each other using techniques suchas molecular beam epitaxy (MBE). The heterostructures in this thesis aregrown by Michael Manfra’s lab at Purdue University [30].Two characteristics of this heterostructure give rise to a 2DEG. Firstly,AlxGa1−xAs has a tunable bandgap ranging from 1.42 - 2.16 eV [31] whilstGaAs has a bandgap of 1.42 eV. Secondly, an n-type dopant layer is sand-wiched between the AlxGa1−xAs layers [32]. The smaller bandgap of GaAsallows the electrons from the donor atoms in the dopant layer to drop intothe GaAs conduction band, resulting in a triangular potential well (Fig. 2.1).62.3. Quantum Point Contact (QPC)This triangular well contains multiple energy levels. The Fermi energy liesbetween the first and second energy level, so that only the first energy levelis occupied. As the second energy level is ∼150meV above the first, it willremain unoccupied as this energy gap is much greater than measurementtemperatures 500mK ≈ 43 µeV and source-drain bias 100 µeV. Hence, theelectron gas is considered two dimensional [33].Efforts are made to keep the heterostructure and 2DEG clean and defect-free. A 7 nm cap of GaAs is placed on top of the heterostructure to pre-vent oxidation. Also, a small lattice mismatch between the GaAs and Al-GaAs [34] layers keeps the number of boundary defects in the 2DEG planelow. The 30 nm AlGaAs buffer layer between the 2DEG and dopants helpsprevent defects near the 2DEG plane. The resulting 2DEG has a mobilityof µe = 2.56× 106 cm2/Vs and electron density n = 2.42× 1011 cm−2Fabrication details on these devices are described in Appendix A. Asan overview, the heterostructure is divided into separate areas with isolated2DEGs by etching away the top layers of the heterostructure, removingthe 2DEG underneath. Contact to the 2DEG is made with ohmic contacts.These are made by annealing a layer of Ni-Au-Ge; the metal diffuses throughthe heterostructure and forms an electrical connection to the 2DEG. Ontop of that, an insulating dielectric of 10 nm Al2O3 is deposited across thesurface of the heterostructure to limit leakage from the metallic gates thatare added afterwards [35]. A thin layer (10 nm) of Au is deposited to formthe ‘inner gates’. The structure and shape of the inner gates have beencarefully designed so that QPCs and quantum dots can be formed in the2DEG. In the second step, a thick layer (100 nm) of Au is deposited (‘outergates’) to connect the inner gates to square bond pads. Wire bonds arethen made from a chip carrier to the square bond pads to connect the fridgewiring and quantum device.2.3 Quantum Point Contact (QPC)In this thesis, a quantum point contact (QPC) is a one-dimensional channelconnected to a source and drain lead (Fig. 2.2). A one-dimensional channel72.3. Quantum Point Contact (QPC)c)a) b)AG2G11086420Conductance (2e2/ h)-1600 -1200 -800 -400VG1 (mV)VG2−700mV−1400mVFigure 2.2: (a) Graphic representation from a top-down view of a quan-tum point contact (QPC). The gold fingers are the metallic gates to whichnegative voltages can be applied. The grey is where the electrons in the2DEG can go, depending on the negative voltage applied to the gates. Thecrossed squares are ohmic contacts to the 2DEG. These contacts allow a biasacross the QPC to be applied, so a conductance through the QPC can bemeasured. (b) Simulation of the electric potential the electrons see in the2DEG due to the negative voltage applied to the gates [36]. With sufficientnegative voltages applied to the gates, the electrons cannot overcome thepotential barrier under the gate and flow between the gates. (c) Measure-ment shows the quantised conductance through a QPC as the voltage onG1 becomes more negative. At sufficiently negative voltages, the potentialbarrier between the gates is large enough that a measurement will not recordany tunnelling events, and the conductance is zero. The pinch-off is shiftedfrom a more negative (green) to a less negative (yellow) value by increasingthe negative voltage applied to G2.82.3. Quantum Point Contact (QPC)in the 2DEG can be engineered by applying sufficient negative voltage ontwo metal gates on the surface of the heterostructure with a gap in between.Using the ohmic contacts, a potential bias is applied across the QPC, and acurrent will flow. The more negative the voltage on the gate, the higher thepotential barrier is for the electrons in the 2DEG. A large enough potentialbarrier can stop the electrons flowing underneath the gate (called ‘deple-tion’). However, due to the gap between the gates, the electrons can stillflow from one side of the QPC to the other. Figure 2.2b shows a simulationof the electric potential in the 2DEGs where the gates are depleted, butelectrons can still tunnel between [36]. Further details on this simulationare described in Appendix B. It usually requires a more negative voltage tostop electrons flowing between the gates (called ‘pinch off’). A QPC lengthis the length of the one-dimensional channel, and the width is the spacebetween gates. In the devices used in this thesis, a QPC can have lengths50 - 350 nm and widths 100 - 350 nm, depending on its usage.The electrons are free to move in the y direction, but if the width of theQPC is comparable to the Fermi wavelength, the allowed energy levels arequantised in the x direction. The confining potential in the x direction can bemodelled as a parabolic potential as seen in Fig. 2.2b. The allowed 1D energylevels resemble solutions to the harmonic oscillator. Without a magneticfield, each occupied energy level contributes 2e2/h to the conductance. Thefactor of 2 comes from the spin degeneracy of the electrons, which can belifted with a magnetic field. When measuring the conductance through aQPC as it is pinched off (Fig. 2.2c), plateaus at integer values of 2e2/h [2] area signature of the quantised conductance. As the temperature is increased,the plateaus become smeared out. A shorter length of QPC also leads tomore narrow plateaus.In our devices, QPCs are utilised to form tunable tunnel barriers be-tween a quantum dot and a lead (described in the next section) and chargesensors [37], which measure the charge around a quantum dot.92.4. Quantum Dot2.4 Quantum Dota)200nmAVP VDEFΓL ΓRUϵ0U + Δb)VLCVNVCSSΓLΓRFigure 2.3: (a) SEM image of the gates used to define a quantum dot.The gold-coloured gates indicate sufficient negative potential is applied suchthat the 2DEG below is depleted. The gates VP or VD have the largest ef-fect on dot energy ϵ0 without changing other energies. The crossed squaresare ohmic contacts which contact the 2DEG. A small bias applied to theohmic contacts allows conductance through the quantum dot to be mea-sured. (b) Coulomb blockade energy diagram showing discrete energy levelsin the quantum dot and continuous energy levels in the leads. The greyboxes represent continuous energy levels of the electrons in the leads. Thewhite rectangles represent tunnel barriers into and out of the quantum dot.The tunnelling rate is denoted by the parameter Γ; the wider (narrower) thebarrier, the smaller (larger) the tunnelling rate. The orange circle representsthe first electron in the quantum dot with spin down. The second electron(purple) will pair with the first spin-down electron, so the energy requiredto add the second electron is only the charging energy U. To add the thirdelectron (not currently in the quantum dot), the charging energy U, plusthe orbital level spacing energy ∆, is required.In this thesis, a quantum dot is a zero-dimensional structure. It can beformed by connecting a potential well to source and drain leads through atunnel barrier [38]. Like a QPC, the potential well is formed by applyingnegative voltages to gates to confine a small region in the 2DEG. Figure 2.3a102.4. Quantum Dotis an example of the gate geometry used for a quantum dot. The tunnelbarriers that connect this isolated region in the 2DEG to source and drainleads are formed by QPCs. The first characteristic of quantum dots is thecharging energy, U (Fig. 2.3b). This is the energy required to add or removean electron from the quantum dot due to the Coulomb force. Under certainconditions, the charge in the quantum dot is quantised and equal to Ne,where N is the total number of electrons and e is the charge of a singleelectron. The two conditions for quantised charge are Rt >> h/e2 and U >>kBT. The first condition is the resistance of the tunnel barriers Rt, shouldbe greater than the resistance quantum, h/e2 = 25.813 kΩ. Qualitatively,the tunnel barriers should be large enough so the electron is located inthe source or drain leads, or quantum dot. The second condition is thatthe charging energy, U, should be larger than the system’s thermal energy.These conditions lead to a quantised charge in the quantum dot. However,the quantised charge is not a unique feature of quantum dots and is seen insmall metallic islands such as Sn particles [39]. The second characteristicof a quantum dot is if the size of the dot is comparable to the de Brogliewavelength, additional contributions to the energy level spacing arise fromthe orbital level spacing, ∆ [38], [40] (Fig. 2.3b). Contributions from orbitallevel spacing are important when ∆ >> kBT. Here the energy level spacingis greater than the thermal energy of the system. The simplest model,which combines charging energy and orbital level spacing, is the constant-interaction model [41].2.4.1 Conductance Through a Quantum DotA quantum dot is formed by applying negative voltages to gates which con-fine a small region in the 2DEG (Fig. 2.3a). Careful gate design is used sothat each gate has a primary function, e.g., varying the tunnel barrier orpotential well depth. In Fig. 2.3a, VLC is the ‘left coupling’ gate and has thestrongest effect on the left tunnel barrier along with VN. VN is the ‘nose’gate and affects both couplings equally. It also has a large effect on thedot energy and can push the location of the electron wave function closer to112.4. Quantum DotEF EF EF1) 2) 3)0.30.20.10.0Conductance ( 2e2/ h)40353025VD (mV)Figure 2.4: Measured conductance through a quantum dot (green trace)as a single electron is added. The corresponding Coulomb blockade energydiagrams show how the energy level of the quantum dot varies. In 1), thereis a single electron in the quantum dot (orange dot) and the next availableenergy level (above the Fermi energy of the source and drain leads) is empty.Here the conductance is zero as no electron in the leads has sufficient energyto tunnel into the quantum dot and off again. In 2), VD is more positive,lowering the dot energy, ϵ0. The dot energy now aligns with