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Local probe of electronic states in high mobility quantum Hall samples Samani Nasab, Mohammad 2017

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Local Probe of Electronic States In High MobilityQuantum Hall SamplesbyMohammad Samani NasabBachelor of Science, York University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)September 2017c©Mohammad Samani Nasab, 2017AbstractThe discovery of the integer quantum Hall effect (IQHE) and the fractional quantumHall effect (FQHE) in a 2-dimensional electron gas (2DEG) have created a newand rich field in condensed matter physics in low dimensions. Almost 35 yearsafter these discoveries, there are still several unanswered questions regarding thenature of various electronic phases formed in such systems. The 2DEG in ultra-high mobility quantum well (QW) samples in large magnetic fields and millikelvintemperatures are studied in this thesis.We developed a reproducible recipe for enhancing the quality of the very fragileFQHE states reliably, which can be used to reset an electrically shocked sample in-situ at low temperatures. We then developed a protocol for measuring the localelectronic density on QW samples using a single-electron transistor (SET). We alsodeveloped a technique for modulating the temperature of the sample at about 10Hz by about 10 mK. We used the electrometer and the fast temperature modulatorto obtain a measure of changes in chemical potential as a function of temperatureoscillations. This quantity can reveal the existence of an enhanced entropy in thestate of the electrons.We investigated theories that predict the non-Abelian state of matter, that fol-lows neither Fermionic nor Bosonic statistics. Non-Abelian quasi-particles areexpected to form as collective excitations in the fractional quantum Hall regime atfilling factor ν = 5/2. The experimental results were incompatible with the non-Abelian theory under investigation.We also studied the nature of localization of electrons in the bulk of the samplewhen the system is in one of the incompressible states near an IQHE plateau. Thenoise characteristics detected by the ultra-sensitive charge sensor implanted on theiisurface of the sample revealed new behaviours not observed in the past. The resultscan be explained by telegraph noise arising from charge carriers jumping from onelocalized potential pocket to another, evolving into 1/ f noise, as the filling factorshifts away from the centre of the integer state.iiiLay SummaryAll elementary particles in nature fall under two broad categories: Fermions andBosons. Electrons are Fermions. Photons are Bosons. Composite particles alsocan be Fermions or Bosons. A He-4 atom (an atomic nucleous with 2 protons and2 neutrons) is a Boson, and a He-3 atom (2 protons and 1 neutron) is a Fermion.Recently, new theories have been proposed that predict the existence of compositeparticles that do not belong to either category. In this thesis, we aimed at findingexperimental evidence for one such theory, which predicts electrons are confinedin a 2-dimensional plane and placed in a strong magnetic field at very low tempera-tures can form composite particles that behave neither like a Boson nor a Fermion.We did not find evidence for such behaviour, but if found, these systems can beuseful for manufacture of quantum computers and information storage systems.ivPrefaceChapter 3 of this thesis is based on the following published work:• Samani, M. et al. Phys. Rev. B 90, 121405 (2014)The general direction of the research presented in this thesis was proposed bymy supervisor, Professor Joshua Folk, and the results are presented in Chapter 6.The results presented in Chapter 3 and Chapter 7 are reports on our findings thatseemed to be interesting as we marched towards our goal of finding evidence forthermodynamic signatures of non-Abelian particles. The fast heating techniquepresented in Chapter 5 was first tried by other members of our group, Dr. SilviaFolk and Nicolaas Rupprecht. I developed and tested the technique further duringmy investigations.The QW samples used in all the works presented in this thesis were grown bymembers of Professor Michael Manfra’s group at Purdue University. The sampleused in Chapter 3 was cleaved from wafer number 6-6-11.2. The samples used inChapter 4, Chapter 5, Chapter 6, and Chapter 7 were cleaved from wafer number3-10-15.2.I designed the location of devices and wires on samples, and performed vari-ous fabrication steps at UBC’s facilities including e-beam lithography, clean-roomwet-cleaning and wet-etching, and thermal metal evaporation. The SETs used inChapter 4, Chapter 6, and Chapter 7 were designed collaboratively with NeerejaSundaresan and Mattias V. Fitzpatrick, members of Professor Andrew Houck’sgroup at Princeton University. The e-beam lithography and thermal metal evapo-ration steps in the fabrication of the SETs were performed at Princeton University.I am responsible for controlling the experiments, data analysis, and preparingvthe conclusions. I prepared the draft of the published work mentioned above and itwas edited by the co-authors.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Realizing The 2-Dimensional Electron Gas . . . . . . . . . . . . 42.2 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . 72.3 The Fractional Quantum Hall Effect And Anyons . . . . . . . . . 132.4 Composite Fermions . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Non-Abelian States and ν = 5/2 . . . . . . . . . . . . . . . . . . 173 Illumination And Annealing . . . . . . . . . . . . . . . . . . . . . . 213.1 Motivation For Large Energy Gaps . . . . . . . . . . . . . . . . . 213.2 The Optimum Recipe . . . . . . . . . . . . . . . . . . . . . . . . 22vii3.3 The Low Density Regime . . . . . . . . . . . . . . . . . . . . . . 293.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 The Single Electron Transistor . . . . . . . . . . . . . . . . . . . . . 314.1 The Principles of Operation . . . . . . . . . . . . . . . . . . . . . 314.2 Characterizing The SET . . . . . . . . . . . . . . . . . . . . . . . 335 Characteristics of The Heater . . . . . . . . . . . . . . . . . . . . . . 395.1 Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2 DC Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 AC Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 Signatures of Non-Abelian States . . . . . . . . . . . . . . . . . . . . 486.1 The Moore and Read State . . . . . . . . . . . . . . . . . . . . . 486.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 506.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4 Stability After Illumination . . . . . . . . . . . . . . . . . . . . . 557 Localization In Strong Magnetic Fields . . . . . . . . . . . . . . . . 587.1 Impurity Induced Localization . . . . . . . . . . . . . . . . . . . 587.2 Wigner Crystallization . . . . . . . . . . . . . . . . . . . . . . . 607.3 Microscopic Probing of The Localized States . . . . . . . . . . . 627.4 Sound Card Measurements . . . . . . . . . . . . . . . . . . . . . 667.5 Temperature Dependence of The Localized States . . . . . . . . . 687.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . 81A.2 SET Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.3 Fridge Wiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88viiiB Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91B.1 The Triple Product Rule of Partial Derivatives . . . . . . . . . . . 91B.2 State Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B.3 Free Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93B.4 Chemical And Electrochemical Potential . . . . . . . . . . . . . . 94ixList of FiguresFigure 2.1 Band diagram and wave-functions in a Si-MOSFET . . . . . . 5Figure 2.2 Measurement geometry and measurements of Rxx and Rxy . . . 9Figure 2.3 Landau fan diagram . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.4 Composite Fermion diagrams for integer and fractional states 16Figure 3.1 Rxx and Rxy traces before and after illumination . . . . . . . . 23Figure 3.2 Evolution of fractional states over annealing time . . . . . . . 24Figure 3.3 Evolution of density and ν = 5/2 gap over annealing time . . 27Figure 3.4 The low-density state after illumination without annealing . . 29Figure 4.1 Schematic cartoon and monograph of SET . . . . . . . . . . . 32Figure 4.2 Coulomb blockade diamond . . . . . . . . . . . . . . . . . . 35Figure 4.3 Comparison between transport and SET measurements as a func-tion of the magnetic field . . . . . . . . . . . . . . . . . . . . 38Figure 5.1 3D cartoon of the sample . . . . . . . . . . . . . . . . . . . . 40Figure 5.2 Temperature measurement using transport features . . . . . . 41Figure 5.3 DC heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 5.4 I-V curves for the heater at various magnetic fields . . . . . . 43Figure 5.5 AC heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 5.6 Heating efficiency as a function of frequency . . . . . . . . . 46Figure 6.1 Rxx between ν = 2 and 3, and AC heating near ν = 3 . . . . . 52Figure 6.2 SET monograph and measurements with AC modulation of2DEG’s voltage . . . . . . . . . . . . . . . . . . . . . . . . . 53xFigure 6.3 ∂µ/∂T near ν = 5/2 . . . . . . . . . . . . . . . . . . . . . . 54Figure 6.4 Noisy SET after illumination . . . . . . . . . . . . . . . . . . 56Figure 7.1 Broadened density of state by impurities near Landau levels . 60Figure 7.2 SET measurements of the insulating bulk near integer states . . 63Figure 7.3 Noise from SET over time in the insulating regime . . . . . . 64Figure 7.4 Frequency response of the sound card . . . . . . . . . . . . . 66Figure 7.5 Sound card measurements over ν = 2 . . . . . . . . . . . . . 67Figure 7.6 Comparison of temperature-dependent transport measurementsand SET measurements . . . . . . . . . . . . . . . . . . . . . 69Figure 7.7 Sound card measurements as a function of temperature . . . . 70Figure A.1 Before and after annealing the ohmic contacts . . . . . . . . . 84Figure A.2 Double angle Al evaporation apparatus . . . . . . . . . . . . 86Figure A.3 Al evaporation while cooling the substrate . . . . . . . . . . . 88Figure A.4 Low-pass filters for the dry-fridge . . . . . . . . . . . . . . . 89Figure B.1 Contour of constant u on the x-y plane . . . . . . . . . . . . . 92xiGlossary2DEG 2-dimensional electron gasQHE quantum Hall effectIQHE integer quantum Hall effectFQHE fractional quantum Hall effectMMA Methyl MethAcrylatePMMA PolyMethyl MethAcrylateRPM revolutions per minuteSEM Scanning Electron MicroscopeMIBK Methyl IsoButyl KetoneIPA IsoPropyl AlcoholSCCM standard cubic centimetres per minuteMOSFET metal oxide semiconductor field effect transistorSET single-electron transistorQW quantum wellMBE molecular beam epitaxySHJ single heterojunctionxiiLED light emitting diodePID proportional-integral-derivativeDAC digital-to-analogRMS root mean squareTLF two-level fluctuatorRTA rapid thermal annealingRC resistor-capacitorWC Wigner crystalxiiiAcknowledgmentsI would like to thank my supervisor Professor Joshua Folk for his unconditionalsupport, incredible patience, and pleasant sense of humour. I would also like tothank him for helping me understand lock-in amplifiers. The idea behind a lock-inamplifier is one of the coolest concepts I have learned so far in my life.I would like to thank my supervisory committee members, Professors WalterHardy, Doug Bonn, Marcel Franz, and Mark Halperin. Their guidance was instru-mental to my success during the roughest months of my work.It was a pleasure to work with Dr. Silvia Folk. I would like to thank her forbeing always helpful, and for teaching me, by example and by instruction, on howto be a good experimentalist. Without her help, this would have been a lot morepainful.I benefited from discussions with my incredible lab members. I learned fromAli Khademi, Ebrahim Sajjadi, Nik Hartman, Christian Olsen, Alexandr Rossokhaty,Rui Yang, Mark Lundeberg, and Matthias Studer on a daily basis.I would like to thank Mirko Moeller for his seemingly innocent questions aboutmy experimental apparatus that made me think deeper and harder about what onearth I was doing. Our extended discussions over dinner about fundamental con-cepts of my work, and his, made me a better physicist, regardless of the effect thosediscussions had on our shared dinners. The several bicycling and hiking trips wewent on together helped me keep my sanity during the time when everything elsewas failing.I would like to acknowledge the incredible support I received from Erica Zachariasduring the 5 years I lived in Vancouver. I could not have finished this work if shehadn’t constantly pulled me up and set me straight on my path. I might have pro-xivvided a few pints of ice-cream for her on a few occasions, but she constantly gaveme encouragement, something that convenient stores don’t usually sell.I would like to thank my friends who provided the emotional support any-one needs in a new city. Tim Cox, Amanda Parker, Kris Burris, Morgan Burris,Katrina Wechselberger, Sophie Comyn, Philippe Sabella Garnier, Samara Pillay,Ryan Shcwartz, Andrew MacDonnald, Tegan MacDonnald, Alan Manning, andStephanie Rabe gave me all the love one needs.And finally, I would like to thank my parents, Zahra and Jafar, for not lettingtheir idea of a “sensible life” get in the way of the love and support that continu-ously flows my way in outrageous amounts, even from very far away.xvChapter 1IntroductionThe process of interchanging two particles, and interchanging them again back totheir original positions, is identical to moving one particle around the other par-ticle, forming a full circle. In three spatial dimensions, if the two interchangesare performed adiabatically, the two interchanges are identical to not moving ei-ther particle at all, since the circle can be reduced to a point in a topologicallycontinuous transformation. This leaves two choices for a single interchange oper-ation: multiplication of the wave-function by +1 and -1. The former gives Bosons,and the latter, Fermions. This simple argument is central to the theory of quan-tum mechanics. Much of chemistry and biology is a consequence of this simpleobservation. The fact that electrons are Fermions forms our understanding of theperiodic table. The coherent propagation of electromagnetic radiation is the resultof photons being Bosons.Composite particles, the bound state