the Fermi energyof the source and drain leads. Resonant tunnelling through the quantum dotgives a maximum in measured conductance. On average, there is a fractionof the electron charge in the quantum dot (partial purple dot). In 3), theVD is made even more positive, and the dot energy ϵ0 is below the thermalbroadening of the electrons in the source and drain leads. The quantum dotis fully occupied (purple dot), and conductance drops to zero.VCSS. VCSS is the ‘charge sensor spine’. The charge sensor is discussed inthe next section, but the proximity of the electron’s wavefunction to VCSS isimportant for the higher sensitivity of the charge sensor. VP is the ‘plunger’gate and primarily controls the dot energy ϵ0. By making this gate less122.4. Quantum Dotnegative, the dot energy decreases or the size of the potential well increases,and at some point, another electron can enter the quantum dot (Fig. 2.4).VD is the ‘dot’ gate. This gate is operated differently from the other gatesas it lies directly above where the electrons in the quantum dot live. Thisgate is most strongly coupled to the dot energy and is used to add or removesingle electrons into the quantum dot with fine control. This gate is usuallyat positive voltage values to help form a nicely shaped potential well, asnegative voltages will form a doughnut-shaped potential.2.4.2 Charge Sensing a Quantum DotA charge sensor is a way to measure changes in the charge in the quantumdot. If correctly tuned, it is very sensitive to the additional charge of anelectron that enters a quantum dot [42]–[44]. It is formed by adding a QPCnear the quantum dot. The charge sensor is formed using the gate VCSQ, inFig. 2.5a. Changing the voltage applied to VCSQ allows the QPC to be tunedso that the current is on a steep slope or plateau (Fig. 2.2c). When used asa charge sensor, the QPC is adjusted to be on a steep slope. In this setting,small changes in nearby potentials (x-axis of a QPC trace) show up as largechanges in the current measured through the QPC [45]. In Fig. 2.5b, theQPC is set up on a steep slope, and VD is swept from negative to positive,lowering the dot energy ϵ0 past the Fermi energy of the leads, so that anelectron tunnels into the dot. The background increase in current is due tothe capacitive coupling of VD. As VD is swept to more positive voltages,the potential in the QPC lowers, and the measured current increases. Oncean electron can tunnel into the quantum dot, the extra negative charge ofthe electron shows up as a small but sharp decrease in current through theQPC, which is called a ‘charge transition’. The derivative of the currentcan be used to identify charge transitions in large scans where there is asignificant change in background current. The steep slope of the chargetransition shows up as a large negative spike in the derivative.132.4. Quantum Dot-240-220-200V P (mV)-150 -100 -50 0VD (mV)N = 0 1 2 3 4 5 6 7 8 9b)c)200nmAVP VDa) VLCVNVCSSVCSQΓLΓR4321Current (nA)40200-20-40VD (mV)Figure 2.5: (a) SEM image of the gates used to define a quantum dot.The gold-coloured gates indicate sufficient negative potential is applied sothat the 2DEG below is depleted. The gates VP or VD have the largest ef-fect on dot energy ϵ0 without changing other energies. The crossed squaresare ohmic contacts which contact the 2DEG. VCSQ forms a QPC near thequantum dot. By setting up the QPC on a steep slope between conductanceplateaus, the current through the QPC is sensitive to nearby changes incharge. (b) Measured current through the charge sensing QPC. As VD be-comes less negative, the current through the QPC increases. The downwardjumps in current result from the negative repulsion from electrons enteringthe quantum dot (called charge transitions). The inset (yellow) shows azoomed-in scan over one of these transitions. (c) 2D scan shows how thequantum dot’s occupation changes as a function of VP and VD. The data isthe differentiated current through the charge sensor. The differential of thesteep slopes in a charge transition shows up as a sharp peak. This is usefulfor locating charge transitions when there is a large change in backgroundcurrent.Figure 2.5c demonstrates how the charge sensor is used to count howmany electrons are in the quantum dot. By making VD more negativeand ensuring the tunnel barriers are small enough for electrons to enter thequantum dot, at some point, there will be a final charge transition signallingthe last electron. A 2D sweep with different gates on either axis reveals therelative cross-capacitance between each gate on the dot energy ϵ0. From142.5. Determining Electron TemperatureFig. 2.5c, a −50mV change on VD removes ∼ 4 electrons, but −50mVchange on VP only removes ∼ 2 electrons.2.5 Determining Electron TemperatureCharge transitions can be used to count the number of electrons in a quan-tum dot [46] or determine the relative cross-capacitance between differentgates on the dot energy.b)a) 0.2320.2300.2280.2260.2240.2220.2200.218Current (nA)-2 -1 0 1 2VP (mV)20mK100mK200mK400mK120100806040200Θ (mV)5004003002001000MC Temperature (mK) DataLinear Fit T > 100mKFigure 2.6: (a) Weakly coupled (thermally broadened) charge transitionis measured at different fridge temperatures. The broadening of a weaklycoupled charge transition is linearly proportional to its temperature. (b)Charge transition’s calculated broadening (Θ) at different fridge tempera-tures. At fridge temperatures >100mK, the increase in the charge transitionbroadening is linear with temperature. This indicates the electrons are wellthermalised to the fridge. A linear fit to fridge temperatures >100mK isextrapolated to base temperature to calculate the real temperature of theelectrons given the measured broadening. Electron temperature is deter-mined to be 20mKHowever, Fig. 2.6 demonstrates how they can also be used to measurethe temperature of electrons in the 2DEG. The broadening of the chargetransition depends on the temperature of the system and the size of thetunnel barriers (strength of coupling Γ). When the coupling strength is less152.5. Determining Electron Temperaturethan the thermal energy of the system (Γ/kBT << 1), the broadening ofthe charge transition depends only on the temperature of the system. Thecharge transition is described as thermally broadened or weakly coupled inthis regime. The current lineshape of a charge transition in this regime is,ICS = Iamp tanh(VP −Vmid2Θ)+ γVP + Iconst (2.1)here, ICS is the current through the charge sensor. Iamp is the amplitude ofthe charge transition, VP is the voltage on the sweep gate used to sweep overthe charge transition, Vmid is the midpoint of the charge transition. Θ =kBTαeis the thermal broadening in units of gate voltage, α ≡ dϵ0dVP is a lever armthat relates changes in a gate voltage to changes in the dot energy ϵ0. γ isthe cross-capacitance between the sweep gate and the current through thecharge sensor. Iconst is a constant current offset.The broadening of a weakly coupled charge transition can be used todetermine the temperature of the electrons in the 2DEG [47]. Figure 2.6ashows the broadening of charge transitions at a range of temperatures. Thisbroadening is linearly proportional to the temperature (Θ ∝ T) of the elec-trons. At high fridge temperatures, the electrons in the 2DEG are in ther-mal equilibrium with the fridge. But as the fridge reaches base temperature(8mK), the electrons in the 2DEG will not be in equilibrium due to poorthermal contact and sources of electrical noise. Also, a source-drain biasacross the quantum dot larger than the temperature of electrons can artifi-cially broaden the charge transition (i.e. 1 µeV = 11.6mK). Additionally,charge motion in the dopant layer shifts the dot energy up or down, re-sulting in the charge transition moving left or right in a scan, this leads toartificial broadening. Charge motion can also primarily affect the currentthrough the charge sensor, rather than the dot energy. This charge motionleads to a sudden increase or decrease in current through the charge sensor(‘charge jump’). This type of charge motion is only problematic when thejump occurs during a charge transition.To determine the proportionality constant between the broadening andtemperature, charge transitions are measured at a range of fridge tempera-162.6. Measurement Procedurestures from 8 - 500mK. Each charge transition is measured quickly (∼0.5 s)and repeated many times (∼300). Any traces with charge jumps near thecharge transition are thrown out, and the remaining traces are centred andaveraged. The combination of quick scans across the charge transition andaveraging, reduces additional broadening from charge motion in the dopantlayer. The amount of charge motion in the dopant layer scales between 1/fand 1/f2 [48]. Each charge transition is then fit using Eq. 2.1, and the calcu-lated Θ is plotted versus the fridge temperature. Figure 2.6b shows a linearfit to Θ between 100 - 500mK, which is used to determine the proportional-ity constant. This constant converts the Θ calculated at base temperatureinto the effective temperature of the electrons in the 2DEG. A base electrontemperature of 20mK was determined in this fridge.2.6 Measurement Procedures2.6.1 Cooldown BiasAt room temperature, +200mV is applied to each of the gates forming thequantum dot and charge sensor (apart from VD). The positive bias repels thepositive charge underneath the gates in the dopant layer. Once the devicereaches 10K, the dopants are frozen in, and the electrons see an effectivepotential in the 2DEG due to the absence of a positive charge. Coolingwith bias has two advantages. Firstly, it reduces the charge noise [49].Secondly, it reduces the amount of negative potential required on the gates.The effective potential from the positive bias is roughly equal to a negativevoltage of equal magnitude when the device is cold (i.e. +200mV cooldownbias ≈ −200mV). This is useful as the fine gates can ‘blow up’ (in reality,the metal gates will lift off a little from the heterostructure surface) fromhigh voltages, rendering the device unmeasurable. The fine gates are alsosensitive to static discharge, so 1MΩ of inline resistance is added to all gates.172.6. Measurement Procedures2.6.2 Gate DividerFine gate control at large voltages is often required when measuring chargetransitions. For example, VP may require a voltage of −500mV, but aresolution of 0.001mV. Digital-to-analog converter (DAC) channels are usedto apply voltage on the gates. The DACs have a range of ±10V, with16-bit resolution. This puts a lower bound on the step size of 0.305mV(20V/216). To achieve a large voltage range with high resolution, two DACsare connected to the same gate. One DAC channel covers a wide voltagerange, while the other is used for fine control by adding a voltage divider inline. Voltage dividers between 2 and 100 000 are commonly used to achievethe required resolution (up to 3 nV). All of the voltages in this thesis havebeen converted to the real voltage applied to the gate (i.e. not the voltageoutput on the DAC). However, for clarity, only the fine gate may be shownin a figure, even though an additional rough gate applies a large voltage tothe same gate.2.6.3 Virtual GateTemperature broadened (weakly coupled) transitions are straightforward tomeasure and fit. The lineshape is described by an analytic function Eq. 2.1where the broadening depends on a single parameter, simplifying the fittingroutine. In addition, the sharpness of the transition makes visual inspectionof the fit trivial. However, the work in this thesis focuses on the gammabroadened (strongly coupled) regime where Γ/kBT > 1. The charge transi-tions in this regime become very spread out, and it can be tricky to inspectthe fit quality visually.One way to tackle this problem is by using a virtual gate. In general,a virtual gate is a name given to a combination of gates that, when varied,change a specific parameter of the quantum dot whilst keeping other param-eters constant. The main parameters to control are the dot energy ϵ0, leftcoupling ΓL, right coupling ΓR and current through the charge sensor ICS.An example usage would be a virtual gate that controls the left couplingΓL only. A combination of VLC and VP would be used for this virtual gate.182.6. Measurement Procedures1.301.251.201.151.101.05Current with Offset (nA)-10 -5 0 5 10VP +VCSQ (mV)| No Virtual Gate| Virtual GateΓ/kBT < 1Γ/kBT > 1Figure 2.7: Data of a weakly (green) and strongly (yellow) coupled chargetransition scanned using a single gate VP (solid line) and a virtual gateVP + VCSQ (dashed line). For clarity, the traces have been offset in thecurrent. VP has a large effect on the dot energy ϵ0. However, it also changesthe current through the charge sensor ICS due to capacitive coupling. Here,a virtual gate changes the dot energy while keeping the current throughthe charge sensor constant. This helps to visually identify strongly coupledcharge transitions which become very broad. More importantly, the chargesensor stays in the linear regime where changes in the potential are linearlyproportional to changes in the current. Note the x-axis is only showing thevalue of VP. As VP is swept negative to positive, VCSQ is swept positive tonegative.192.6. Measurement ProceduresVLC to primarily changed the left coupling ΓL and VP to offset any changesto the dot energy ϵ0.In the context of this thesis, only a single type of virtual gate is used.One that changes the dot energy ϵ0, but keeps the current through thecharge sensor ICS, constant. Figure 2.7 shows the effect of this virtual gateon a weakly coupled (Γ/kBT < 1) and strongly coupled (Γ/kBT > 1) chargetransition. Only two gates (VP and VCSQ) are used to form this virtual gate.By sweeping VCSQ in the opposite direction of VP, the cross capacitive effectof VP on the charge sensor is largely removed.20Chapter 3Conductance in the KondoRegimeNı´ dhe´anfaidh smaoineamh an treabhadh duit.Irish proverb3.1 IntroductionThis chapter serves as an introduction to the Kondo effect. The history ofthe Kondo effect began with a measurement that showed a strange behaviour(resistivity minimum with decreasing temperature) in impure gold wires [14].It took thirty years before Jun Kondo explained this effect, hence the name‘Kondo effect’ [15]. After another thirty years, advancements in technologyrevealed the quantum dot as an exciting platform to explore the Kondoeffect due to its high degree of in-situ tunability [50].3.2 Kondo Effect in Bulk MaterialsIn the 1930s, it was found that the resistivity would surprisingly increasein impure gold wires at low temperatures [14]. This was unexpected as dueto electron-phonon scattering, the resistivity should decrease with T5 beforesaturating at some non-zero resistivity. Data from the original experimentis plotted in Fig. 3.1 with expected electron-phonon dependence. Over thecoming years, it was found that metals with magnetic impurities had asimilar behaviour [51].213.2. Kondo Effect in Bulk Materialsexpected saturationρphonons ∝ T526.926.826.726.626.5104 ρ(T) /ρ(273)54321Temperature (K)Figure 3.1: Temperature dependence of the resistivity, ρ. In simple metals,the resistivity initially decreases as T 5 due to electron-phonon scattering.However, due to the Kondo effect, the resistivity reaches a minimum beforeincreasing logarithmically with decreasing temperature. The data used forthis figure was obtained from [14].It took until 1964 for a theoretical explanation of the resistivity min-imum [15]. Jun Kondo used the s-d model, which couples a metal (non-magnetic, s-band) to a magnetic impurity (unfilled d-level). By looking atthe second Born approximation, a logarithmic term appeared, which addeda large correction to the resistivity at low T. This contribution comes froma process where spin exchange interactions occur. This work showed that amagnetic impurity at low temperatures can give rise to an alternative scat-tering process, involving a temporary exchange of spin states between themagnetic impurity and surrounding conduction electrons.Figure 3.2 illustrates how an exchange of spin states can build a coher-ent state between the magnetic impurity and the free conduction electrons.It begins with Fig. 3.2a, where conduction electrons surround a magneticimpurity. Figure 3.2b shows a scattering event that may or may not resultin the spin-flip of the conduction electron and the magnetic impurity. Thisleaves the two particles entangled. Continued scattering events between the223.3. Kondo Effect in Quantum Dotsa) b) c)Figure 3.2: (a) A single magnetic impurity (black arrow) is surroundedby conduction electrons (white arrow). (b) Conduction electrons scatteroff the magnetic impurity, resulting in a possible spin flip of both particles.This leaves the particles entangled. (c) Continued scattering events build amacroscopic coherent state known as a ‘Kondo singlet’.magnetic impurity and conduction electrons will result in the macroscopicstate shown in Fig. 3.2c. This is known as a ‘Kondo singlet’ or ‘Kondocloud’. It is found that only a single parameter is needed to describe thelow-temperature properties and formation of the Kondo singlet, the Kondotemperature, TK. Further discussion on the Kondo effect theory is out ofthis thesis’s scope, but excellent summaries are found here [52]–[54].3.3 Kondo Effect in Quantum DotsUntil the late 1990s, theorists primarily explored the Kondo effect due tothe difficulty of experimentally controlling the Kondo state [51], [52]. How-ever, after advances in quantum dot design, the potential for precise in-situcontrol was irresistible, and the first measurement of a Kondo singlet wasin 1998 [50]. Around this time a scanning tunneling microscope (STM) wasalso used to study the Kondo effect [55].In quantum dots, an odd number of electrons results in an unpairedelectron that acts as the magnetic impurity in bulk metals. Like bulk metals,the conduction electrons in the 2DEG can interact with the net spin in thequantum dot. The advantage of studying the Kondo effect in quantum dotscomes from the in-situ control over the coupling strength, voltage bias, and233.3. Kondo Effect in Quantum DotsΓRΓLΓRΓLUϵ0ΓRΓLΓRΓLb)d)a)c)Figure 3.3: Coulomb blockade energy diagram illustrates the formation ofa Kondo singlet in a quantum dot. Here, the dot acts as a localised spin.The dark grey represents the continuous energy level of electrons in theleads. The white rectangles represent tunnel barriers between the quantumdot and leads. The rate of tunnelling is denoted by the parameter Γ; thewider (narrower) the barrier, the smaller (larger) the rate of tunnelling. (a)The quantum dot has a one-spin degenerate energy level with dot energy ϵ0,occupied by a single electron. The charging energy, U, separates the nextenergy level. According to Fig. 2.4, zero conductance is expected throughthe quantum dot. However, (b,c) depict a possible virtual tunnelling event.Here, the spin-up electron tunnels out of the dot, and a spin-down electrontunnels into the dot within a short time. (d) Many virtual tunnelling eventsinvolving possible spin flips lead to a correlated state, the Kondo singlet.This state is pictured as a narrow density of states formed at the Fermienergy of the leads. If a Kondo singlet has been formed, enhanced conduc-tance is measured through the quantum dot, even as the dot energy ϵ0 isbelow the energy level of the leads. Note the formation of the Kondo singletrequires an odd number of electrons in the quantum dot.243.3. Kondo Effect in Quantum Dotsenergy level of the magnetic impurity.The formation of a Kondo singlet in quantum dots is shown in Fig. 3.3.Suppose the dot energy of the quantum dot is far below the Fermi energy ofthe leads. In that case, the electron is not expected to tunnel out of the dotas it does not have the required energy to reach the available empty energylevels in the leads (Fig. 2.4). However, a second-order tunnelling process canoccur resulting in a non-zero conductance, even as the dot energy is belowthe Fermi energy of the leads. If the electron in the quantum dot tunnelsinto the leads (Fig. 3.3b) another electron must tunnel into the quantum dotwithin a timescale limited by Heisenberg’s uncertainty principle (Fig. 3.3c).This second-order process can result in a spin flip of the impurity. Many ofthese virtual tunnelling events will lead to the formation of a singlet stateillustrated in Fig. 3.3d. This results in a narrow density of states formed atthe Fermi energy of the leads.Unlike the resistivity increase in bulk metals, the presence of a Kondosinglet leads to enhanced conductance through the quantum dot. Figure 3.4cshows a measurement of conductance at a range of temperatures. At odd oc-cupation, a Kondo singlet is formed and conductance through the quantumdot increases with decreasing temperature. This enhancement of conduc-tance is dependent on a single parameter which sets the energy scale, theKondo temperature TK. The Kondo temperature is given by Haldane [56],[57] as,TK =√ℏΓU2kBeπϵ0(ϵ0+U)/ℏΓU (3.1)Here ϵ0 is the dot energy, U the charging energy and Γ is the coupling of thedot to both source and drain leads, Γ = ΓL + ΓR. An important realisationfrom Eq. 3.1, is that the Kondo temperature is not a constant value butis dependent on parameters that can be controlled in an experiment. InFig. 