of multiple, more elementary, particles,can also be Fermions or Bosons. Two electrons can form a Cooper pair, a Boson;Cooper pairs conduct electrical current without dissipation, whereas conductingelectrons in a metal dissipate energy. Helium-4 atoms are Bosons and can form asuperfluid, whereas helium-3 atoms are Fermions, which form Fermi fluids. Simi-lar to electrons, a pair of helium-3 atoms can form Cooper pairs, which are bosonsand form superfluids [2].In two spatial dimensions, the wave-function of particles does not have to fol-low the interchange symmetry of three dimensional space. The path of a particle1that goes around another particle cannot be deformed into a point without crossingthe other particle or cutting the path [3, 4]. The possibility, therefore, exists intwo dimensions for breaking the Boson-Fermion dichotomy. Moving such parti-cles around each other can be described by braiding groups, which are, in general,non-Abelian groups. The composite particles in the fractional quantum Hall ef-fect (FQHE) regime, realized in a 2-dimensional electron gas (2DEG), provide theappropriate environment for observation of such a fascinating possibility [5].Experimentally detecting the signature of such particles was the main goal ofmy research and is the primary focus of this thesis. The idea, as proposed by Sternand Cooper [6], is to detect the enhanced entropy of these particles by measuringthe change in chemical potential as a function of a changing temperature (∂µ/∂T ).A clear answer to this question turned out to be quite difficult to obtain, but severalinteresting and important results emerged from my attempts to detect non-Abelianparticles.The bulk of this thesis is dedicated to the attempt to detect non-abelian parti-cles in the FQHE regime. The observation of FQHE in a laboratory is possible onlyunder very stringent conditions. The sample must be of an extremely high quality,its temperature should be kept below 1 K, and it has to be placed in a high magneticfield, typically in the range of a few Tesla. It turns out other than these environ-mental variables, the process of cooling down the sample also plays a key role indetermining the strength of these states. A reliable and reproducible cool-downwas missing in the community. In Chapter 3, we described a technique we devel-oped for reliably obtaining strong FQHE features, and resetting the sample to a wellknown state in case an electric shock or other perturbations in the environment af-fect the sample during measurements. The technique consists of illuminating thesurface of the sample with red light at low temperatures followed by annealing thesample to a few kelvins. Unfortunately this technique did not turn out to be usefulfor the measurements of ∂µ/∂T .In order to detect the chemical potential, we used a single-electron transistor(SET), which is an exquisitely sensitive electrometer. They have been used in thepast [7] in the quantum Hall effect (QHE) regime to study the properties of thebulk of the sample. In Chapter 4, we describe the characteristics of this device ingeneral and the various measurements that helped us calibrate and characterize our2particular device.To measure ∂µ/∂T , we also need to be able to oscillate the temperature ofthe sample. This was achieved by an on-chip heater which we used to heat up thesample periodically. This technique was developed in our lab, and enabled us tooscillate the temperature by 10 mK at 100 Hz or lower. The technique is describedin Chapter 5.The attempt to measure non-Abelian characteristics of FQHE quasiparticles cul-minates in chapter Chapter 6, where we describe the results of the ∂µ/∂T mea-surements. The signature of non-Abelian particles, the expected enhanced entropy,was ultimately not observed, but we think the results were not conclusive becauseduring the measurements, strong FQHE features were not observed. The illumi-nation and annealing technique described in Chapter 3 essentially renders the SETunusable for unknown reasons and therefore strong FQHE were not obtainable atthe same time as SET measurements were possible.However, the sophisticated samples that were developed for the ∂µ/∂T mea-surements can be used for a much wider range of experiments than just the searchfor non-Abelian entropy. In particular, the ability to detect chemical potentialchanges on a microscopic length scale, in extremely high quality samples madeit possible to investigate some unusual effects, probably related to correlations,that emerge in the integer quantum Hall regime even without the illumination thatwas crucial for FQHE experiments. These final measurements of integer states arepresented in chapter Chapter 7, and offer a taste of the type of experiment that willbe possible in the future based on the experimental advances that were developedin this thesis.3Chapter 2Theoretical BackgroundIn this Chapter, we go over some of the theoretical backgrounds, and the ingredi-ents of the experiments that are necessary to understand the upcoming chapters.We start by explaining how particles, electrons in our case, can be confinedin a two dimensional space in a laboratory environment. We then review some ofthe basic concepts of the QHE, and the composite Fermion theory within the QHEregime. Finally, we discuss the theory behind one of the prime candidates for astate with non-Abelian statistics, the ν = 5/2 state, and one proposal for observingsignatures of such a state [6], based on which the experiments in this thesis weredesigned and carried out.2.1 Realizing The 2-Dimensional Electron GasThe integer quantum Hall effect (IQHE) was first observed in a metal oxide semi-conductor field effect transistor (MOSFET) in 1980s[8]. A MOSFET consists of ap-doped Si substrate; an insulating layer of SiO2 is placed on top of the substrate,and it is capped by a metallic layer (see Figure 2.1.(a)). When a positive voltageVg is applied to the metallic gate, the metal and the p-Si behave like the two platesof a capacitor. Extra positive charge accumulates on the gate, which causes thevalence and conduction bands to bend in the p-Si region. If Vg is high enough, theconduction band of the p-Si will cross the Fermi energy, and a thin (≈10 nm) layerof conduction electrons is formed right at the boundary between p-Si and SiO2.4These electrons are free to move along the interface in the x-y plane. Because thisis a negatively charged region in a p-type semiconductor, it is commonly referredto as the inversion layer [9, Chapter 1].zEFMetal SiO2 p-SiVg(a)zE0EFE1ψ0ψ1≈ 180 eV(b)EFFigure 2.1: (a) A silicon MOSFET consisting of a metal layer, an insulatinglayer of SiO2, and p-doped Si. A positive voltage Vg is applied to themetal gate, which causes the valence (green) and the conduction (blue)bands to bend in the Si. If Vg is large enough, the conduction band of Sicrosses the Fermi energy. (b) The region near p-Si/SiO2 interface, closeto the Fermi energy is zoomed in after Vg is increased to a level that con-duction band of p-Si crosses the Fermi energy close to the interface. Thecrossing of the conduction band and the Fermi energy in (a) can be ap-proximated as a triangular potential. The electronic wave-functions areshown in red for the lowest two energy states. At normal temperatures,only the bound state with energy E0 is occupied. The electrons can onlymove along the plane of the SiO2/p-Si interface, in 2 dimensions.Figure 2.1.(b) is a closer look at the inversion layer. If we assume infinitepotential on the oxide side, and a constant electric field, or linear potential, throughthe semiconductor for the first 10 nm, the first two energy levels in the z-directioncan be found to be 180 eV or 2000 K apart from each other for a typical MOSFET[10]. Therefore at low enough temperatures, the electrons are confined in the z5direction to the first energy level E0, and their motion is restricted to the plane ofthe interface; they form a 2DEG at the interface between the SiO2 and p-Si.A major problem with a MOSFET-based 2DEG is that, because the donors areinside the semiconductor, and very close to the 2DEG layer, they introduce a largerandom potential, and therefore decrease the mobility of 2DEG electrons. Thisproblem was addressed by the advent of GaAs structures in condensed matterphysics in which very clean samples and sharp heterostructures can be engineered,resulting in high mobility samples. The observation of the FQHE [11] was firstmade in a GaAs/AlxGa1−xAs sample, grown using molecular beam epitaxy (MBE).MBE technology for two decades since was the dominant driver of condensed mat-ter physics in two dimensions. The MBE is a technique for growing crystals ona substrate by thermally evaporating the ingredients, and placing the substrate onthe line of sight. The growth rate is so low that the thin film layers can form awell-defined single crystal structure. Using MBE, large area single crystal filmscan be grown with very few extended defects. The growth rate is usually slow (onthe order of one monolayer per second), therefore abrupt interfaces can be made.The ultra-high vacuum (on the order of 10−12 mbar) in the state-of-the-art MBEmachines ensures low incorporation of impurities in the crystal. Also, MBE ma-chines are generally equipped with real time feed-back mechanisms on the state ofthe process. [12]The simplest method for constructing a 2DEG in GaAs/AlGaAs is a single het-erojunction (SHJ). It consists of a thick layer of GaAs (say 1µm), with 250 nmof AlGaAs on top of it. The AlGaAs is δ -doped with Si atoms 70 nm from theGaAs/AlGaAs interface. δ -doping refers to the confinement of dopant atoms dur-ing the growth in a single atomic layer in a two dimensional plane. (See 13 for anexample.) The AlGaAs layer is capped by a thin layer of GaAs. Since Si atoms,which have 4 valence electrons, replace Ga atoms with 3 valence electrons, they actas electron donors. Both GaAs and AlGaAs are insulators, that is, their Fermi en-ergy sits between the conduction and the valence band. The energy gap in AlGaAs,however, is larger than that of GaAs. The existence of the donors in the AlGaAscauses the conduction band in GaAs bend enough to cross the Fermi energy, andthe 2DEG is formed at the boundary of GaAs and AlGaAs. The set-back distance,the distance between the donors and the 2DEG in this design, greatly reduces the6random potential experienced by the 2DEG and enhances the electrons’ mobility.In general, the mobility of electrons in a 2DEG increases with increasing elec-tronic density [12, 14]. The density of electrons can be increased by increasing theconcentration of the Si dopants, and bringing the doping layer closer to the 2DEG.Bringing the dopants close, however, amplifies the effect of the random potentialintroduced by the ionized donors. As a result, the mobility does not increase forelectron densities above 2.2× 1011/cm2 in a SHJ. This problem can be solved byconstructing a quantum well (QW) with AlGaAs barriers on both sides of a nar-row GaAs channel. The AlGaAs barriers are then symmetrically δ -doped with Si.δ -doping with a large distance between the doping layer and channel is used toensure the effect of the potential created by the dopants is minimal in the channel.This is further enforced by the symmetry between two doping layer. In very highmobility samples, the main source of scattering is, therefore, not the dopants; it israther the uniformly distributed charged impurities close to the 2DEG channel thatwere unintentionally added during the growth process. The samples used in all ofthe experiments in this thesis are QW structures, grown using the MBE techniquein Professor Michael Manfra’s group at Purdue University. Details on the moredelicate aspects of the high quality QWs can be found in Reference [12].It is worth noting that mobility is not the determining factor in the developmentof the delicate FQHE states, even though correlations certainly exist. Crystal growthtechniques are also not the only factors that determine the quality of FQHE states.The cool-down procedure also plays a significant role. The relationship betweencool-down procedures and the quality of the FQHE state at ν = 5/2 is investigatedextensively in Chapter 3.2.2 The Quantum Hall EffectConsider a long and very thin piece of metal of width w and length L that is placedin the x-y plane with an electric current I flowing along L in the x direction (seeFigure 2.2.(a)). Classically, the following matrix equation connects the electricfield and the current density:7E= ρi (2.1)and in 2 dimensions (ExEy)=(ρxx ρxyρyx ρyy)(ixiy)(2.2)where E is the electric field vector, i is the 2D current density vector, and ρ isthe resistivity tensor, which is the inverse of the conductivity tensor. Assuming anisotropic sample implies ρxx = ρyy and ρxy = −ρyx. We further assume that thecurrent is uniform and only flows in the x direction: ix = I/w and iy = 0. Then thelongitudinal resistance Rxx and Hall resistance Rxy areRxx =VxxI=ExLixw= ρxxLw(2.3)Rxy =VxyI=Eywixw= ρxy (2.4)Note that in 2D, the Hall resistivity is equal to the Hall resistance.We now turn on a magnetic field B = Bz in the positive z direction. Accordingto classical electromagnetism, a Lorentz force is now applied to the electrons, anddeflects them to the −y side of the bar in Figure 2.2.(a). This process continuesuntil the electric field created by the excess charge on one side of the bar is counter-balanced by the Lorentz force:e|v×B|= evxBz = evB = eEy (2.5)where v is the velocity of electrons, and e is the charge of an electron. This effect,first reported by Edwin H. Hall in 1879 [15], is called the classical Hall effect.Replacing I = nevw where n is the sheet density of electrons and Ey = ρxyI/w, weobtain:Rxy =VxyI= B/ne (2.6)8LwFigure 2.2: (a) Schematic view of a Hall bar. The red rectangle representsthe 2DEG. The green squares are ohmic contacts. A current, I, runsfrom the left contact to the right, and the voltage difference is measuredbetween the two bottom contacts (Vxx) and two contacts on oppositesides of the current (Vxy). The magnetic field, marked as B, is pointingout of the plane. (b) Classically, Rxx is independent of the magneticfield, and it would look like the dashed blue line, and Rxy grows linearlywith magnetic field and would look like the dashed green line. Thesolid lines show the results of these measurements at 13 mK in a 2DEGsample.9For a given material (constant n), and a constant current, classically we expect theHall voltage Vxy to increase linearly with the magnetic field B. We also expect thelongitudinal voltage Vxx to be independent of the magnetic field. At room tem-perature and low magnetic field, this prediction works well. The dashed lines inFigure 2.2.(b) show the results of a measurement based on this theory, and the solidlines are the result of an actual measurement at 13 mK between 0.5 and 6 T, whichis obviously in sharp contrast with the classical Hall model.In order to begin to understand these results, we need to invoke quantum me-chanics. The quantum Hall effect was first observed in 1980 by Klaus von Klitzinget. al [8] for which von Klitzing was awarded the Nobel Prize in physics in 1985[16].First notice that the Hall resistance is not linear in B; instead it increases indiscrete steps. The locations of these steps are determined by factors of h/e2 ≈25,813Ω where h is the Planck constant:Rxy =he21ν(2.7)Here ν is a dimensionless number that is called the filling factor. It is an integerfor the IQHE and a simple fraction for FQHE. At the same field where the Hallresistance is on a step, the longitudinal resistance, as well as the longitudinal con-ductance, vanish. The current flows without local dissipation.What is astonishing is that the values of these steps are independent of thematerial; they only depend on two universal constants, the Planck constant and theelementary charge. These values are also surprisingly precise. They have beenmeasured to the accuracy of 1 in 109 [17, 18].The longitudinal resistance, which vanishes at low temperatures, can rise tofinite values by increasing the temperature [19]. The rise of Rxx to finite valuescan be understood using Arrhenius theory of reaction rate, originally developed tocalculate the energy barrier of a chemical reaction using the rate of reaction as afunction of temperature. The modern version of the theory, however, is applied tomany situations where there is an energy gap between two distinct states of parti-cles. Since at the ground state of a QHE state, Rxx = 0 and at the first excited state,10Rxx is finite, measuring Rxx in some range of temperature can give us a measure ofthe energy gap between the two energy states:Rxx ∝ e∆/2kBT (2.8)where ∆ is the gap between two lowest energy levels of the 2d electrons.Even though there are still many open questions regarding the details of Fig-ure 2.2.(b), the plateaus at integer [20] and most odd-denominator fractions of νare well understood. Without going into details, we introduce some of the maintheories that describe these states and are going to be useful for understanding thelater chapters.The Hamiltonian of an electron in the x-y plane in a perpendicular magneticfield that produces the vector potential A can be written asH =12me(p− eA)2+gµBsB = pi22me+gµBsB (2.9)where p is the canonical momentum operator, g is the g-factor of a free electron,a number close to 2 which can be significantly enhanced for electrons in a semi-conductor, µB is the Bohr magneton, s is the quantum number of the spin of anelectron, that is ±1/2, and pi = p− eA is the dynamical momentum of an electronthat is proportional to its velocity. The first term of this Hamiltonian has the samealgebraic form as the simple harmonic oscillator. If we define the ladder operatorsasa† =1√2h¯(pix− ipiy) (2.10)a =1√2h¯(pix+ ipiy) (2.11)then the first term of the Hamiltonian isH = h¯ω(a†a+12) (2.12)11Figure 2.3: The black lines represent the energy levels of the Hamiltonian inEquation 2.14. The first two black lines represent N = 0, the second tworepresent N = 1, and so on. There are two lines per Landau level becausethe system is fully spin-polarized, and each spin has a slightly differentenergy. The chemical potential, the red zigzag, oscillates around theFermi energy, in order to keep the electrochemical potential fixed at theFermi energy, as the magnetic field and consequently the degenerecy ofeach Landau level increase.where ω = eB/me and the energy levels, called Landau levels, are 1EN = h¯ω(N+12) (2.13)The separation of energy between two Landau levels at high magnetic fieldsis in the order of 0.1 meV or 1 K2. Therefore we assume the electrons occupyingdifferent Landau levels don’t mix at temperatures of the experiments, usually be-low 50 mK. The electrical properties of the system are determined solely by theelectrons in the highest occupied Landau level. Also in the limit of high magneticfield, the electrons are fully spin-polarized. We can, therefore, assume the differentspins in the Hamiltonian of Equation 2.9 do not mix. For these reasons, historically1See [9, Chapter 2] for derivations.2At B=1 T, h¯ω = h¯eB/me ≈ 0.1 meV or 1K.12much of the theoretical analysis has been done on the lowest filling factor. The re-sults, to a good approximation, carry forward to higher filling factors. The energylevels of the system, including the spin-splitting energy, then becomesE±N = h¯ω(N+12)± 12gµBB (2.14)where the+ and− correspond to the spins anti-parallel and parallel to the magneticfield, respectively. Both terms are proportional to the magnetic field so they pro-duce straight lines as a function of B. The chemical potential jumps from one levelto another as the field is increased. These energy levels are depicted in Figure 2.3.The wave-function of these electrons is localized in a circle of radius equal tothe magnetic length `=√h¯/eB. In a naive picture, these localized electrons forman insulating bulk; no current flows to the sides of the Hall bar, and Vxy will be on aplateau. All the current from the source to the drain flows on the edges of the Hallbar, without dissipation, when the bulk is an insulator, and that causes the Vxx tovanish. This is how the plateaus in Rxy and the vanishing Rxx in Figure 2.2.(b) areexplained in IQHE.2.3 The Fractional Quantum Hall Effect And AnyonsThe single-particle picture of non-interacting electrons described above only ex-plains the plateaus in Hall resistance at integer values of the filling factor. An arrayof plateaus at fractional multiples of the quantum resistance, h/e2, have been ob-served, which require a different explanation. Almost all of these fractions haveodd denominators. One exception is the ν = 5/2 state. Of the odd denominatorfractions, most of them belong to a series of the form p/(2p+ 1) where p is aninteger.Shortly after the discovery of the FQHE, Laughlin pointed out [21] that theexcitations in the electronic states correspond to fractionally charged particles. Theanyonic statistics of these states follow immediately from the fractionally chargedparticles. It is worth reviewing Laughlin’s simple and elegant argument carefully.Consider a 2DEG shaped like a disk in the x-y plane with a perpendicular mag-netic field in the z direction with a hole somewhere in the disk through which a thin13and infinitely long solenoid is passed. Assume the system is initially in the groundstate, and that it is in a FQHE state of filling factor ν . If the solenoid is slowlyturned on (slower than all the energy gaps in the system), the adiabatic theorem en-sures that the system will remain in an energy eigenstate. The electrons will feel anelectric field E in the azimuthal direction because of the increasing flux φ throughthe diskE=12pir∂φ∂ tθˆ (2.15)where r is the distance from the centre of the solenoid, and θˆ is the unit vector inazimuthal direction. Since at filling factor νRxy = ρxy =1νhe2(2.16)and the electrons are in a fractional quantum Hall state, the current density i isi=e2νh|E|rˆ (2.17)Integrating the current that goes through any circle of radius r over time givesthe total amount of charge Q(t) that has left the interior of the circle in time tQ(t) =∫ t02pirirdt ′ =∫ t02pire2νh12pir∂φ(t ′)∂ t ′dt ′ =e2νhφ(t) (2.18)When φ(t) is equal to the magnetic flux quantum φ0 = h/e, the charge trans-ferred is equal to eν . The ground state, however, could have evolved from the initialground state to a different ground state due to the extra flux that is now piercingthe system. The extra flux causes the vector potential A, whose various derivativesconstitute the electric and magnetic field, to evolve into A+δA. According to theAharanov-Bohm effect [22], the extra phase picked up by a particle that circledaround the solenoid of magnetic flux φ is1h¯∮eδA ·dl= 2pi φφ0(2.19)where the integral is calculated around a closed loop containing the solenoid. If14the particle does not complete a closed loop, the extra phase is 0. Notice thatif φ(t) = φ0, the system has not picked up any phase compared to the originalground state; only a charge eν has moved from the interior edge of the disk to itsexterior. We call this new state, the solenoid and the transferred charge, a quasi-particle or a quasi-hole, depending on the sign of the charge that was transferred,although we will not be concerned with sign of the charge of these particles. Theexternal magnetic field can play the role of the infinitely long solenoid. When themagnetic field deviates from its value at filling factor ν , it pierces the sample witha magnetic flux quantum. An integer number, j, of quasi-particles can be createdif the magnetic field deviation ∆B from the filling factor ν is∆B = jφ0a(2.20)where a is the surface area of the sample.Note that in this argument, it was assumed that the system has only one groundstate. The adiabatic theorem then guarantees that the system will remain in thatground state, up to a complex phase, if the flux is introduced slowly. The violationof this assumption, as we will see later, is the source of non-Abelian statistics,where the ground state is highly degenerate.Given the charge of each quasi-particle eν , if one quasi-particle encircles an-other one, the Berry phase its wave-function picks up is 2piν [23]; remember thateach quasi-particle has a magnetic flux quantum embedded in it. But winding ofone quasi-particle around another is the same as two consecutive exchange oper-ations. Therefore under the exchange operator of two quasi-particles, the wave-function of the system picks up a phase of piν , which makes the quasi-particlesanyons [23].Since the anyons are defined by the quantum mechanical phase that they ac-cumulate, interferometers have been used to observe their signatures. Fabry-Pe´rot[24] and Mach-Zehnder [25] interferometers have been used to probe the fractionalstatistics of various FQHE states.The theory discussed above is capable of explaining the FQHE at filling factors1/p and their particle-hole inverted states 1−1/p, for an odd integer p, in the low-15est Landau level. There have been attempts to extend this theory into a hierarchyof states [26, 27]. The theory that explains the fractions of the form p/(2p+ 1),which is the form of the most observed fractions, is the composite Fermion theory[28, 29].2.4 Composite FermionsThe composite Fermion theory attempts to map the FQHE state onto IQHE states[28]. Given the observed similarities between the two phenomena, a plateau inHall resistance accompanied by a vanishing longitudinal resistance, the existenceof such a mapping seems reasonable.Figure 2.4: Image adapted from Reference [29] (a) Fermi sea of electronswhen B = 0. (b) The Fermi sea splits into Landau levels separated bythe cyclotron energy h¯ω (ν = 3). (c) At ν = 1 all electrons occupy thefirst filling factor, and the system of non-interacting electrons is highlydegenerate. The green and orange lines indicate that (d) is the energylevels of (c) blown up. (d) The degeneracy is lifted by the interaction be-tween the electrons. They arrange themselves into composite Fermions,and form a Fermi sea of composite Fermions when B∗ = 0. (e) When B∗deviates from 0, the composite Fermions split into Landau levels, likebefore. ν∗ = 1/3 or ν = 7/3.A composite Fermion is defined as the combination of an electron and a vortex.A vortex is an object that produces a Aharanov-Bohm phase of 2pi for a particlecircling a closed loop around it. A more useful way of thinking about compositeFermions is the bound state of an electron and an even number of magnetic flux16quanta, but this picture should not be understood literally; no flux is bound to anelectron. In this picture, electrons bind to 2p magnetic flux quanta, with p aninteger, and turn into composite Fermions, which, unlike electrons in the lowestLandau level, only weakly interact with each other. These composite Fermionsonly feel the remainder of the original magnetic field B∗:B∗ = B−2pnφ0 (2.21)where n is the sheet density of electrons as before. The filling factor ν then trans-forms to ν∗ = nφ0/|B∗|ν =ν∗2pν∗±1 (2.22)The + and − correspond to B∗ parallel and anti-parallel to B respectively.What we gain by doing this transformation is a mean field approximation forthe highly degenerate electrons in ν < 1 filling factor. Now even without consid-ering the interactions between the composite Fermions, the degeneracy is lifted forinteger values of ν∗. With the reduced interaction, we are justified to treat com-posite Fermions as non-interacting particles as a first approximation. They form aFermi sea of their own when B∗ vanishes, or when ν = 1/2p, and they form Lan-dau levels when B∗ does not vanish. See Figure 2.4 for a pictorial representationof the transformation. In this manner, the odd-denominator fractions of the formν =q2pq±1 (2.23)are accounted for. p and q are both integer numbers. The ν = 5/2 requires adifferent explanation.2.5 Non-Abelian States and ν = 5/2So far, we have extended quantum statistics beyond Bosons and Fermions, toanyons, where the exchange operation introduces an arbitrary phase to the wave-function of the system. The next extension gives the non-Abelian anyons, which17originate from the violation of the assumption about the function of the adiabatictheorem. In the discussion of anyons, we stated that if magnetic flux is increasedslowly, the system remains in the ground state. In the case of degenerate groundstates, however, the adiabatic theorem only guarantees that the system will remainin a ground state, if it was originally in a ground state, but might evolve from oneground state to another. The effect of the motion of non-Abelian particles is there-fore not a phase factor, but a unitary transformation that takes the system fromone ground state to another. These unitary transformations can only depend on thetopology of the path taken, and in general, they