3.4a, the dot energy is controlled by VP or VD and the coupling byVLC, VN or VCSS. The expression for the Kondo temperature in Eq. 3.1,only holds in the ‘Kondo regime’. The Kondo regime is characterised byϵ˜0 << −0.5 where ϵ˜ ≡ ϵ0/Γ. Qualitatively, this regime is satisfied when253.3. Kondo Effect in Quantum DotsΓRΓLa) b)c)VP VDΓLΓRVLCVNVCSSN=odd N=even0.60.50.40.30.20.10.0Conductance ( 2e2/ h)25201510VD (mV) 42mK 50mK 100mK 200mK 300mKFigure 3.4: (a) SEM image of the gates used to define a quantum dot.The dot contains an odd number of electrons, and the tunnel barriers aretuned to form a Kondo singlet. (b) Coulomb blockade energy diagrampicture of a Kondo singlet. As the dot energy falls below the energy levelof the leads, there is enhanced conductance due to virtual tunnelling eventsthrough the quantum dot. (c) Data showing the temperature dependence ofconductance through a quantum dot when a Kondo singlet is formed. In theeven occupied sides, conductance decreases as the temperature is lowered.With odd occupation, a Kondo singlet forms, and conductance increaseswith decreasing temperature.the full charge of the electron is in the quantum dot. Consequently, asthe dot energy is raised so that only a fraction of the electron charge islocalised in the quantum dot, this expression for the Kondo temperature263.3. Kondo Effect in Quantum Dotsbreaks down. However, the quantum dot still mainly exhibits a net spin,and Kondo enhancement remains. Importantly, the Kondo temperatureincreases as the dot energy approaches the Fermi energy of the leads [58],although it may not be exponential in form. This ‘turning on’ of the Kondoeffect close to the Fermi energy of the leads is the basis for measurementsin the following Chapter 4.Many early measurements on the Kondo effect in quantum dots used verystrong couplings, so the system temperature remained below the Kondo tem-perature deep into the Kondo regime [17]. Such studies examined tempera-ture dependence of the conductance in the Coulomb blockade valley wherethe conductance dependence on temperature is non-monotonic and has aminimum at finite temperatures [59]. The Kondo temperature (Eq. 3.1)sets a new many-body scale in the Kondo regime. The Kondo temperaturecan be used to normalise the conductance such that it is independent ofother energy scales Γ, U, and ϵ˜0 [16]. This normalisation is given as,G(T) = G0(T′2KT2 +T′2K)s(3.2)Here T′2K = TK/√21/s − 1 and s ≈ 0.20 for a spin 1/2 system in the Kondoregime [58]. The parameter s determines the steepness of the conductancedrop with increasing temperature. The one-parameter scaling of the con-ductance in the Kondo regime is used as one of the demonstrations of theKondo effect. Other measurements include a zero bias peak in the Coulombblockade valley [17] and a splitting of this zero bias peak with magneticfield [60].Other efforts have been made to explore the Kondo effect with less strongcoupling. In this regime, the conductance in the middle of the Coulombblockade valley does not increase with decreasing temperature [58]. How-ever, the dot energy of the quantum dot can be tuned to increase the Kondotemperature. As the dot energy approaches the Fermi energy of the leads,a conductance enhancement due to the Kondo effect can be recovered atsufficiently low system temperatures. The next Chapter 4 will focus on thisregime of weaker coupling.27Chapter 4Occupation ResolvedConductance of the KondoEffectPo bitveˇ je kazˇdy´ genera´l.Czech proverbTo my knowledge, the following work in this chapter is the first time thatconductance and charge transitions have been simultaneously measured toresolve the conductance enhancement due to the Kondo effect as a func-tion of the quantum dot occupation in an experiment. The conductanceenhancement from Kondo is most pronounced when the quantum dot isstrongly coupled (Γ/kBT >> 1) to the source and drain leads. Previousmeasurements of the Kondo effect focus on the conductance enhancementbetween Coulomb peaks. However, when the coupling is strong, chargetransitions broaden and become challenging to convert into the occupation.The following work operates within the narrow spectrum bounded by strongcoupling, where the Kondo effect is observable, and weak coupling, wherecharge transitions are easily measured.4.1 IntroductionPrevious measurements of the Kondo effect in quantum dots study the tem-perature dependence of conductance between Coulomb peaks. It is in thisso-called Kondo regime that the full charge of the electron is in the dot,284.1. Introductionso there is a definite net spin, and the theory is well understood [16], [17].In this regime, the Kondo temperature (Eq. 3.1) sets a new, many-bodyscale and is used to universally scale the conductance (Eq. 3.2). This one-parameter scaling of conductance in the Kondo regime is used as one ofthe demonstrations of the Kondo effect. When the coupling between thequantum dot and source and drain leads is reduced, the Kondo temperaturedrops below the system temperature, and the conductance enhancementis not seen between Coulomb peaks. Only a handful of experiments havestudied conductance with less strong coupling [58]. Such studies observedthe parameter s from Eq. 3.2 and found that in the Kondo regime (the fullcharge of the electron is in the dot), s was constant and equal to 0.20. How-ever, as the shoulder of a Coulomb peak was approached (ϵ˜0 > −0.5), svaried rapidly. Qualitatively, this regime is entered when the dot energy ϵ0approaches the Fermi energy of the leads, and only a fraction of the electroncharge is in the dot.Our aim was to measure the Kondo effect with a coupling strength suchthat the conductance between Coulomb peaks was zero. This regime, char-acterised by a relatively small Γ, results in a small Kondo temperature be-tween Coulomb peaks where no conductance enhancement would be mea-sured. However, the Kondo temperature will increase as the dot energyapproaches the Fermi energy of the leads (moves towards the shoulder of aCoulomb peak). Hence, for relatively small coupling strengths, the Kondoeffect can still be seen as a very small conductance enhancement on the oddoccupied shoulder of a Coulomb peak. As the conductance enhancement inthis regime is very small, it is insufficient to investigate the Kondo effectwith a conductance measurement only, for two reasons. Firstly, entropyshifts the occupation of the quantum dot with temperature. This effecthas been taken advantage of in a recently developed entropy measurementtechnique [6], [28], [61]. This shift in occupation can be ignored when thecoupling is strong, and conductance enhancement between Coulomb peaksis large. Secondly, charge motion in the dopant layer can shift the conduc-tance left or right with respect to the gate voltage in a scan. The shiftingfrom charge motion renders it impossible to offset the effects of entropy by294.2. Occupation Resolved Conductance Methodretroactively shifting conductance. For these two reasons, it is necessaryto simultaneously measure a second ‘reference’ signal alongside the conduc-tance. This reference signal can then be used to offset the above effects andallow for comparison of the conductance across multiple temperatures.We measure the charge in the quantum dot simultaneously with conduc-tance. Assuming a linear relation between the measured change in chargeand the real occupation of the quantum dot, the charge transition is con-verted into an occupation. The occupation is then used to plot the conduc-tance as a function of occupation. A shifting of the conductance maximumto a higher occupation is a signature of the Kondo effect. As the universalscaling of conductance does not hold in this regime, a comparison to theorycalculations (described in an upcoming section) is required to verify Kondoenhancement.4.2 Occupation Resolved Conductance MethodThe conductance and charge of the quantum dot are simultaneously mea-sured to investigate the small enhancement of conductance due to the Kondoeffect with relatively small coupling. On the hardware side, a simultaneousmeasurement of the charge and conductance is trivial as two different cur-rent amplifiers are simply connected to the device. The first current am-plifier measures the conductance through the quantum dot and the other,current through the charge sensor. Each current amplifier is connected totwo analog-to-digital converters (ADCs), which sample data points simul-taneously. Conductance is measured using a pseudo-lockin technique. Herea DAC channel voltage biases the current amplifier using a ±1 µV squarewave. The underlying signal is recovered in analysis by multiplying with asine wave of equal frequency and averaging all the points per square wavecycle. This technique mitigates the effects of bias drift in the current am-plifier so that a reliable 1 µV source-drain bias is applied across the dot. Aconstant 100 µV DC bias is applied across the charge sensor. Large source-drain bias across the charge sensor can de-phase the Kondo singlet [62],however, a range of 50-250 µV source-drain biases across the charge sensor304.2. Occupation Resolved Conductance Methodwere tested with no significant effect.It is much trickier to tune the quantum dot into a regime where boththe conductance and charge transition can be compared to theory. Whenweakly coupled (Γ/kBT << 1), the charge transitions have a sharp drop incurrent and can be fit analytically using Eq. 2.1. However, a weakly coupledCoulomb peaks amplitude drops with the strength of coupling [40] as,G0 ∝ ΓL · ΓRΓL + ΓR(4.1)Here G0 is the conductance maximum. This decrease in conductance ampli-tude means that the conductance signal can be lost within the backgroundnoise. Additionally, as the Kondo temperature decreases with reduced cou-pling, the conductance enhancement due to Kondo can become immeasur-able.In a strongly coupled regime (Γ/kBT >> 1), the line shape for a Coulombpeak is given by the Breit-Wigner formula [63]. The conductance is maxi-mum when the dot energy is in line with the Fermi energy of the leads anddepends on the coupling as,G0 ∝ ΓL · ΓR(ΓL + ΓR)2 (4.2)In both weakly and strongly coupled regimes, it is evident that the conduc-tance reaches a maximum when the tunnel barriers are symmetric (ΓL =ΓR). Strong coupling also has the added benefit of an increase in Kondotemperature, leading to greater enhanced conductance. However, chargetransitions become very broadened and difficult to measure (Fig. 2.7). Whenstrongly coupled, large sweeps in gate voltage