do not commute with each other,which is why these states are called non-Abelian states.The first FQHE state suspected of having non-Abelian statistics was the ν = 5/2state, as suggested by Moore and Read [30]. The Moore-Read state will be the basisof the analyses and the experiments presented in this thesis. Before describing thisstate, however, it is worth noting that, there exist other candidates for the ν = 5/2state, some predict Abelian statistics: the K = 8 state [31], and the (3, 3, 1) state[32], and some predict non-Abelian statistics: level-2 SU(2) theory or SU(2)2 [33],and Majorana-gapped theories that are relatives of the Moore-Read states, exceptfor different edge-current characteristics [31].At ν = 5/2, the first two spin-polarized Landau levels are completely full, andinert. The third level, however, is half-full. We can now apply the compositeFermion approach, and combine electrons with magnetic flux quanta until the ef-fective magnetic field B∗ is zero. This happens at exactly half-integer filling factors.The composite Fermions then form a Fermi sea, and at a sufficiently low temper-ature, it becomes energetically favourable for them to form Cooper-pairs and con-dense into a superconductor. Since the electrons are spin-polarized in each Landaulevel, the composite Fermions are also spin-polarized, therefore they cannot forms-wave Cooper-pairs [34]. The simplest alternative is a p-wave. The ν = 5/2 state,in this picture, is a p-wave superconductor of Cooper-paired composite Fermions.Now assume the magnetic field is increased by a small amount. In a normalodd-denominator composite Fermion system, that increase produces a hole in anotherwise full Landau level. In the case of ν = 5/2, however, a superconductingvortex accommodates the residual magnetic field. These vortexes are the non-Abelian quasi-particles of the ν = 5/2 state. Each vortex screens half of a magnetic18flux quantum, and since each electron is attached to two flux quanta, the charge ofeach vortex is e/4. Experiments have shown the quasi-particles of the ν = 5/2state carry this fractional charge [35, 36].The vortexes can be thought of as Majorana zero mode states, that can be spa-tially separated [37]. This implies that:1. If there are N vortexes in the system, the degeneracy of the ground state isΩ= 2N2 −1 (2.24)2. The adiabatic motion, or “braiding”, of well-separated vortexes correspondsto a unitary transformation that takes the system from one ground state intoanother one. The destination ground state, up to an Abelian phase factor,depends only on the topology of the braiding of the vortexes with respect toone another.Various experiments with interferometers have been proposed [38–40], and tothe best of my knowledge only one such experiment has actually been carried out[41], to test the braiding effect of the quasi-particles. The results of the experimentwere consistent with the non-Abelian statistics of the Moore-Read state.Another approach to probe the non-Abelian statistics is through the lens of ther-modynamics. The exponential degeneracy of Equation 2.24 implies an enhancedlow-temperature entropy in the system, which can be detected experimentally. Sev-eral proposals have been put forward for probing the non-Abelian entropy of the5/2 state [6, 42–44]. A measurement based on the thermoelectric response of the2DEG near fractional states has been performed, with no conclusive results regard-ing the non-Abelian statistics of the ν = 5/2 state [45].We attempted to probe the low-temperature non-Abelian entropy of the ν =5/2 state. To observe the enhanced entropy of the non-Abelian states, we attemptedto follow the proposal in Reference [6] and measure the quantity ∂µ/∂T near theν = 5/2 state. The details of the measurements are discussed in Chapter 6. Ourresults did not support the non-Abelian theory of the 5/2 state, but for reasons wediscuss in Chapter 6, we believe that the results are inconclusive.Before diving into the details of the measurement described above, however,19we decided to solve a problem that was essential for the reliable observation of theν = 5/2 state. The ν = 5/2 state is a very fragile state. It was discovered severalyears after the discovery of the FQHE, in a high mobility GaAs/AlGaAs SHJ sampleat temperatures well below 100 mK [46]. However cooling a high quality sample tomillikelvin temperatures does not produce consistent mobility and energy gaps inthe sample. Low temperature illumination is often used to optimize the outcome. InChapter 3, we investigate the effect of low-temperature illumination and annealingon QW samples in order to optimize the energy gap at the 5/2 state. The results ofthis investigations were published in Reference [1].We fabricated a SET on the surface of the sample and used it as an ultra-sensitive electrometer to measure the local chemical potential underneath the de-vice and on-chip heaters to modulate the temperature locally. These two devicesare discussed in more details in chapters Chapter 4 and Chapter 5.Finally, in Chapter 7, we will discuss the results of SET measurements near in-teger plateaus where the bulk of the sample is insulating. We discuss the possibilityof the observation of a Wigner crystal (WC) in this region.20Chapter 3Illumination And AnnealingThis chapter describes a recipe for obtaining fully developed FQHE states in QWsamples. Maximizing the activation energy gap at ν = 5/2 was the objective ofthese investigations [1].3.1 Motivation For Large Energy GapsThe FQHE state at filling factor ν = 5/2 has been the subject of extensive experi-mental and theoretical scrutiny since its discovery over 25 years ago [46]. The 5/2state, and its conjugate at ν = 7/2, are unique among more than 80 other observedFQHE states in that they are the only incompressible states to have been observedat even denominator filling factors in a single layer 2DEG. A primary reason forsuch widespread interest in the 5/2 state is the expectation that it may possess non-Abelian quasi-particle statistics.Like many of the more exotic FQHE states, the ν = 5/2 state is very fragile.The incompressible state develops fully only in samples with mobility well above106 cm2/Vs, and only at temperatures below 50 mK. The fragility of the 5/2 statemakes experimental measurements very challenging, and has limited progress inunderstanding its intrinsic microscopic description. The same can be said of otherweak FQHE states, such as the one at ν = 12/5. The 12/5 state may also be non-Abelian [47], but its activation energy gap is even smaller than that of 5/2 state[48]. It is therefore desirable to obtain samples in which FQHE states are more21robust.One route to improving FQHE activation energy gaps is to optimize the MBEgrowth recipe [12, 49, 50]. The highest mobility samples today are GaAs/AlGaAsQWs with silicon δ -doping placed in secondary, extremely narrow, QWs on bothsides of the primary QW. These secondary wells are often referred to as “dopingwells”. This strategy makes it easier for charges to equilibrate among the dopantsfor improved screening. Furthermore, the placement of dopants in separate dopingwells—that is, in a GaAs layer as opposed to AlGaAs/AlAs barrier—increases thechance that the dopant energy levels will be shallow, not far below the chemicalpotential. This increases their ionization fraction dramatically which in turn re-duces the disorder potential due to random ionization states. The sample used inthe present experiments is of this doping well variety.Although crystal growers have made substantial progress in optimizing thequality of the 2DEG samples for FQHE characteristics, it is well known that thegrowth recipe is not the only parameter that affects the activation energy gap [49].It is widely believed that the illumination by light above the band gap of GaAs fa-cilitates redistribution of charge such that random potential fluctuations caused bydisorder are effectively screened. Illumination during the cool-down process fromroom temperature to millikelvins plays an important role, including the timing ofturning the light off. Historically samples have been illuminated down to temper-atures around 4.2 K, but the physical parameters that set this temperature have notbeen studied in detail.3.2 The Optimum RecipeIn this Chapter, we investigate a protocol involving illumination and thermal an-nealing whereby the density, mobility, and FQHE quality of a GaAs doping well2DEG can be optimized after cooldown in a dilution refrigerator. Illumination atmillikelvin temperatures is found initially to lower the carrier density of the 2DEGby more than a factor of 30 during the illumination. Subsequent warming of the2DEG to temperatures above 2 K causes the density to return to its nominal value,and generates superb FQHE characteristics. By optimizing heating parameters, anactivation energy gap ∆5/2 = 600± 10 mK for the ν = 5/2 state is obtained, one22of the highest values reported. The outcome is found to be extremely repeatable:the 2DEG can be brought to the same high quality state cool-down after cool-down.It can even be reset in situ after its mobility and FQHE characteristics are severelydegraded by an electrostatic shock.510 ν = 3ν = 4(a) T = 30± 1mK before treatmentRxy(kΩ)050100150(b) ν = 5/2Rxx(Ω)0510 ν = 3ν = 4(c) T = 30± 1mK after treatmentRxy(kΩ)1 2 3 4 5 6050100150(d)5/2Magnetic Field (T)Rxx(Ω)Figure 3.1: Hall resistance Rxy (a), and longitudinal magnetoresistance Rxx(b), up to filling factor ν = 2 immediately after cooldown in the dark.(c) Rxy and (d) Rxx after illuminating the sample at base temperature for30 minutes, and annealing the sample at 2.25 K for 20 minutes.The QW under study in this work was grown by the doping well strategy de-scribed in Reference [12], yielding a maximum carrier density of 3.1×1011cm−2and low temperature mobility of 1.5× 107 cm2/Vs. Eight indium pads were an-nealed into the 5×5 mm2 sample to establish ohmic contact to the 2DEG; the con-231012T=30mKRxy(kΩ)4min8min20min2.2 2.4 2.6 2.800.51Filling FactorRxx(kΩ)Figure 3.2: The longitudinal and Hall resistance after 30 minutes of shiningred LED light and 4, 8, and 20 minutes of annealing time at 2.25 Kas a function of filling factor. FQHE states are partially developed andreentrant integer quantum Hall states are asymmetric after 4 minutesof annealing. These characteristics improve by more annealing. Thequality reaches its maximum after 20 minutes.tacts are midway along the edges and at the corners (Figure 3.2.(a) inset). The sam-ple was thermally sunk to the gold-plated base of a ceramic chip carrier by meltedindium. Gold bond wires connected each indium contact to chip carrier contacts,and an additional 14 bond wires connected the carrier base to well-cooled measure-ment wires. Two resistors mounted to the back side of the chip carrier were usedfor rapid control of the sample temperature: a metal film resistor (“local heater”)was for heating, while a nearby carbon resistor (“local thermometer”) monitoredthe temperature [51].The sample was cooled in a dilution refrigerator with a base temperature around15 mK, a process that took 6 hours to reach 4.2 K and an additional 6 hours to reachthe base temperature. During each cooldown, the local thermometer was calibratedagainst a RuO2 resistor mounted on the mixing chamber. The same chip was cooled24down from room temperature 4 times, with essentially identical results.A multi-mode optical fibre was used to shine light onto the sample from a redLED located outside the refrigerator. Below-bandgap light was found to have noeffect. With a power of 140 mW applied to the LED (80 mA at 1.7 V), 6± 2µW of optical power reached the bottom of the fibre, as estimated from the 15 mKrise from base temperature in the mixing chamber temperature during illumination.Geometrical considerations suggest an optical intensity of 120±50 nW/mm2 at thelocation of the sample. The data presented here are taken for the sample broughtfrom room temperature without fibre illumination during the cooldown.After arriving at base temperature, and prior to illumination, the longitudinal(Rxx) and Hall (Rxy) magnetoresistance were measured at 30 mK using a lock-inamplifier at 53.4 Hz (Figure 3.1.(a),(b)). The bias current was kept low enough toensure that it was not heating the sample, typically 100 nA below 100 mT and 2 nAat high field. The temperature (30 mK) was selected to provide a repeatable base-line for very low temperature FQHE characteristics, cold enough that most stateswere fully or almost fully developed (Rxx → 0) but warm enough that we couldreturn easily to this temperature without waiting many hours for thermal equilibra-tion.The carrier density determined from the Hall slope was 90% of the nominaldensity, although variations of several percent were common from cool-down tocool-down; the Drude mobility was µ = 1.45±0.05×107 cm2/Vs. FQHE featuresin the ν = 2−4 range were only weakly developed; no reentrant integer quantumHall states were visible (Figure 3.1 (a),(b)) [52, 53].Next, the sample was illuminated for 30 minutes, during which the chip carriertemperature (monitored by the local thermometer) warmed to 600 mK due to theoptical power. The light was turned off, then the sample was annealed using thelocal heater at a temperature between 2.2 K and 6.5 K for a period of time between4 and 48 minutes, before cooling again to 30 mK. Figure 3.1.(c) and (d) show Rxxand Rxy after illumination and 20 minutes of annealing at 2.25 K. Compared tosample characteristics immediately after cool-down in the dark, carrier density andmobility have only slightly increased (µ = 1.49± 0.05× 107 cm2/Vs) but manymore fractional and reentrant states are visible.The improvement of FQHE quality as a function of annealing time was grad-25ual, taking tens of minutes or longer for annealing temperatures below 2.1 K, butless than 4 minutes above 2.5 K. When annealing at 2.25 K, for example, FQHEquality increased with annealing time for the first 10-15 minutes, after which thequality was saturated at its highest level. Figure 3.2 shows longitudinal and Hallresistances at 30 mK between filling factors ν = 2 and 3, for annealing times of 4,8, and 20 minutes at 2.25 K. Before each anneal step, the sample was reset by a30 minute illumination. After only a 4 minute anneal the FQHE features are poorlyformed, but after 8 minutes the features as they appear at 30 mK are close to theirsaturated form, as seen by comparing to the 20 minute data.In order to quantify the improvement in sample quality with annealing temper-ature and time, the carrier density and activation energy gap of the ν = 5/2 state,∆5/2, were recorded for many different anneal parameters (Figure 3.3.(a),(b)). Thecarrier density, ns, was extracted from the magnetic field location of the ν = 5/2minimum by the relation ν = nsh/eB, where h is the Planck constant and e is theelementary charge. The activation energy gap was extracted by fitting the temper-ature dependence of the minimum in Rxx near ν = 5/2 to the Arrhenius formula,Rxx(T ) ∝ exp[−∆5/22kBT], (3.1)for the Boltzmann constant kB, see, for example, Figure 3.3.(a) (inset). Fits wereperformed for sample temperatures between T=60 mK and T=100 mK, controlledby the mixing chamber heater and RuO2. At lower temperatures the value of Rxxwas too low to be measured accurately, while at higher temperatures Rxx no longerfollowed the exponential dependence of equation (Equation 3.1). After each tem-perature change, the resistance of the sample was monitored over time to ensurethat thermal equilibration with the mixing chamber was complete. This took lessthan 2 minutes for T>70 mK, rising to 10 minutes for T = 60 mK.Comparing a 4 minute anneal at several different temperatures, it is interestingto note that the annealing speed increases dramatically as the anneal temperatureincreases from 2.2 to 2.5 K. Above 2.6 K, the process was so fast that full equili-bration was obtained within 4 minutes, the shortest anneal time we performed. Nodegradation in any of the measured sample qualities was observed for anneals up to6.5 K, the highest annealing temperature accessible in this experiment. The 2.25 K26200400600 (a)10 12 14 16−2−101/T (1/K)ln( R xx(T)Rxx(0.1K))4 min20 min5/2Gap(mK)0 4 8 12 16 20 24 28 32 36 40 44 483030.53131.5 (b)Annealing Time (min)Density(×1010/cm2)2.20± 0.06K2.25± 0.06K2.40± 0.06K2.50± 0.06K2.67± 0.06K0 20 40 60 80 100 120 140024(c)Magnetic Field (mT)Rxx(Ω)Figure 3.3: ∆5/2 (a) and carrier density (b) increase gradually with annealingtime, at a rate that increases dramatically with annealing temperature.Before each data point, the sample was illuminated for 30 minutes. Insetto (a) is example Arrhenius plot for two annealing times at 2.25 K. (c)4 minutes of annealing at 2.20 K yields Shubnikov-de Haas oscillationswith strong beats, suggesting inhomogeneous carrier density. Since theexperiments were intended for high magnetic fields, weak paramagneticmaterials were not necessarily avoided, which could explain the strangebehaviour near zero field.27data in Figure 3.3 also indicate that very long anneals do not degrade the ν = 5/2gap, or change the carrier density beyond its saturated value.Carrier density, ns, equilibrates on a timescale similar to the evolution of ∆5/2.On the other hand, the carrier density increased by only 3% from 4-minute to 20-minute anneal times at 2.25 K, whereas ∆5/2 increased by nearly a factor of 3.Apparently, the improvement in ∆5/2 for longer anneals is not simply the result ofhigher densities, and not the result of higher Drude mobility (the observed increasefrom 1.45 to 1.49×107cm2/Vs is within the measurement error). Below, we spec-ulate that the higher reported ∆5/2 values may instead reflect improvements in thesample homogeneity.The values for ∆5/2 reported in Figure 3.3 are simply results of a fit to equation(Equation 3.1), and may not represent a homogeneous gap across the sample. Al-though the fits to Equation 3.1 were always reasonable over the range 60-100 mK,the shape of the dip in Rxx(B) near ν = 5/2 was often asymmetric when annealingtimes were insufficient—not simply less deep as one might expect for a weakly-gapped state. Misshapen features in Rxx may indicate a spatially varying chargedensity, such that macroscopic measurements of resistivity represent averages overmany regions in the sample, each with a slightly different filling factor. This wouldbe consistent with the strong beats observed in Shubnikov de Haas oscillationsfor short annealing times (Figure 3.3.(c)), and provides some evidence that longerannealing times enhance the density homogeneity.A microscopic description of the process by which dopants equilibrate to screenpotential fluctuations has not, to our knowledge, been worked out. However, theextreme temperature dependence of the annealing rate may indicate a thermallyactivated diffusion process, where charges hop between localized sites in or nearthe doping well with a diffusion constant D ∝ T · exp[−∆H/kBT ], where ∆H is theenergy barrier to hopping. Noting that, in Figure 3.3(a),(b), the values are similarfor [4 minutes, 2.5 K] and [8 minutes, 2.25 K], and also for [4 minutes, 2.4 K]and [8 minutes, 2.2 K], we can estimate D(T = 2.5 K) ≈ 2D(T = 2.25 K), andD(T = 2.4 K)≈ 2D(T = 2.2 K), giving ∆H between 13 and 16 K.280 200 400 600 800 1,0001234(a)Time (min)Density(×1010cm−2)0 0.5 1 1.5 2 2.5 3 3.5020406080ν = 1ν = 2/3ν = 2ν = 1/3(b)Magnetic Field (T)Rxy(kΩ)After Illumination35 minutes later100 minutes later200 minutes laterFigure 3.4: (a) After illumination is switched off, the density increases overtime (T = 30 mK). Illumination was switched off after the first pointbut before the second point on the graph. (b) Evolution of Hall tracesafter illumination at base temperature, with no subsequent annealing.The first trace (dashed red line) begins 5 minutes after illumination wasswitched off.3.3 The Low Density RegimeAlthough the carrier density was within a few percent of its maximum value af-ter only 4 minutes of annealing (Figure 3.3.(b)), the carrier density immediatelyafter illumination was a factor of 30 lower. Similar effects have previously beenobserved in GaAs heterostructures [54]. In our experiment, the density remainedvery low (less than 4.4× 1010/cm2) for many hours. The time dependence of the29increase in density after illumination was monitored using the slope of the Hallresistance near zero field, and is shown in Figure 3.4.(a).Figure 3.4.(b) shows four consecutive Hall resistance traces, taken after 30minutes of illumination with no annealing. The first trace was begun a few minutesafter illumination was switched off (the sample had returned to less than 50 mKby this time), then subsequent traces were taken over the next three hours. Notethat ν = 1 occurs for B ∼ 1 T in Figure 3.4, compared to 13 T where the ν =1 would appear for annealed samples in Figure 3.1.(c). The Drude mobility, asdetermined by the low field Rxx, was 4×106 cm2/Vs for the last trace when densitywas 4.4×1010/cm2. Several of the ν = 1/3 ladder of fractional states can be clearlyseen, increasing in strength for the later traces.3.4 ConclusionIn conclusion, we found that optimal FQHE characteristics of a doping well 2DEGcan be obtained by illumination at low temperature, followed by annealing at tem-peratures between 2.2 K and 6.5 K. The time scale for annealing depends sensi-tively on the temperature, but too-long anneals appear not to degrade FQHE charac-teristics. Illumination can also be used to reset the sample to a well-defined initialstate with very low density. The FQHE characteristics in this initial state are verypoorly developed (Figure 3.4), but gradually improve with annealing. This tech-nique can be used to compare FQHE characteristics of QW samples grown underdifferent MBE protocols, without having to rely on a precise cooldown recipe thatis inherently difficult to control.30Chapter 4The Single Electron TransistorSET devices were used to measure the local chemical potential of the 2DEG inexperiments described in the following Chapters. In this Chapter, we go over someof the basics of the operation principles of SETs, as well as the results of some ofthe basic measurements on a 2DEG in the quantum Hall regime.4.1 The Principles of OperationSETs are commonly used as extremely sensitive charge sensors in semiconduc-tor physics [7, 36, 55]. Their charge sensitivity has been shown to be a few10−5e/√Hz. That means a charge variation equal to 10−5 of the charge of oneelectron in a measurement time of about 1 second can be detected. This level ofprecision is due to the fact that SETs exploit the quantum mechanical phenomenonof tunnelling through a metal-insulator-metal junction. The idea for the fabricationof SETs was first proposed in 1985 [56], and was first realized and measured in Belllabs in 1987 [57].SETs are normally made of a metal source, drain, and island, which are sep-arated by insulating tunnel junctions, made of metal-oxides (see Figure 4.1.(a)).The current flows from the source to the drain by tunnelling through the insulatingtunnel junctions. The junctions have an effective resistance, RT , which depends onthe tunnelling transmission coefficient, T , of the electrons’ wave-function whichvanishes exponentially with the thickness of the junction, and the number of inde-31Figure 4.1: (a) A cartoon image of an SET. A voltage Vg is applied to thesource. The current, that goes from the source through the tunnel junc-tions and the island is measured between the drain and the ground. Theelectrons have to tunnel through the tunnel junctions, and the conduc-tivity between the source and the drain is modulated by the gate voltage.In the sample investigated in this thesis, source, drain, and the island aremade of aluminum. The tunnel junctions are aluminum oxide, the gateis the 2DEG, and the insulating layer between the island and the gate is180 nm of GaAs that sits between the SET and the 2DEG. (b) Coulombblockade oscillations. G= ∂ I/∂Vsd is the differential conductance. ∆Vgcorresponds to one extra electron sitting on the island. (c) SEM mono-graph of an SET. The island is the horizontal line which is about 5 µmlong, and the intersections between the island and each lead is 80×80nm2.32pendent wave modes, M = A j/λ 2F , penetrating the barrier. Here A j is the area ofthe junction, and λF is the Fermi wavelength of the electrons [58].If T M  1, the charge passing through the junctions is quantized. Thereforeto avoid the effect of quantum mechanical fluctuations on the conductance of thetransistor, RT has to be larger than the quantum of resistance (also known as thevon Klitzing constant) RK = h/e2 ≈ 25.8 kΩ. A typical tunnel junction resistancefor our SETs was 100-150 kΩ.The island is coupled to a gate via an insulating layer through which no tun-nelling is allowed. The island is also capacitively coupled with the source and thedrain. The total capacitance of the island is therefore CΣ =Cg + 2CJ , where CJ isthe capacitance between the island and the source or drain through the junction,and Cg is the capacitance between the gate and the island. The charging energy,the energy it takes to add one extra electron to the island, is EC = e2/CΣ. If thesize of the island is sufficiently small, the charging energy can become larger thanthe thermal fluctuations of the system, EC  kBT . A typical charging energy foran SET is between 0.5-2.5 K. At temperatures well below the charging energy, theconductance through the junctions will be suppressed by the Coulomb blockade ef-fect [59], unless the voltage applied between the source and the drain, Vsd is largerthan the energy gap between the highest occupied electronic state and the lowestunoccupied state on the island. The electrostatic potential of the island can thenbe tuned by changing the gate voltage Vg, to allow the passage of current. In thisregime, the conductance through the SET oscillates as Vg is increased and one avail-able electronic state moves between the electrostatic potential of the source and thedrain. The high sensitivity of the SET current to gate voltage is what makes SETa very sensitive electrometer. In Figure 4.1.(b), each oscillation in the differentialconductance G = ∂ I/∂Vsd corresponds to one extra electron sitting on the islandof the SET. The steep slope of G is its most sensitive region to variations of Vg.4.2 Characterizing The SETIn our sample, the SET simply sits on the surface of the sample, and the 2DEG actsas the gate, which is 180 nm below the surface. Therefore the insulating layer isGaAs that is between the island and the 2DEG. Figure 4.1.(b) is an SEM image of33an SET used in measurements presented in this thesis. The source, the drain, andthe island in the SETs are made of aluminum. The tunnel barriers are thin layersof aluminum oxide that separate the source and the island, and the island and thedrain.The SET is fabricated using electron-beam lithography, and the double-angleevaporation technique [60]. In this technique, aluminum is grown on the sampleby thermal evaporation while placing the sample in the line of sight. Then oxygenis gently introduced into the evaporation chamber in order to create a thin layer ofaluminum oxide. Then the angle between the line normal to sample relative to theevaporation source is changed, and another layer of aluminum is evaporated on topof the oxide layer [55]. See Section A.2 for more details on the fabrication process.1.Figure 4.2.(a) shows a circuit diagram used for the characterization measure-ment of the SET. Vg and Vsd are tuned and differential conductance, G = ∂ I/∂Vsdis measured using a lock-in amplifier. Figure 4.2.