are required to cover a rangewhere the quantum dot is fully unoccupied to occupied. This can result inthe charge sensor being pushed into a non-linear regime due to the cross-capacitive coupling from the sweep gate. A non-linear relationship betweenthe current through the charge sensor and the addition of charge into thequantum dot makes extraction of dot occupation from the charge transitionunreliable. Hence, a virtual gate shown in Fig. 2.7, that keeps the current314.2. Occupation Resolved Conductance Methodthrough the charge sensor constant is required. The exact ratio of gatesused to form a virtual gate can drastically change the underlying shape ofthe charge transition, which will be discussed in an upcoming section.4.2.1 Numerical Renormalisation Group (NRG)The dot is carefully tuned to a coupling regime where the conductance en-hancement due to Kondo is expected, and charge transitions are reliablymeasured. As previous tests of the Kondo effect are not possible in thisregime, a comparison to Numerical Renormalisation Group (NRG) calcula-tions [64] is required. Our theory collaborators (Yigal Meir, Yaakov Klee-orin, and Andrew Mitchell) provide these calculations. Two 2D datasetscorresponding to conductance and occupation are received. The columnsare energy scaled by Γ (energy/Γ), so the x-axis is unitless. The rows inthe 2D datasets correspond to a different Γ/kBT value. We confirmed withYaakov that the lineshape does not change if both T and Γ increase together.The lineshape represents the ratio Γ/kBT, not Γ or T individually.To compare NRG to data, Γ/kBT should be reliably determined so thecorrect row from NRG is used. Then, the correct scaling is applied to convertthe NRG into measurement units. Conductance and charge transitions havesimilar scaling parameters. Amplitude, x-offset and lever arm which scalesthe NRG x-axis in units of mV. The occupation NRG has extra parame-ters: a y-offset which comes from the current through the charge sensor,a linear term which comes from the cross-capacitance between the sweepgate and charge sensor, and an occupation-dependent linear term. Thisoccupation-dependent linear term reflects a change in the cross-capacitanceas an electron enters the quantum dot. In Fig. 4.5, the charge transitionslopes are different between the unoccupied (left) and occupied (right) sides.We argue that the additional charge of an electron in the quantum dot cangive rise to such a change in cross-capacitance. Many of these parametershave low cross-correlation when fitting data to NRG, meaning the globalminimum of the minimiser is reliably reached. However, this is not true forthe parameters Γ/kBT and lever arm. Changes to Γ/kBT can be offset by324.2. Occupation Resolved Conductance Methoda change in lever arm, and fitting a single trace allowing both parameters(Γ/kBT and lever arm) to vary freely is unreliable.4.2.2 Conductance Global Fit to NRGIn the temperature broadened regime (Γ/kBT < 1), Γ/kBT and lever armare decoupled as the broadening of the conductance or charge transition isfrom temperature only. To access the temperature broadened regime fromthe gamma broadened regime (Γ/kBT > 1), the temperature of the fridge isincreased until Γ/kBT ≲ 1. Figure 4.1 shows data that is taken at multipletemperature setpoints across this range, so that Γ/kBT and lever arm can bereliably determined. Conductance and charge transitions are simultaneouslymeasured at each temperature. However, only the conductance data is usedto determine Γ/kBT and lever arm, as it is generally cleaner. A global fitto the conductance data, including each temperature setpoint, is used inFig. 4.1a. Here Γ and lever arm are allowed to vary, but held fixed betweentemperatures, and T is held fixed to the calculated electron temperaturesfrom Fig. 2.6. The other parameters (amplitude and x-offset) used to fit theNRG to conductance data are allowed to vary freely. The fitting range isthe full width at 90% the maximum conductance. This removes any bias ofpicking a ‘good’ fitting range.4.2.3 Charge Transition Fit to NRGEach charge transition is fit separately to NRG, where Γ/kBT and leverarm determined from the conductance global fit, are held fixed. All otherparameters (amplitude, x-offset, y-offset, linear and occupation-dependentlinear) are allowed to vary freely (Fig. 4.1b). The fitting range of the chargetransitions is maximised up to, but excluding charge jumps.334.2. Occupation Resolved Conductance Methoda)b)0.50.40.30.20.10.0Conductance (2e2/ h)-8 -6 -4 -2 0 2 4 6 8VP +VCSQ (mV)20mK90mK275mK400mK0.450.400.350.30Current with offset (nA)-20 -10 0 10 20VP +VCSQ (mV)20mK90mK275mK400mKFigure 4.1: (a) Conductance as a single electron enters a strongly coupled(Γ/kBT > 1) quantum dot at four different temperatures. The x-locationof the conductance maximum has not been shifted and is the original x-location of the measured data. The fits (in grey) are from a global fit toNRG, where the Γ and lever arm parameters are held fixed across all fourtemperatures. (b) Charge transitions are measured simultaneously with theconductance. Each charge transition is separately fit to NRG, where the Γand lever arm parameters are held fixed to the values determined from theglobal fit to conductance.4.2.4 Determining OccupationThe charge transition cannot be used as a reference to compare with con-ductance due to changing amplitude and linear terms with changes in dot344.2. Occupation Resolved Conductance Methoda)b)1.41.21.00.80.60.40.2Occupation with offset (1e)-20 -10 0 10 20VP +VCSQ (mV)20mK90mK275mK400mK0.50.40.30.20.10.0Conductance (2e2/ h)1.00.80.60.40.20.0Occupation (1e)20mK90mK275mK400mKFigure 4.2: (a) Charge transitions are converted to occupation by remov-ing the relevant fit parameters. These are amplitude, current offset, cross-capacitance of virtual gate, and occupation-dependent cross-capacitance.(b) Conductance versus occupation is used to show the enhanced conduc-tance due to Kondo. As temperature decreases, the conductance maximumoccurs at greater occupation. The NRG (grey) conductance versus occupa-tion corresponding to the determined Γ/kBT is plotted on top of the data,where good agreement is found at each temperature.settings. However, charge transitions can be converted into an occupation,allowing for the comparison of conductance enhancement between dot set-tings. The fit parameters from the NRG fit to the charge transitions are usedto remove relevant terms. The current offset (y-offset) and linear term are354.3. Kondo Effect with Varying Coupling Strengthremoved trivially. The occupation-dependent linear term is removed by mul-tiplying this linear term with the correct NRG occupation row correspondingto Γ/kBT. Lastly, the charge transition is re-scaled by the amplitude. NRGonly has to be shifted by the x-offset and scaled by the lever arm to make acomparison with data (Fig. 4.2a).4.2.5 Conductance versus OccupationPlotting conductance versus occupation overcomes the issue of conductanceshifting due to entropy or charge motion. Figure 4.2b shows the conductancedata plotted against the determined occupation. Conductance enhancementdue to Kondo can be seen from a shift of the conductance maximum to occu-pation greater than 0.5 (N > 0.5). This shift in the conductance maximumindicates that even as the dot energy of the quantum dot falls below theenergy level of the leads, there is enhanced conductance due to the forma-tion of a Kondo singlet. As the temperature is lowered in Fig. 4.2b, theconductance maximum shifts to higher occupation. This is because the sys-tem temperature falls below the Kondo temperature, and the Kondo singletremains formed. As the conductance and occupation data are fit to differentranges, it is essential to ensure that the conductance and occupation datapoints plotted against each other correspond to the same gate voltage. TheNRG conductance versus occupation in Fig. 4.2b, is the unscaled NRG pro-vided by the theorists corresponding to Γ/kBT. There is good agreementbetween the data and NRG at each temperature.4.3 Kondo Effect with Varying CouplingStrengthIn Fig. 4.2b, the Kondo enhancement dependence on the system’s temper-ature was confirmed. At high temperatures T > TK, there is no Kondosinglet, and so the conductance maximum is near N = 0.5. But at low tem-peratures T < TK, a Kondo singlet is formed, and the conductance maxi-mum shifts to higher occupation N > 0.5. This displays the dependence of364.3. Kondo Effect with Varying Coupling Strengththe Kondo temperature on the dot energy. The Kondo temperature risesas the dot energy gets close to the leads. However, the Kondo temperaturealso depends on the coupling strength between the quantum dot and leads(Eq. 3.1). As strongly coupled quantum dots have larger Kondo tempera-tures, the conductance maximum is expected to occur at higher occupationthan in weakly coupled quantum dots.0.50.40.30.20.10.0Conductance (2e2/ h)1.00.80.60.40.20.0Occupation (1e)Γ/kBT = 20.5Γ/kBT = 9.7Γ/kBT = 2.5Figure 4.3: Conductance versus occupation in a weak (green) and strong(blue) coupling regime. Each trace is taken at 20mK. The coupling strengthΓ/kBT was determined from a global fit to multiple temperatures. TheNRG (grey) conductance versus occupation corresponding to the determinedΓ/kBT is plotted on top of the data. Good agreement is found at eachcoupling strength.Figure 4.3 shows 20mK traces of conductance versus occupation at threedifferent coupling strengths. Γ/kBT is determined using the fitting routinedescribed previously and is used to plot the corresponding NRG alongsidethe data. Expectedly, the more strongly coupled data shows a greater shiftin the conductance maximum to higher occupation. Good agreement withNRG is observed at each coupling.374.4. Kondo Effect with Varying Charge Sensor Current4.4 Kondo Effect with Varying Charge SensorCurrentThis new method of plotting conductance versus occupation not only re-quires a reliable determination of Γ/kBT and lever arm from conductancedata, but also clean charge transitions that can be converted into an occupa-tion. The shape of the charge transition can vary depending on the choiceof virtual gate and the charge sensor QPC setpoint. In the next section,different current setpoints of the charge sensor QPC are measured and theresulting conductance versus occupation is tested for agreement with NRG.4.4.1 Varying Charge TransitionFigure 4.4 shows a charge sensor trace (green) and the resulting chargetransition (yellow) at a range of current setpoints through the charge sensor.The current (y-axis) has not been scaled differently between the chargesensor trace and charge transitions. However, the sweep gate (x-axis) of thecharge sensor and charge transitions are different. The charge transitionsweep gate axis has been scaled for clarity.The shape of the charge transition varies significantly as the currentsetpoint through the charge sensor changes. Typically, the charge sensor isset to its steepest slope where the derivative is the largest. Here, the chargesensor is most sensitive as small changes in the potential result in largechanges in the current and the resulting amplitude of the charge transitionis greatest. However, charge transitions can be measured at any currentsetpoint through the charge sensor. As long as the change in current throughthe charge sensor is linear with respect to the change in voltage. Thiscan be achieved with a virtual gate, which keeps the current through thecharge sensor constant. Yet, at each charge sensor current setpoint, thecharge transition slopes vary differently on the unoccupied and occupiedsides. Comparing the slopes on the unoccupied (left) side, at low currents,they point downwards, then point upwards with increasing current, and thenpoint downwards again at the highest current setpoint. It is important to384.4. Kondo Effect with Varying Charge Sensor Currentshow that the conduction versus occupation agrees with NRG, irrespectiveof the current setpoint through the charge sensor. Otherwise, it may bepossible to ‘pick’ a current setpoint through the charges sensor that eithershows or does not show agreement with NRG. This would render the methodunreliable.1.00.80.60.40.20.0Current (nA)-195 -190 -185 -180 -175 -170 -165 -160VP +VCSQ (mV) QPC Trace Charge Transi/onFigure 4.4: Current through the charge sensor (green) from pinch-off to thestart of the first conductance plateau. Corresponding charge transition (yel-low) at each current setpoint through the charge sensor. Note the x-axis isdifferent between the underlying charge sensor trace and charge transitions.The charge transitions x-axis has been scaled the same amount for clarity.The charge transitions vary dramatically as the current through the chargesensor is changed. The left and right slopes curve upwards, downwards, inthe same direction, or opposite each other.4.4.2 Conductance versus OccupationFigure 4.5a shows charge transitions taken at 20mK for three different cur-rent setpoints through the charge sensor. For clarity, each charge transitionhas been offset in the current. The shape of the charge transition clearlydepends on the current through the charge sensor. The unoccupied (left)side of the high current (blue) charge transition slopes down, whilst the lowcurrent (green) charge transition slopes upwards. This suggests that the op-394.4. Kondo Effect with Varying Charge Sensor Current1.101.051.000.950.900.85Current with offset (nA)-20 -10 0 10 20VP +VCSQ (mV)Charge sensor current1.6 nA1.0 nA0.4 nA0.80.60.40.20.0Conductance with offset ( 2e2/ h)1.00.80.60.40.20.0Occupation (1e)Charge sensor current1.6 nA1.0 nA0.4 nAa)b)Figure 4.5: (a) Charge transitions are measured with high (blue) andlow (green) current through the charge sensor. For clarity, the transitionsare offset in the current. The x-axis uses the same virtual gate for eachcharge transition. The slopes on either side of the charge transitions varywith the current through the charge sensor, suggesting the best virtual gatealso changes. (b) Conductance versus occupation (Γ/kBT = 21) at differentcurrent setpoints through the charge sensor. The traces are offset for clarity.Each trace is taken at 20mK. The coupling strength Γ/kBT was determinedfrom a global fit to conductance at multiple temperatures. The NRG (grey)conductance versus occupation, corresponding to the determined Γ/kBT,is plotted on top of the data. Good agreement is found at each currentsetpoint.404.5. Kondo Effect with Varying Coupling Symmetrytimal virtual gate to keep the charge sensor current constant also changes.At each current setpoint, conductance was simultaneously measured at arange of temperatures. A global fit to the conductance was used to deter-mine Γ/kBT and lever arm. Each charge transition was converted into anoccupation, and the conductance versus occupation is plotted with a com-parison to NRG (Fig. 4.5b). For clarity, the conductance versus occupationtraces have each been offset by 0.1 2e2/h. Excellent agreement between dataand NRG is found at each current setpoint.4.5 Kondo Effect with Varying CouplingSymmetryAll previous conductance data shown in this thesis was taken with symmetriccoupling (ΓR = ΓL). Previous measurements of the Kondo effect also tunethe quantum dot to be symmetrically coupled with the source and drainleads. However, some experimental studies have investigated the Kondoeffect with asymmetric coupling (ΓR ̸= ΓL) [65]. It was found that the char-acteristic zero bias peak between Coulomb peaks shifted to a nonzero bias.However, this effect would require a strong energy dependence of the tun-nel barriers (i.e., Γ = Γ(E)) in a NRG calculation, which is not assumed inour current NRG calculations. Hence, a measurement of conductance withnonzero bias cannot be currently compared to NRG. Although we may notquantitatively corroborate these previous findings, exploring coupling sym-metry is interesting as it effectively tests how the Kondo enhancement varieswhen a quantum dot is coupled to two leads (symmetric case) versus a singlelead (asymmetric case). In practice, the limit of a single lead regime cannotbe reached, as conductance will not be measured through the quantum dot.4.5.1 Symmetric to Asymmetric CouplingThe width of a strongly coupled Coulomb peak is controlled by Γ. WhereΓ is the sum of the individual tunnelling rates, Γ = ΓL + ΓR. The conduc-tance amplitude in both the weakly (Eq. 4.1) and strongly (Eq. 4.2) coupled414.5. Kondo Effect with Varying Coupling Symmetry-240-235-230-225-220-215V CSS (mV)-140 -120 -100 -80 -60VLC (mV)G0 = 0.03G0 = 0.09G0 = 0.35G0 = 0.590.550.500.450.400.35Current with offset (nA)-20 -10 0 10 20VP +VCSQ (mV)G0 = 0.59G0 = 0.35G0 = 0.09G0 = 0.03a)b)Figure 4.6: (a) 2D scan of conductance through the quantum dot varyingthe two coupling gates VCSS, and VLC. In the top left, the dot is morecoupled to the right lead than the left as VLC is made more negative. Inthe middle of the scan (where the conductance is maximum), the coupling issymmetric ΓR = ΓL (b) Charge transitions measured with different ratios ofcoupling between the two leads in the dot. G0 = 0.59 is symmetric coupling(pink), G0 = 0.03 is asymmetric coupling (green). Charge transitions weremeasured with roughly the same current setpoint through the charge sensor(∼ 0.4 nA), but for clarity, they have been offset in the current. The x-axisuses the same virtual gate for each charge transition.regimes depend on Γ. Hence, the conductance amplitude can be used as amarker of how asymmetrically coupled the quantum dot is to the leads.424.5. Kondo Effect with Varying Coupling SymmetryTo vary the coupling symmetry in a quantum dot, a 2D scan measuringthe conductance with gates that control a separate coupling on each axisis used. In Fig. 4.6a, VCSS (y-axis) mainly controls ΓR, and VLC (x-axis)mainly controls ΓL. Four points are picked from symmetric coupling (middleof scan) to asymmetric coupling (top left of scan), where the dot becomesmore strongly coupled to the right lead.Charge transitions were simultaneously measured so that conductanceversus occupation could be compared with the corresponding NRG. Chargetransitions for each coupling setpoint were measured with ∼ 0.4 nA throughthe charge sensor (Fig. 4.6b). Each charge transition did not have chargejumps near the transition and was deemed a reliable measure of the occu-pation. Interestingly, the asymmetrically coupled transition (green) is morebroadened than the symmetrically coupled transition (pink). Γ/kBT de-termined from the global fit to conductance varied from 20.5 for symmetriccoupling to 26.0 for asymmetric coupling. However, a global fit to the chargetransitions finds 19.6 for symmetric coupling and 45.0 for asymmetric cou-pling (Table 4.1). This discrepancy between the Γ/kBT value determinedfrom the global fits to conductance and charge transitions is surprising. Con-versations with our theorist collaborators suggest that NRG only dependson the overall Γ, and not the ratio between ΓL and ΓR.Γ/kBT from Global FitG0 (2e2/h) Conductance Fit Charge Transition Fit0.59 20.5 19.60.35 22.0 21.60.09 24.4 30.90.03 26.0 45.0Table 4.1: Γ/kBT determined from a global fit to conductance and sepa-rately a global fit to charge transitions at different ratios of coupling sym-metry. At symmetric coupling (G0 = 0.59), the determined Γ/kBT fromconductance and charge transitions are similar. However, at asymmetriccoupling (G0 = 0.03), the Γ/kBT from a global fit to charge transitions ismuch greater than from a global fit to conductance.434.5. Kondo Effect with Varying Coupling Symmetry4.5.2 Conductance versus OccupationFigure 4.7a shows conductance versus occupation taken at 20mK at thefour different ratios of coupling symmetry found in Fig. 4.6a. There is goodagreement with NRG in the symmetric coupling (pink). However, as thecoupling becomes more asymmetric, the conductance amplitude decreases(Eq. 4.2), and agreement with NRG is difficult to determine. Hence, forclarity, Fig. 4.7b shows the same data with an added scaled and offset. Thecorresponding NRG was also scaled and offset the same as the data. Adisagreement between asymmetric coupling (green) data and NRG becomesmuch clearer. When asymmetrically coupled, the conductance maximumof the data is shifted to the right of the NRG. This suggests the Γ/kBTdetermined from the global fit to conductance is lower than expected. Thisdiscrepancy with NRG is surprising as conversations with our theorist col-laborators suggest that NRG only depends on the overall Γ, and not theratio between ΓL and ΓR.Previous experiments have investigated the use of a charge sensor as anoise source to de-phase the Kondo singlet [62]. In these studies, the biasacross the charge sensor was 1200 µV, twelve times the bias that is used forthe charge sensor in this thesis (100 µV) and the suppression of the measuredconductance was ∼ 6%. A more recent theoretical study found that forweak coupling between the charge sensor and quantum dot, the spectralweight of the Kondo resonance is reduced, leading to a suppression of theconductance. However, the width of the resonance is not affected, suggestingthe absence of dephasing [66]. Although this de-phasing mechanism couldresult in the conductance