(b) is the result of such a mea-surement. There are a couple of features in this figure that are worth mentioning:1. The dark regions, commonly referred to as Coulomb blockade diamonds,are the regions where the conductance through the island is blocked by theCoulomb blockade effect. The charging energy is half the width of the di-amond: EC = e∆Vsd/2 = 43µeV where ∆Vsd is the width of the red region[61].2. The shape of the Coulomb diamonds is not symmetric, which suggests thatthe source and the drain junctions have different capacitances.3. There are discontinuities in the measurement that seem random, for exam-ple at Vg ≈ 525 and 625µV. The location of these discontinuities is not afunction of gate voltage or the bias voltage, or any other controllable pa-rameter of the measurement [62]; they appear on an average time-scale of1Even though SETs were successfully fabricated in our lab (see Section A.2), since the double-angle evaporation and the controlled introduction of oxygen were done using hand-made apparatus,the yield for working SETs were low. The SETs that were used for the measurements presented herewere fabricated by Professor Andrew Houck’s group in the department of electrical engineering atPrinceton University.34Figure 4.2: (a) The circuit diagram used for characterizing the SET. A DCgate voltage Vg is applied between the 2DEG and the ground, and Vsd isapplied between the source and the ground. An AC voltage is appliedbetween the source and the ground using a lock-in amplifier at a certainfrequency. The differential current between the drain and the ground ismeasured at the same frequency. (b) A 2-dimensional measurement ofdifferential conductance G as a function of gate voltage Vg and source-drain voltage Vsd at B = 5 T and mixing chamber temperature below 15mK.35once every several minutes, independent of other measurement parameters.We suspect they are due to random charge fluctuations in the donor layerof the QW. Since a typical scan over a Coulomb blockade peak takes lessthan one minute, these random jumps do not cause a serious problem to ourmeasurements, even though they make it hard to have any absolute senseof chemical potential over large regions of the parameter space, specially ifthe system passes through an insulating state where the SET signal does notfollow Coulomb blockade oscillations. In other words, after we scan over aninteger, we don’t know if a random jump has occured, or the system’s chem-ical potential was shifted by a certain amount, because during the passage,the SET was insensitive to the sample’s chemical potential.4. The centre of the Coulomb blockade diamonds is not at Vsd = 0. We wantto operate close to zero DC bias through the SET because 1) a non-zero biascan heat up the SET since there is more DC bias running through the SETas Vg is scanned, and 2) the SET is more sensitive to gate voltage variationsalong the vertical line that goes through the centre of the diamonds; this linecorresponds to zero DC bias through the SET. Therefore in all the measure-ments that follow, we applied a DC voltage bias to the source to cancel thisundesired current. Looking at Figure 4.2.(b), we set Vsd =−40µV.5. Aluminum is a superconductor below 1.2 K and 10 mT [63]. All SET mea-surements are performed above 100 mT, in order to destroy aluminum’s su-perconductivity.The differential conductance measurements were performed using a lock-inamplifier at frequencies between 70 - 250 Hz, and amplitude of a few µV to avoidheating while maximizing the measurement sensitivity.It is important to emphasize that the SET is an electric charge sensor; it can beused to measure the electrostatic potential on the gate, which can be changed byeither applying a voltage to the gate directly, or by manipulating the environmentin such a way that the electrostatic potential of the gate is changed. Examples ofenvironmental changes relevant to our discussions include changing the magneticfield, or the temperature. A change in the magnetic field changes the chemical36potential of the 2DEG according to Equation 2.14, and since the 2DEG is kept at aconstant voltage, electric current can flow in or out of it, which then changes theelectrostatic potential under the SET. Since the SET experiences the same field, thechemical potential of the body of the SET should not be affected for a meaningfulmeasurement. This is the case because the SET is made of bulk aluminum in non-superconducting state [7]. A change in temperature can broaden the width of theCoulomb blockade peaks of Figure 4.1.(b). How much this broadening affects themeasurements will be discussed later when temperature modulation is investigated.See Section B.4 for a more detailed discussion about what the SET measures.The gate voltage Vg in the circuit diagram of Figure 4.2.(a) only works as a gateto the SET if the bulk of the sample is conducting. If the bulk is insulating, whichis the case for integer filling factors, the voltage applied to the ohmic contact onthe left does not change the potential close to the SET. The top plot of Figure 4.3 isRxx and Rxy covering filling factors 3 to 5, while the bottom plot is a 2-dimensionalgraph of G as a function of the magnetic field and Vg. Notice that:1. The SET signal looks like noise when the magnetic field is near the centre ofan integer filling factor. This is because the potential right beneath the SETdoes does not equilibrate with Vg applied to an ohmic contact far away fromthe SET. The bulk is insulating. This noisy region will be investigated morecarefully in Chapter 7.2. Far from integer filling factors, the SET signal oscillates as a function of themagnetic field and Vg. The slopes of the lines on the B-Vg plane can beunderstood by the dependence of the chemical potential of the Landau levelsas a function of magnetic field. See Equation 2.14 and Figure 2.3.37345 Filling Factor−2040100160Rxx(Ω)2.5 3.0 3.5 4.0 4.5Magnetic Field (T)0100200Vg(µV)46810Rxy(kΩ)2.03.6G(µA/V)Figure 4.3: Top is Rxx (green trace on the left axis) and Rxy (red trace on theright axis) as a function of the magnetic field (bottom axis) or fillingfactor (top axis). The bottom image is the differential conductance Gof the SET over the same magnetic field, or filling factor, as a functionof the gate voltage Vg. When the bulk is conducting, the slopes of thelines follow Equation 2.14 and Figure 2.3. When the bulk is insulating,near integer values of the filling factor, the SET does not respond to thechange in Vg.38Chapter 5Characteristics of The HeaterIn order to benefit from low noise measurements offered by a lock-in amplifier, wedeveloped a technique for modulating the temperature of the sample at a frequencyin the range of 1-100 Hz. This was achieved by fabricating on-chip heaters. Twotypes of heaters were used:1. A thin and long aluminum wire (70 nm thick, 3-5 µm wide, and 10-12 mmlong), thermally evaporated on the surface of the sample. In the magneticfields of interest, aluminum is in its non-superconducting state, and behavesas a normal Joule heater. These wires had a few kΩ of electrical resistance.2. The 2DEG was used as a heater. A current runs from one ohmic contact tothe other, and heat is dissipated in the 2DEG. This technique was used in thefinal measurements, and in all of the results presented in this thesis, unlessotherwise specified. See Figure 5.1 for a 3-dimensional cartoon representa-tion of the sample. Two heaters are placed on two sides of the Hall bar toensure a more uniform distribution of heat through the Hall bar. For heating,a voltage is applied to one side of the heater, while the other side is grounded.I worked with two different dilution fridges for my experiments: a “wet-fridge”made by Oxford InstrumentsTM [64], and a “dry-fridge”, or cryo-free fridge, madeby BlueFors CryogenicsTM [65]. The wet-fridge was an older system, with a heav-ily customized and tested cold-finger, which achieved low electronic temperatures,but was much more tedious to operate. The dry-fridge was a newer system which39Figure 5.1: (a) A 3D model of the sample equipped with an SET and twoheaters on both sides of the Hall bar. (b) A cartoon representation of thesample, used in the future for simplicity.was effortless to operate, but the electronic filters still needed work in order toachieve temperatures close to the mixing chamber temperature1.5.1 ThermometryTo measure the temperature of the electrons in the 2DEG, we used the shape of thepeak in Rxx near the IQHE plateau at ν = 4. The peak, depicted in Figure 5.2.(a),moves towards lower magnetic fields as the temperature of the mixing chamberis raised. The temperature of the mixing chamber is controlled using a heaterattached to the mixing chamber, and a proportional-integral-derivative (PID) digitaltemperature controller. The temperature of the mixing chamber is measured usinga RuO2 resistor [66]. The field at which Rxx crosses 200Ω, the black dashed linein Figure 5.2.(a), are plotted in Figure 5.2.(b). A line is fitted through the pointsthat correspond to mixing chamber temperatures above 30 mK (green line). Thepoint at which this line crosses the red horizontal line which specifies the field ofthe lowest temperature point is taken to be the lowest temperature achieved on the2DEG. This is based on the assumption that the curve keeps moving linearly at1See Section A.3 for more details on filters used in both fridges.404.40 4.45 4.50 4.55 4.60 4.65Magnetic Field (T)0100200300400500600700Rxx(Ω)(a)13.0 mK15.1 mK20.0 mK30.0 mK40.0 mK50.0 mK60.0 mK69.9 mK80.0 mK89.9 mK99.9 mK20 40 60 80 100Mixing Chamber Temperature (mK)4.484.494.504.514.524.534.544.554.56MagneticField(T)(b)Figure 5.2: (a) Rxx near ν = 4 is used as an on-chip thermometer which mea-sures the electronic temperature of the 2DEG directly. (b) The field atwhich the curve in (a) crosses Rxx = 200Ω is plotted and used to estimatethe base temperature of the 2DEG to be 21.5 ± 2 mK.lower and lower temperatures, but after a certain point, the electronic temperatureof the 2DEG is not in equilibrium with the temperature of the mixing chamber.For the data shown in Figure 5.2, it was found that the base temperature of the2DEG is 21.5± 2 mK. This number can be different from cool-down to cool-down,and the wet-fridge generally gave lower electronic temperatures. Base tempera-tures as high as 50 mK were observed on the dry-fridge; 20 mK was the lowestobserved so far.5.2 DC HeatingThe simplest way to test the effect of heating on the sample is to apply a DC voltageto the heaters, and measure Rxx at various voltages. In Figure 5.3.(a), the resultsshown in Figure 5.2.(a) are repeated for comparison. The magnetic field is then setto 4.55 T (dashed line), and a DC voltage is applied to the heaters using a digital-414.40 4.45 4.50 4.55 4.60 4.65Magnetic Field (T)0100200300400500600700Rxx(Ω)(a)13.0 mK15.1 mK20.0 mK30.0 mK40.0 mK50.0 mK60.0 mK69.9 mK80.0 mK89.9 mK99.9 mK−30 −20 −10 0 10 20 30DC Voltage (mV)0100200300400500600700Rxx(Ω)(b) B = 4.55 TFigure 5.3: (a) Rxx near ν = 4 is used as an on-chip thermometer which mea-sures the electronic temperature of the 2DEG directly. (b) A DC voltageis applied to the heater at B=4.55 Tto-analog (DAC) voltage supply. Rxx is measured using a lock-in amplifier at 225Hz and is shown in Figure 5.3.(b). The y axes of (a) and (b) have the same scales,making the comparison easier.In Figure 5.3.(b), notice that when the applied voltage is zero, the value of Rxxis the same as the point where the dashed line crosses the lowest temperature curve.The curve is perfectly symmetric on both sides of 0 which implies that applyinga positive voltage to the heater has exactly the same effect on the temperature asapplying a negative voltage. This is not surprising because the power dissipated ina resistor is P =V 2/R. The symmetry around zero also implies that the time delaybetween when voltage is applied to the heater and when temperature is changed inthe 2DEG is much shorter than the time difference between two consecutive datapoints, which is about 300 ms. In fact Figure 5.3.(b) shows two plots on top ofeach other corresponding to scanning the voltage from positive to negative (blue)and from negative to positive (green). The two plots are almost perfectly on top ofeach other and only one curve is visible.42Figure 5.4: (a) The conductance of the heater as a function of the magneticfield between the filling factors ν = 2 and 3 is measured using a lock-inamplifier. (b) I-V curves of the heater at various magnetic fields. A DCvoltage is applied to one side of the heater and a DC current is measuredon the other side.As the voltage deviates from zero, Rxx increases, corresponding to warmercurves of Figure 5.3.(a), until a maximum is reached, which presumably corre-sponds to the 80 mK curve. After this point, increasing the temperature, causes Rxxto decrease.When a 2DEG is used as a heater, its resistance changes as the magnetic fieldchanges. Roughly speaking, the resistance of the heater is proportional to Rxx. InFigure 5.4.(a), the conductance of the heater is measured using a lock-in betweenthe filling factors ν = 2 and 3. Around ν = 5/2 or B= 5.2 T, the conductance doesnot fluctuate by more than a factor of 2. Figure 5.4.(b) is the I-V characteristicsof the heater at various magnetic fields. Again around ν = 5/2, the I-V curves arelinear with roughly similar slopes.Therefore, to modulate the temperature around ν = 5/2 at a constant powerP, a voltage V is applied to the heater, and a current I is measured. Every timethe magnetic field is changed, V is adjusted such that the product P = V I is keptconstant. As long as the field is not near an integer plateau, which is the case for43ν=5/2, this technique works smoothly.5.3 AC HeatingIn order to benefit from the precision offered by the lock-in amplifier’s technology,an AC voltage was used to modulate the temperature of the sample. Of courseAC signals introduce their own set of complications, which will be discussed inthis section. The lock-in amplifiers used for all of the experiments discussed herewere model SR830 [67] made by Stanford Research Systems. In the followingdiscussions, some of the features specific to this model will be referenced, but theprinciples discussed should be applicable to all modern digital lock-in amplifiers.A lock-in amplifier, LA1, was used to apply a sinusoidal voltage V with rootmean square (RMS) amplitude V0 and frequency f1 to the heaterV =√2V0 sin(2pi f1t) (5.1)If the resistance of the heater R is constant, which is the case at constant magneticfield, the power P applied to the heater isP =V 2R=2V 20 sin2(2pi f1t)R=V 20R[1− cos(2pi(2 f1)t)] (5.2)As discussed before, the change in temperature follows the change in the appliedpower. If the effect of temperature on some other quantity, say the transport quan-tity Rxx, is desired, another lock-in amplifier, LA2, can be used to measure Rxx ata different frequency f2 which should be chosen to be much larger than f1. Thevalue measured by LA2 can be read in several ways: it is displayed on the machine,it can be digitally transferred to a computer using a communication port, or it canbe measured on an output channel as a voltage. This voltage is updated at a highfrequency of 256 kHz [67, page 4-13], although not every value is independent ofthe previous value, and its magnitude Vout is equal toVout =Vs×10Vsens(5.3)444.40 4.42 4.44 4.46 4.48 4.50 4.52 4.54 4.56Magnetic Field (T)−40−20020406080∆Rxx(Ω)2nW1nW700pW400pW300pW200pWFigure 5.5: The change AC heating causes on Rxx as a function of magneticfield at various powers. The power is kept constant throughout the mag-netic field sweep by reading the output current from the heater, andadjusting the modulating voltage accordingly. The signals shown arethe negative of the out-of-phase signal from the lock-in that modulatesthe heater, detecting at twice the frequency relative to its reference fre-quency.where sens is the sensitivity range of LA2.Now if this output is connected to the input port of LA1, then the effect oftemperature modulation on Rxx can be measured using LA1. Care should be takenwhen setting the parameters of the lock-ins LA1 and LA2, and interpreting theresults.1. From Equation 5.2, notice that the frequency at which the power, and hencethe temperature, is oscillating is 2 f1, and not f1. Therefore LA1 should be setto detect the signal at the second harmonic compared to its reference signal.2. Also notice that the sin output voltage of LA1 in Equation 5.1 has turned intoa cos function in Equation 5.2 with a negative sign. Therefore the relevantsignal to be read from LA1 is the negative of the out-of-phase signal. Thetime independent term will be ignored by LA1.3. It takes LA2 some time to produce a truly new value on its output. This time450 5 10 15 20f1 (Hz)−80−70−60−50−40−30−20−100∆Rxx(Ω)Figure 5.6: The change AC heating causes on Rxx as a function of magneticfield at various powers. The power is kept constant throughout the mag-netic field sweep by reading the output current from the heater, andadjusting the modulating voltage accordingly. The signals shown arethe negative of the out-of-phase signal from the lock-in that modulatesthe heater, detecting at twice the frequency relative to its reference fre-quency.depends on its time-constant and the slope of the low-pass filter. See [67,page 3-21].As a concrete example, the following settings on the lock-in amplifiers arereasonable: LA1 has reference frequency of f1 = 7 Hz and is set to detect thesecond harmonic at 14 Hz. Its time constant is 100 ms and the low-pass filter has aslope of 18 dB/oct, and the negative of its out-of-phase signal is detected. LA2 hasreference frequency of f2 = 225 Hz, and it is set to detect the signal at the samefrequency, with a time-constant of 3 ms and the low-pass filter slope of 18 db/oct.The results of such a measurement are shown in Figure 5.5. Compare the changein Rxx with the values in Figure 5.3.(a).The next question to answer is, how fast the temperature can be modulated.To answer this question, the magnetic field was set to 4.5 T, where the effect ofthe temperature modulation has a peak. At 2 nW, ∆Rxx is measured as before fordifferent frequencies f1 of LA1. The results are shown in Figure 5.6. Rememberthe actual frequency at which the temperature oscillates is 2 f1. Above f1 = 12 Hz,the effect of temperature modulation on Rxx reduces to half of its DC-limit value.46f1 = 7 Hz is chosen for the rest of the measurements.It is worth mentioning that in a similar measurement, performed on a differentQW sample in the wet-fridge, the frequency response did not collapse appreciablyup to a few hundred Hz. This difference can be explained by poor thermal contactsbetween the sample and the mixing chamber. The sample takes longer to cool,after it is heated up by the heater, and therefore the amplitude of the oscillation insample’s temperature deteriorates at higher frequencies because of this cooling lag.In chapter Chapter 6, we use this technique to modulate the temperature ofthe sample, but instead of measuring the change in Rxx, we look at the change inchemical potential in the sample’s bulk, by looking at the differential conductanceof an SET.47Chapter 6Signatures of Non-Abelian StatesDuring the last 3 decades, it has been shown that composite quasiparticles con-fined in a 2 dimensional space can behave in ways that are compatible with neitherBosonic nor Fermionic statistics [24, 25, 68]. For start, the exchange of two suchquasiparticles can multiply the wavefunction by a phase factor that does not haveto be pi or 2pi , for fermions or bosons; it can be any number, and hence these par-ticles are called anyons [23, 69]. The more interesting case, however, belongs tonon-Abelian quasiparticles where the exchange operation shifts the system fromone ground state to a different ground state [30, 33], but conclusive experimentalevidence has not been obtained to support the existence of such states.6.1 The Moore and Read StateFor a system to possess non-Abelian quasiparticles, an energy gap must separatethe ground state from excited states, the ground state must be degenerate with adegeneracy that grows exponentially with the number of quasiparticles, the degen-eracy must be insensitive to small external perturbations, and the transformationfrom one ground state to another should be determined only by the topology of thequasiparticles’ trajectories [70]. Topologically protected degenerate ground statesthat are insensitive to external perturbations make a promising platform for a topo-logical quantum computer. Finding a system with non-Abelian quasiparticles in alaboratory is therefore highly desirable.48The Moore and Read state [30] is the primary candidate for describing theground-state of the quasiparticles of the fractional quantum Hall state at the fillingfactor ν = 5/2, which possesses non-Abelian statistics. It has been proposed [6]that the signature of non-Abelian quasiparticles of ν = 5/2 can be found in theircontribution to low-temperature entropy of the 2DEG. More specifically, in theMoore and Read state, the multiplicity of the ground state isΩ= 2Nqp/2−1 (6.1)where Nqp is the number of quasiparticles. The Moore and Read quasiparticles ofthe 5/2 state contribute sMR to the entropy densitysMR = kB(4∣∣∣∣n− 5n02∣∣∣∣) 12 ln(2) (6.2)where n is the electron number density and n0 = eB/h . Using a Maxwell relation1,the differential of chemical potential µMR as a function of temperature T is obtained(∂µMR∂T)n,B=−dsMRdnqp(∂nqp∂n)=∓kB2ln(2) (6.3)which is a quantity that can be measured in a laboratory, and can reveal the possi-bility of an enhanced entropy [6]. In the vicinity of the 5/2 state, this quantity isdiscontinuous as a function of the magnetic field. In Equation 6.3, − and + cor-respond to ν larger than 5/2 for quasiparticles and smaller than 5/2 for quasiholesrespectively; in this picture, there are no quasiparticles when ν = 5/2.Therefore by modulating the temperature of the 2DEG by a small amount ∆T ,and measuring the corresponding change in the chemical potential ∆µ , the ratio∆µ/∆T can be obtained which then gives a measure of ∂µ/∂T . This quantityis expected to be discontinuous as the magnetic field is scanned over ν = 5/2.Modulating the temperature by 10 mK, should change ∂µ/∂T from −1.4kB× 10mK ≈ -1 µeV, to +1 µeV as ν = 5/2 is crossed over.1See Appendix B for derivation and discussions.496.2 Experimental SetupTo make this measurement, we fabricated a device on a GaAs/AlGaAs QW (Fig-ure 5.1), equipped with a Hall bar, a pair of on-chip heaters for modulating thetemperature, An SET [56] was implanted on top of the Hall bar, in Professor An-drew Houck’s group2, which can measure the electronic density of the 2DEG rightbelow the island of the SET [7, 36]. The conductance through the tunnel junctionsof the SET is extremely sensitive to the amount of electric charge that is sittingon the island, which is capacitively coupled to the area of the 2DEG right belowit. The chemical potential of the 2DEG, µ , is the voltage drop between the 2DEGand the ground3. By keeping µ constant, any change in the system that affects itsenergy causes charge carriers to flow in or out of the 2DEG, which changes theelectronic density and can be detected by the SET. We can then adjust µ to returnto the previously measured density. The magnitude of the adjustment then gives usa measure of the change in the chemical potential at constant density.The QW under study in this work was grown by the doping well strategy de-scribed in reference [12], yielding a maximum carrier density of 3.1× 1015/m2,180 nm below the surface. Using electron-microscope lithography techniques,100×100µm ohmic contacts, made of nickel, gold, and germanium (green squaresin Figure 5.1) were placed on the surface, and annealed in order to make con-tact with the 2DEG. The borders of the Hall bar and the heaters were definedusing e-beam lithography and wet-etching. SET contacts (yellow squares), made ofchromium and gold capped with 10 nm of palladium, were placed on the samplein the next e-beam stage. The SETs were then fabricated using e-beam lithographyand double-angle evaporation of aluminum [55, 60]. The purpose of the palladiumlayer was to avoid the formation of gold-aluminum alloy known to have unreliableelectronic characteristics [71]. Gold wires were used to bond a chip-carrier to theohmic and SET contacts.The sample was cooled in a cryogen-free dilution refrigerator with a base tem-perature of 13 mK. Using a lock-in amplifier, we applied 2nA of AC current be-2The fabrication was done by Neereja Sundaresan and Mattias V. Fitzpatrick, two graduate stu-dents at Professor Andrew Houck’s group.3For a more detailed discussion on the meaning of chemical potential, see Section B.4.50tween the two ohmic contacts on the far sides of the Hall bar, and measured thelongitudinal voltage between two other ohmic contacts between the filling factorsν = 2 and 3, with the results shown in Figure 6.1.(a). The peak in the longitudinalvoltage on the side of the ν = 3 plateau was used as an on-chip thermometer, as inSection 5.2.A second lock-in amplifier was then used to apply an AC voltage to the ohmiccontacts on the top and the bottom rectangles, which were used as on-chip heaters,and the difference in longitudinal voltage due to the modulation of temperaturewas measured using this lock-in, as in Section 5.3. In Figure 6.1.(c), we can seethat applying 2 nW of power modulates the longitudinal voltage at 4.46 T by about-0.08 µV. Looking at Figure 6.1.(b), this is of the order of 10 mK difference intemperature. To detect the signatures of non-Abelian states, 10 mK of oscillationsin temperature is enough to modulate the chemical potential by 1 µV in the 2DEG(see the discussions below Equation 6.3). As we show below, 1 µV of oscillationsin chemical potential is detectable using the SET.The SET used in this experiment is shown in Figure 4.1.(c), and is fabricatedon top of a Hall bar. The long horizontal line is the island of the SET which iscapacitively coupled to the electrons in the 2DEG. The area of the island is 5 µm×80 nm, and it is sitting on the surface which is 180 nm above the 2DEG layer. Thecapacitance between the island and the 2DEG is then estimated to be Cg = 0.3 fF. Atthe base temperature, the differential conductance G of the SET between the sourceand the drain is measured using a lock-in amplifier at 223 Hz (Figure 6.2.(a)). InFigure 6.2.(b) a DC voltage Vg is applied to the 2DEG, which acts as a gate voltagefor the SET. The observed Coulomb blockade oscillation has a period of ∆Vg = 210µV which corresponds to the potential difference on the 2DEG that allows for oneextra electron to sit on the island of the SET. We can then estimate the capacitancebetween the 2DEG and the island to be Cg = 0.75 fF, which is in order of magnitudeagreement with our previous estimate based on geometry.We also confirmed that the SET is capable of detecting changes in the 2DEG’schemical potential that are much smaller than 1 µV, the expected signal associatedwith non-Abelian entropy with a temperature rise of 10 mK. In Figure 6.2.(c) the2DEG voltage was modulated on top of the DC voltage Vg with amplitude of 1 µVat 14 Hz, and the change of G was detected at that frequency, using a second lock-514.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0Magnetic Field (T)0.00.51.01.52.0Vxx(µV)(a) Mixing Chamber Temperature=13 mKν=5/24.44 4.45 4.46 4.47 4.48 4.49 4.50 4.51 4.52 4.53Magnetic Field (T)0.00.20.40.60.81.0Vxx(µV)(b)15 mK19 mK29 mK39 mK50 mK59 mK70 mK79 mK89 mK99 mK110 mK119 mK4.40 4.42 4.44 4.46 4.48 4.50 4.52 4.54 4.56Magnetic Field (T)−0.08−0.06−0.04−0.020.000.020.04∆Vxx(µV)(c)2nW1nW700pW400pW300pW200pWFigure 6.1: (a) Longitudinal voltage, measured with 2nA of current bias be-tween filling factors ν = 4 and 3. (b) The peak in the longitudinal volt-age near filling factor ν = 4 as a function of mixing chamber’s temper-ature. (c) The change in the longitudinal voltage as the temperature wasmodulated at 14 Hz, using the on-chip heaters.521 μmFigure 6.2: (a) The circuit diagram used for the measurements in (b) and (c).(b) Differential conductance G is measured using a lock-in amplifier asa function of DC voltage Vg applied to the 2DEG. ∆Vg = 210µV is theperiod of Coulomb blockade oscillations. (c) The gate voltage is mod-ulated by 1 µV on top of the DC voltage Vg at 14 Hz, and the changein G is detected at that frequency using a second lock-in amplifier (redtrace corresponding to the right axis as indicated by the red arrow). Thedifferential conductance G is measured simultaneously (green trace).in amplifier while scanning Vg. The observed signal (red trace) is very similar tothe expected ∂G/∂Vg, also shown in dashed red. G was measured simultaneouslyand is superimposed on the graph for comparison (green). The steep sides of theCoulomb blockade peak are the regions where charge sensitivity is maximum, andthe ∂G/∂Vg signal rises to at least a factor of 5 above the noise level.6.3 ResultsSo far we have shown that we possess the tools required to modulate the electronictemperature of the 2DEG by about 10 mK, and to detect less than 1 µV of change53in its chemical potential. In Figure 6.3.(a) a lock-in amplifier was used to apply 2nW of AC power to the heaters at 14 Hz. The 2DEG’s potential over one Coulombblockade peak and the magnetic field over the range that covers the centre of theν = 5/2 at B = 5.12 T were scanned.0 100 200 300Vg(µV)5.085.105.125.145.165.18MagneticField(T)(a)0 100 200 300Vg(µV)406080100120140160(b)30150∆G(pA/V)0.00.20.40.60.81.01.21.4G(µA/V)Figure 6.3: (a) The change in differential conductance of the SET as a resultof modulating the temperature of the sample with 2 nW of AC power at14 Hz, scanned over the 2DEG’s DC voltage Vg and the magnetic field.The centre of ν = 5/2 is at B = 5.12 T. (b) A cut of the plot in (a) atB = 5.10 T (red dashed line) shown in red on the left axis. The greenline is G on the right axis, measured simultaneously.Looking at Figure 6.3.