maximum being shifted to a lower occupation,it would not explain a shift to a higher occupation or the dependence onsymmetric coupling.Instead of a change in the shape or location of the conductance, it ispossible the charge transition behaves differently between coupling symme-tries. Numerous studies have investigated the tunnelling rates onto andoff the quantum dot [67]–[69], with differing barrier symmetries [70], [71].However, these studies required the source-drain bias to be larger than the444.5. Kondo Effect with Varying Coupling Symmetrya)b)0.50.40.30.20.10.0Conductance ( 2e2/ h)1.00.80.60.40.20.0Occupation (1e)G0 = 0.59G0 = 0.35G0 = 0.09G0 = 0.032.01.51.00.50.0Conductance (arb.)1.00.80.60.40.20.0Occupation (1e)G0 = 0.59G0 = 0.35G0 = 0.09G0 = 0.03Figure 4.7: (a) Conductance versus occupation (Γ/kBT = 21) with differ-ent ratios of coupling symmetry between the two leads of the dot. As theasymmetry is increased, the conductance decreases. (b) Same data as in(a), except the traces are offset and scaled for clarity. Symmetric couplingagrees well with NRG calculations. However, the asymmetrically coupleddata is shifted to the right of the predicted NRG. This suggests that Γ/kBTextracted from the global fit to conductance is lower than expected.temperature broadening, so the results cannot be extrapolated to our regimeof zero bias.An early study on the shape of the charge transition with symmetriccoupling found that in the absence of a magnetic field, charge transitionsare unaffected by the onset of Kondo correlations [72]. It was argued that454.5. Kondo Effect with Varying Coupling Symmetrythe conductance enhancement between Coulomb peaks results from a largernumber of electrons traversing the quantum dot, each dwelling there for ashorter time. However, coupling symmetry was not explored in this experi-ment.An interesting insight comes from very recent theoretical studies of thedifferential conductance through a quantum dot and its dependence onasymmetry in the tunnel barriers and source-drain bias [73], [74]. It wasfound that higher-order corrections to the differential conductance includenon-linear terms that depend strongly on the tunnelling and bias asym-metries. To qualitatively explain how this might occur, for ΓL > ΓR, thenumber of electrons entering the quantum dot from the left side becomeslarger than the number of electrons leaving the right side. Therefore, thenumber of electrons in the dot increases and the repulsive interaction in-creases ϵ˜0. In contrast, ϵ˜0 would decrease for ΓL < ΓR. This modification toϵ˜0 can result in increased or decreased conductance. However, corrections tothe differential conductance require an applied bias across the quantum dot.Although 1 µV bias is applied across the dot, we consider this effectivelyzero bias as it is much smaller than the thermal broadening.It should be noted that the exploration of coupling symmetry in thisthesis only came from two overnight measurements in a single cooldownand a second cooldown is required to verify the findings. Additionally, asubsequent cooldown would allow for a more focused exploration of tunnelcoupling and bias symmetry.46Chapter 5Conclusion“Nothing in life is to be feared, it is only to be understood. Now is thetime to understand more, so that we may fear less.”Marie CurieIn conclusion, a method has been demonstrated to study the conductanceenhancement from the Kondo effect, in a more weakly coupled regime thanhas been previously explored. Previous studies have predominantly focusedon very strong coupling, where the Kondo temperature exceeds the sys-tem temperature, even between Coulomb peaks. Weaker coupling reveals amarkedly different behaviour: the Kondo temperature is significantly lowerthan the system temperature between Coulomb peaks, resulting in a lackof conductance enhancement. Nonetheless, as the dot energy approachesthe Fermi energy of the source and drain leads, the Kondo temperature in-creases exponentially. Consequently, even under relatively weak coupling, asmall conductance enhancement persists on the shoulder of odd occupationCoulomb peaks.To reliably measure this small conductance enhancement, a new methodis required to disentangle the effects of charge motion, entropy, and Kondoenhancement on the shape and location of the conductance. The approachrelies on two key components. The first component is a simultaneous mea-surement of a conductance and charge transition. The charge transition isthen used to determine the occupation of the quantum dot. This occupationis used as a reference to remove the effects of charge motion and entropy.When conductance is plotted against the occupation, a shift in the maximumconduction to higher occupation suggests Kondo enhancement. The secondcomponent is measuring conductance at a range of temperatures that cross47Chapter 5. Conclusioninto the temperature broadened regime, Γ/kBT ≲ 1. A global fit to NRG,including each temperature setpoint is used to reliably determine Γ/kBTand lever arm. The determined Γ/kBT is then used to compare data withNRG to test agreement.Agreement between data and corresponding NRG is seen at couplingstrengths Γ/kBT = 20.5, Γ/kBT = 9.7 and Γ/kBT = 2.5. The most stronglycoupled data (Γ/kBT = 20.5) shows the greatest shift in conductance maxi-mum towards higher occupation. A reliable measurement of the charge tran-sition is critical to determine the occupation of the quantum dot. Hence,various current setpoints through the charge sensor were tested to ensureindependence from the charge sensor’s settings. Despite differences in theunderlying charge transition shape across current setpoints, agreement withNRG was consistently observed. To investigate the influence of couplingsymmetry on Kondo enhancement, four ratios of coupling symmetry weremeasured. Good agreement with NRG was found with symmetric coupling.However, a shift towards greater occupation than predicted by NRG wasobserved as coupling symmetry became more asymmetric. This discrepancysuggests a difference between the Γ/kBT value determined from the globalfits to conductance and that which best matches the corresponding chargetransitions. To validate this observation, separate global fits to conductanceand charge transitions were performed. Under symmetric coupling, the de-termined Γ/kBT values agree. However, as asymmetry increases, the Γ/kBTobtained from the charge transitions is significantly larger than that fromthe conductance fits. The discrepancy with NRG is surprising as conversa-tions with our theorist collaborators suggest that NRG only depends on theoverall Γ, and not the ratio between ΓL and ΓR.To my knowledge, only one other experiment has investigated the effectsof coupling symmetry on the Kondo effect [65]. It was found that the char-acteristic zero bias peak between Coulomb peaks shifted to a nonzero bias.However, this effect would require a strong energy dependence of the tunnelbarriers (i.e., Γ = Γ(E)) in a NRG calculation, which is not assumed in ourcurrent NRG calculations.An interesting insight into our observed discrepancy comes from very48Chapter 5. Conclusionrecent theoretical studies of the differential conductance through a quantumdot and its dependence on asymmetry in the tunnel barriers and source-drainbias [73], [74]. It was found that higher-order corrections to the differentialconductance include non-linear terms that depend strongly on the tunnellingand bias asymmetries. However, corrections to the differential conductancein this paper require an applied bias across the quantum dot. As the mea-surements in this thesis were conducted in an effective zero bias limit (1 µVbias), it is unclear whether the discrepancy can be explained by this theory.A further cooldown would allow for a more focused exploration of tunnelcoupling symmetry, bias symmetry and total applied bias. If our data qual-itatively matches these recent theory predictions, a new calculation of ourNRG, including bias and tunnel coupling symmetry may be required.Finally, a recent study observed the change in entropy as an electronentered a quantum dot (coupled to a single lead) with similar couplingstrengths as those in this thesis [28]. 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It should also be noted that I made no contributionto the nano fabrication recipes and simply followed the procedure developedby previous students. The following summary is largely adapted from OwenSheekey’s thesis, a past student from the Quantum Devices Group.A.1 SummaryThe GaAs substrates are grown by Michael Manfra’s group at Purdue Uni-versity. There are two key features to devices built on GaAs/AlGaAs 2DEGs– ohmic contact to the 2DEG and gating structures. In overview, devicesused for this project use a thin film of Al2O3 to electrically isolate top gatesfabricated from Au/Ti. Ohmic contacts are made using annealed Ni/Au/Ge.The general process of preparation of a sample goes as follows:1. Gallium removal from back of wafers.2. Ohmic contact: Lithography, evaporation, annealing.3. Mesa etching: Lithography, H2SO4 etching.4. Atomic layer deposition: Al2O3.5. Gating (2 steps): Lithography, evaporation.60A.2. Recipes6. Wire bondingA.2 RecipesA.2.1 Gallium RemovalThis is a recipe developed by Dr. Silvia Lu¨scher Folk.1. Cleave a full wafer into a quarter or half wafer. Blow off any dustbunnies from the surface before starting.2. Spin a layer of AZ1518 resists at 4000RPM for 40 s.3. Bake the resist for 2minutes at 100◦C.4. Put a clean wipe on a hotplate and set to 50◦C. Put wafer face down(gallium side up) onto the clean wipe.5. Wipe off gallium with q tips. Keep wiping until it is all gone.6. Spin and bake another layer of AZ1518 (4000RPM 40 s and bake 100◦C2minutes).7. Etch 2minutes in full strength HCl. Quench etch by transferring toDI water.8. Rinse well in DI water. Blow dry.9. Squirt down with Acetone to strip resist from the surface and imme-diately rinse with IPA.10. Soak 3 minutes each in Toluene, Acetone, and IPA. Rinse with IPAand blow dry between each solvent. Spray down with DI water andblow dry.61A.2. RecipesA.2.2 Mesa Etch1. Solvent clean – Acetone, IPA. Rinse in DI, blow dry with N2.2. Prebake chip for 60 s at 110◦C.3. AZ5214-E in positive mode, ramp up 500 RPM for 2 s, spin at 4444RPM for 40 s with 60 s softbake at 100◦C.4. Photo lithography in MLA150 Heidelberg (Maskless Aligner) : Expose90mJ/cm2, defocus -15. Develop in MIF 300, 50 s stop develop in DI.6. Hardbake for 60 s at 120◦C.7. O2 plasma etch in Plasma etch PE 50, 60s.8. Etch in 30mL diluted Sulfuric acid (700 Water:3 H2S04) + 2mL H2O2at 18◦C for 50 s9. Stop etch in DI water. Remaining resist can be removed with Acetone,IPA and DI.A.2.3 Ohmic ContactLithography and EvaporationAt the end of my Master’s studies, there was a sizeable effort (that I wasnot involved in) to develop low resistance ohmic contacts by Vahid Mohavedand Dr. Silvia Lu¨scher Folk. Here is the most up to date recipe.1. Solvent clean – Acetone, IPA, Ultrasound. Rinse in DI, blow dry withN2.2. Dehydration bake chip for 1minutes at 110◦C.3. AZ5214-E in image reversal mode, ramp up 500 RPM for 2 s, spin at4444 RPM for 50 s with 60 s softbake at 90◦C.62A.2. Recipes4. Photo lithography in MLA150 Heidelberg (Maskless Aligner) : Expose40mJ/cm2, defocus +15. IR bake for 30 s at 110◦C.6. Photo lithography in MLA150 Heidelberg : Flood exposure 222mJ/cm2,defocus +17. Develop in AZ300 MIF , 40 s stop develop in DI.8. Hardbake for 60 s at 110◦C.9. O2 plasma etch in Plasma etch PE 50, 15 s.10. Dip in 37% HCI for 20 s. Rinse with DI for 120 s11. Evaporate the following:Metal Thickness RateNi 7 nm 1.0 A˙s−1Ge 80 nm 1.3 A˙s−1Au 160 nm 2.0 A˙s−1Ni 36 nm 1.6 A˙s−1Au 80 nm 1.7 A˙s−112. Liftoff in Acetone at 70◦C.13. Rinse in Acetone, IPA and blowdry with N2.AnnealingThe following process was followed on the rapid thermal annealer made “inhouse” for annealing GaAs and other substrates. The basic idea is to usea bulb in a near vacuum (with some amount of forming gas – H2 + N2) toheat the sample to a high temperature for a limited amount of time. Themetal film melts and diffuses through the GaAs substrate to make contactwith the 2DEG, usually 30− 200 nm below the surface.1. Pump down should reach lower limit, 0.133mbar.63A.2. Recipes2. Set regulator to 2.5 psi (forming gas).3. Open Swagelock valve until pressure reads 10mbar.4. Close speedivalve until pressure reads 200mbar.5. Turn on bulb to roughly 34%, it takes about 5− 10minutes6. Wait at 350◦C for 2minutes7. Go to 450◦C as fast as you can, hold it there for 40 s.8. Cool below 300◦C fast by opening the speedivalve on the pump lineall the way, and opening the gas flow.9. Cool to 50◦C, then vent and let cool to room temperature.A.2.4 GatingGating is the process of adding metal to the top of the GaAs/AlGaAs het-erostructure, which are then used to form and control the potential land-scape in the 2DEG below (Fig. A.1). The overall recipe for gating was notvaried during my fabrication time. However, if long periods have passed be-tween fabrication (a few months), it is necessary to test an array of exposuredoses for the E beam step as this parameter can vary over long periods oftime and system re starts.Inner Gates1. Solvent clean - Acetone, IPA. Rinse in DI, blow dry with N2.2. Prebake chip for 3minutes at 180◦C.3. Spin PMMA A2 495K at 5555RPM for 50 s with 3minutes bake at180◦C4. Spin PMMA A1 950K at 3333RPM for 50 s with 5minutes bake at180◦C64A.2. Recipes5. E beam lithography in Jeol JBX 8100FS : Dose 900 µC/cm26. Develop in IPA:DI, 7:3 40 s at 18◦C, stop by directly drying with N2.7. Evaporate Ti 3 nm, 1.5 A˙s−1 then Au 12 nm 4 A˙s−1.8. Liftoff in Acetone at room temperature.b)e)d) f)a) c)GaAsPMMAPMMAAuClean SpinDevelop Evaporate Lift-offExpose (EBL)Figure A.1: (a) GaAs/AlGaAs heterostructure is cleaned in an Ultra Sonic(US) bath with acetone/IPA/DI. (b) Two different layers of PMMA arespun ontop of the heterostructure. (c) The gate design is exposed to anelectron beam to breakup the PMMA. (d) The chip is developed in anIPA:DI solution create an undercut and remove the exposed PMMA. (e)2:12 nm of Ti:Au are evaporated ontop of the chip. (f) The chip is rinsedin acetone to remove remaining PMMA and any metal attached to it. Thechip is ready to be placed onto a chip carrier, wirebonded and measured.Outer Gates and Bondpads1. Solvent clean – Acetone, IPA. Rinse in DI, blow dry with N22.2. Prebake chip for 3minutes at 180◦C.65A.2. Recipes3. Spin PMMA A8 495K at 4000RPM for 40 s with 3minutes bake at180◦C4. Spin PMMA A4 490K at 4000RPM for 40 s with 5minutes bake at180◦C5. E beam lithography in Jeol JBX 8100FS : Dose 1100 µC/cm26. Develop in IPA:DI, 7:3 40 s at 18◦C, stop in DI.7. O2 plasma etch in Plasma etch PE 50, 1minutes.8. Evaporate Ti 10 nm, 2 A˙s−1 then Au 100 nm, 4 A˙s−1.9. Liftoff in Acetone at room temperature.A.2.5 Wire BondingThe final step in the fabrication process is to attach the chip to a chip carrierand connect the gates and ohmic contacts to bond pads (Fig. A.2). The chipcarrier is added to the fridge and connected to the fridge wiring, which leadsto a break-out box outside the fridge.1. Stick the chip to the chip carrier using a small dab of PMMA A8 495K.Bake for 5minutes at 60◦C2. Wirebond bond pads on chip carrier to gates and ohmic contacts onthe chip:Bond 1 Bond 2Wire material Al AlBond strength 260 300Bond duration 30ms 30ms66A.2. Recipesb)a)5mmFigure A.2: (a) Chip carrier stuck to a platform under the wirebonder.(b) Close-up of the Al wire bonds that connect the device gates to a chipcarrier.67Appendix BSoftwareB.1 Open-Source ThesisAll LATEXfiles, data and code used to build the figures in this thesis areavailable on GitHub.The code can be found here:https://github.com/johann997/QDEV_masters_thesis.B.2 Data Acquisition and AnalysisI was fortunate to join FolkLab after Timothy Child had largely rewrittenand significantly improved the acquisition code. However, code is never ina final state, and over the years, I contributed to the repository with newscan functions, analysis, and function dependency visualisation.The code can be found here:https://github.com/folk-lab/IgorAcqFor analysis, I built upon some initial code provided by Josh and Silvia.Currently, the code used to fit data to NRG lives in a separate repositoryfrom the acquisition code.The code can be found here:https://github.com/johann997/KondoConductanceAnalysisB.3 Device SimulationThe code can be found here:https://github.com/johann997/2deg_yodels68B.3. Device Simulationx (μm)y(μm)9.6 9.8 10.0 10.2 10.4 10.60.60.81.01.41.2x (μm)−60 −40 −20 0 20 40 60−60−40−200204060−80y(μm)a) b)Figure B.1: (a) Plot of the uploaded .dxf. Each polyline in the .dxf is savedas a new trace (hence, the varying colours). (b) Discretised matrix from azoom in on the uploaded design file. The matrix was defined by specifyingthe min, max and x, y coordinates and the number of points to discretise inthe x and y direction (here numptsy and numptsy = 250). This approachis quite rudimentary and not optimised, but is sufficient at capturing thedevice design geometry.Fabrication and measurement of quantum devices can be costly and time-consuming. As part of a class project, I developed an easy-to-install, open-source application that allows users to upload a .dxf file and calculate theelectric potential from user-determined gate voltages. kwant [75] is used tocalculate transport properties by applying a tight-binding approximation. Atight-binding model reduces computation so that users can simulate pinch-off or charge stability diagrams in ‘real time’. This app aids the designprocess by providing quick, semi-quantitative information on design choices.For user convenience, drag-and-drop functionality is used. Upon upload-ing the .dxf file, ezdxf Python library is used to extract the polylines. Usersspecify device areas by defining min, max and x, y coordinates, and pointdiscretisation (Fig. B.1).The electric potential is calculated using the method developed byDavies [36]. This method offers the advantage of computational efficiency.69B.3. Device SimulationV (eV)x (μm)y(μm)a)x (μm)y(μm)V (eV)b)Figure B.2: (a) 3d map of the electrostatic potential landscape 90 nm belowthe surface of the heterostructure, calculated using the bare potentials onthe gates on top of the heterostructure. (b) Varied potential landscape bychanging the voltages on the gates.A single ‘geometric factor’ is initially calculated and the resulting potentialis simply the product of the geometric factor with the gate voltages. Real-time updates of the plotted potential are achieved using sliders (Fig. B.2).The disadvantage of this method is that screening terms are not included,and hence, the calculation can only be used as a qualitative visualisation ofthe electric potential profile.−2.0−1.9−1.8−1.6−1.7−2.1a)x (μm)9.8 10.0 10.2 10.4 10.6 10.8y(μm)k−3 1−2 −1 0 320.000.300.250.150.200.100.05Energy(eV)b)Figure B.3: (a) Gate design of a QPC on top of the kwant [75], tightbinding system. A square lattice with lattice constant 5 nm is chosen. Theinfinite leads are shown in red on the left and right axes. (b) Band structurecalculation of this tight binding system. The user can calculate transportproperties at the different allowed energy levels in the band structure.70B.3. Device Simulationkwant is utilised to build a tight-binding model which can be solvedto provide access to transport properties [75]. This method assumes elec-trons are tightly bound to their respective atoms, with interactions fromneighbouring atoms considered as perturbations. A kwant system com-prises a finite scattering region described by a scattering Hamiltonian HSand infinite regions called leads (these represent the ohmic contacts in anexperiment).x (μm)9.8 10.0 10.2 10.4 10.6−2.0−1.9−1.8−1.6−1.7−2.1b) E = 0.010 eVTop Gate, Bottom Gate Voltage (V)−0.06 −0.05 −0.04 −0.03 −0.02 −0.01 00286410121416Conductance(2e2/h)c)0.008 eV0.010 eVE = 0.080 eV9.8 10.0 10.2 10.4 10.6x (μm)y(μm)−2.0−1.9−1.8−1.6−1.7−2.1a)y(μm)Figure B.4: (a) Calculated wavefunction squared at lower band energysuch that a single channel propagates through the QPC. (b) Calculatedwavefunction squared at increased band energy so that two channels prop-agate through the QPC. (c) Calculated conductance through the QPC attwo energy bands as the voltage on both QPC gates is varied. Conductancedrops to zero at lower gate potentials. Both curves show the characteristicQPC conductance plateaus at integer multiples of 2e2/h.In the app, the kwant system is generated by defining the lattice con-stant and lead locations. A plot of the system is shown in Fig. B.3a andcorresponding band structure in Fig. B.3b. An energy level is selected tocompute transport properties. A plot of the wavefunction squared is shownin Fig. B.4a, where a single channel is shown to propagate through the QPC.The calculated conductance at varying gate potentials is shown in Fig. B.4c,71B.3. Device Simulationwhere the characteristic QPC conductance plateaus at integer multiples of2e2/h are found.72
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Folk, Joshua
Occupation resolved conductance of a few electron quantum dot : a test for Kondo correlations in an intermediate coupling regime
Text