(a), first notice that the step function predicted by Refer-ence [6] is not present in the picture. The amplitude of the signal is less than 150pA/V, which is about one order of magnitude smaller than the amplitude of ∆G inFigure 6.2.(d) corresponding to 1 µV of change in the chemical potential of the2DEG. Therefore the effect of modulating the electronic temperature using a 2 nWAC power, or ≈10 mK, around ν = 5/2 is at least one order of magnitude smallerthan modulating the chemical potential directly by 1 µV.The slopes of the lines in Figure 6.3.(a) follow:∂µ∂B= (N+12)h¯em∗±g∗µB (6.4)54for the Landau level N = 2 4. Here m∗ is the effective mass of electrons in the2DEG, g∗ is the effective g-factor, and N is the Landau level number [72]. The plusand minus signs correspond to anti-parallel and parallel spin-polarized electrons,respectively, populating the highest filling factor. Comparing these slopes in dif-ferent filling factors with Equation 6.4, the effective mass can be calculated to bem∗ = 0.07me and the effective g-factor to be g∗ = 15.1, consistent with previousexperiments [73].In Figure 6.3.(b), the red plot is a cross section of the 2 dimensional scan of∆G in Figure 6.3.(a) across the line indicated by the red dashed line at B = 5.10 T,superimposed on G, measured simultaneously across the same line. There are twopeaks in ∆G for each peak in G with different amplitudes. The existence of twopeaks, instead of one peak and one valley as in Figure 6.2.(d), can be explained bythe change of temperature of the SET itself. When an SET gets hotter, the widthof the Coulomb blockade peak gets larger while its height stays roughly the same.The amplitude of the signal rises on the sides of the peak and stays the same ontop of the peak as temperature is modulated resulting in two peaks in ∆G one oneach side. If this effect is combined with a small change in chemical potentialwhich causes the peak to move in one direction on Vg axis, which creates a peakand a valley in ∆G, we end up with two peaks with different heights that we seein Figure 6.3.(b). This difference in heights also remains the same throughout therange of the magnetic field shown.6.4 Stability After IlluminationIt can be argued that since the ν = 5/2 state is not fully developed in this sample, asit is clear from Figure 6.1.(a), it is not surprising that the signature of non-Abelianquasiparticles was not observed. It is, however, worth mentioning that the samplewas cooled down from room temperature to the base temperature in the dark. Re-liable techniques exist for improving fractional states in the quantum Hall regimein a QW such as the one described in Chapter 3, which involve shining light at afrequency below the band-gap of GaAs at temperatures below 10 K. Unfortunately4See the arguments around Equation 2.1455shining light at low temperatures makes the SET unusable for measurements ontime scales available to our equipment.0 100 200 300 400Vg(µV)0.00.20.40.60.8G(µA/V)(a)0 100 200 300Time (s)0.00.20.40.60.8G(µA/V)(b) Vg = 00 2 4 6 8 10Time difference between two jumps (s)050100150200250300CountsCharacteristic stability time =1.1 seconds(c)Figure 6.4: (a) The SET signal G after shining light. (b) G over time at Vg = 0.The state of the SET jumps randomly as a function of time. (c) Thehistogram of time difference between jump events over half an hour atVg = 0.Figure 6.4.(a) is the SET’s differential conductance G, after shining light usinga red LED for 10 min at base temperature, and increasing the temperature to 4 Kfor 20 min. The Coulomb blockade peaks are lost in many random jumps in thephase of the oscillations [62]. As Figure 6.4.(b) shows, these jumps are randomlydistributed in time, at constant 2DEG voltage Vg. Since the SET’s signal are sensi-tive to nearby charges, we speculate that this dramatic change in the stability of theSET signal is due to the fact that after shining and annealing, charge carriers can56jump between donors and the 2DEG more easily. Note that even before shining andannealing, such jumps in Coulomb blockade phase were observed but at a muchlower frequency. In Figure 6.3.(a) right below B = 5.14 T, such a jump is visible.The stability time, however, goes from several tens of minutes before shining/an-nealing, to about 1 second after. Figure 6.4.(c) is the histogram of time periodsbetween jump events. Note that the time constant of the lock-in amplifier was 100ms, which eliminated the events that were faster than 500 ms.In order to conclusively answer the question of whether at ν = 5/2 the quasi-particles reveal any peculiar thermodynamic behaviour consistent with the Moore-Read state using the technique described above, one would need samples that fullydeveloped fractional states at low temperatures with an activation energy gap thatis larger than 100 mK, without the need for any treatment, such as shining light,that might render the SET unusable. Such samples do exist. In fact, in one of ourcool-downs in the wet-fridge, after a few weeks of keeping a sample at the basetemperature without any low-temperature illumination, we observed high qualityFQHE states in transport measurements. The problem is, these results were not re-produced on the sample under investigations in this study. The difficulty is bringingan SET and a heater close to the Hall bar without undermining the quality of thesample.57Chapter 7Localization In Strong MagneticFieldsSET measurements presented in this thesis did not discriminate between variousQHE states anywhere in the range of magnetic fields investigated, except near in-teger filling factors where the bulk of the sample becomes insulating and does notrespond to the voltage applied to the ohmic contacts on the edge. The insulat-ing bulk is formed because electrons in the 2DEG are localized. Two theories forlocalized electrons in the 2DEG in a QHE regime have been provided. The first the-ory considers non-interacting electrons in a random potential landscape, createdby impurities in the crystal. The second is based on Coulomb interactions betweenelectrons in an otherwise flat potential.7.1 Impurity Induced LocalizationThe scaling theory of localization, sometimes referred to as Anderson localization[74], predicts that in a one or two dimensional system where electrons do not in-teract with each other and where impurities create a sizable randomly distributedpotential, the resistivity of the system increases as its size gets larger. For a macro-scopic sample size, the electrons always localize. This theory, however, cannotbe the complete picture in the QHE regime. The formation of the quantum Hallplateaus require extended states to exist in the bulk of the sample [75].58Figure 7.1.(a) shows the relationship between Landau levels and Hall and lon-gitudinal conductivity in a 2DEG as a function of the density n. The top part ofthe figure is the density of states, broadened by the random potential imposed byimpurities. The shaded region corresponds to localized electrons on the tails of theLandau peaks. The centre of the peak contains extended states that are necessaryfor the development of IQHE states.Built upon the scaling theory of localization, other theories exist that considera non-interacting electron Hamiltonian in a magnetic field plus a randomly dis-tributed potential, made by impuritiesH =12m∗(p− eA)2+V (r) (7.1)where V (r) is a spatially distributed random potential.In the limit of high magnetic field, and assuming that the random potential iswhite noise1, Equation 7.1 can be solved exactly [76]. Figure 7.1.(b) shows theexact results of these investigations. The localization length ξ near each Landauband diverges according to a power lawξ = ξN |E−EN |−νN (7.2)In the lowest Landau level E0, the exponent was found to be approximately ν0 ≈ 2[76]. When ξ is larger than the sample size L, the electronic states are extended.Other random potentials also have been considered for which exact solutions donot exist, and numerical techniques were adopted [77–79], but the story stays thesame qualitatively.An intuitive picture of the microscopic mechanism of localization is obtainedby assuming the correlation length of the potential disorder to be much larger thanthe magnetic length [79], which is a correct assumption for sufficiently large mag-netic fields. The electrons perform rapid cyclotron motions on equipotential linesof the disorder landscape. These equipotential lines form closed loops aroundpeaks and valleys of the disorder potential, but when two such loops get closer1If we assume spatial correlation in random potential has a random distribution 〈V (r)V (r′)〉 =W f (r− r′), the function f (x) controls the nature of the random potential. White noise correspondsto f (x) = δ (x).59Figure 7.1: (a) Adapted from Reference [75]. Top: the density of states, mid-dle: longitudinal resistivity, bottom: Hall resistivity, as functions of thedensity n. The shaded region in the density of state corresponds to lo-calized electrons. (b) Adapted from Reference [76]. Exact results forHamiltonian Equation 7.1 in the limit of strong magnetic field and whitenoise random potential. Energy is in units of h¯ω . Green lines are thedensity of states for unperturbed Hamiltonian. Solid lines are the broad-ened density of states due to the random potential. The dashed lines arelocalization lengths. The de-localized states exist when the localizationlength is larger than the size of the sample L.to each other than the magnetic length, electrons can tunnel from one loop to an-other, effectively connecting the loops. Far from the centre of a Landau band, theseloops remain closed and the electrons are localized, but near the centre, the loopsconnect and extended states emerge.7.2 Wigner CrystallizationIn extremely clean samples, such as the ones investigated in this thesis, whereelectron mobility is larger than 107 m2/Vs and electron mean free path is close to1 mm, the effect of the random potential due to impurities is minimal. The signa-60tures of localized states in the transport measurements, however, do not diminishin such samples. Collective electronic states, induced by Coulomb interactions, arebelieved to form crystalline incompressible structures.At very low filling factors (high magnetic fields), WCs [80] are known to be-come energetically favourable [81]. Direct observation of WCs, however, remainsan open challenge for experimental investigations. The most unambiguous tech-nique for observing WCs would be the detection of diffraction peaks from the crys-tal, but because the 2DEG is generally a few 100 nm below the surface of thecrystal, this approach does not seem to be feasible [82]. The observation of WCs’signatures in transport measurements has largely relied on the fact that the disor-der potential creates local distortions in the crystal, which in turn create pinningmodes for the WC. The WC is pinned by one peak in the potential, but can also bepinned by another peak nearby. The result is a fast jump back and forth betweenthe two pinning sites, characterized by an oscillation frequency. This frequencyis estimated to be in the vicinity of 1 GHz. Frequency-dependant transport mea-surements at low filling factors [83, 84] have revealed resonance frequencies in therange of 0.2-8.0 GHz, consistent with the theoretical predictions. More recently,similar resonance frequencies have been observed for filling factors 1, 2 [85], and3 [86].Transport measurements, however, are inherently limited, because the excita-tions and measurements are both happening on a macroscopic level, and from theedges of the sample. They cannot, for example, reveal the WC’s order or lattice con-stant. Very recently, a novel experiment has been performed on a 2-layer 2DEG,10 nm apart, where one layer with a very low density hosts the WC, and the otherlayer is hosting a composite fermion liquid and is used as a probe for the otherlayer [87]. Even though the high-density layer probes the bulk of the low-density2DEG, it is still limited in resolving microscopic features of the system such as theshape and the size of the crystalline domains.Localization at higher Landau levels, where the FQHE states are not as strongas in the first Landau level, are explained differently. It has been suggested thatnovel electronic crystalline phases can form at higher Landau levels that break therotational symmetry by forming bubble phases or stripe phases [88]. These statesare charge density waves, where the high density electron regions form bubbles or61stripes in a sea of low density states. The bubble phases are expected to form nearinteger filling factors, and each lattice site can host one or more extra electrons.Stripe phases form near half-integer filling factors, they encompass the length ofthe sample, and are responsible for anisotropic longitudinal resistance across a Hallbar that were experimentally observed near half-integer filling factors at excitedLandau levels [89]. These phases are predicted to exist as a result of Coulombinteractions between electrons in the absence of a random potential [90].7.3 Microscopic Probing of The Localized StatesUsing an SET seems to be a useful technique for studying the microscopic crys-talline structure of electrons in a 2DEG. Several experiments have used SETs tolearn more about the electronic properties of the bulk of a 2DEG such as the metal-to-insulator phase transition induced by lowering the density [72, 91], and a moredirect characterization of localized states [92] that confirms the single-particle pic-ture is only valid in the limit of high disorder potential.We used the SET mounted on the sample to study the localized states nearinteger filling factors. Figure 7.2.(a) and (c) compare the transport measurementsRxx and Rxy with a two-dimensional measurement of SET’s differential conductanceG as a function of the voltage Vg applied to the 2DEG and magnetic field. Themagnetic field axis is shared between the two plots. The localized states aroundfilling factors 3, 4, and 5 are clearly distinguishable from the conducting statesthat are far from integer filling factors. The insulating regions do not respond tothe changes in the gate voltage and look like vertical noisy lines. The conductingregions, farther from integer filling factors, are modulated both by the gate voltageand the magnetic field, according to Equation 2.14 and Figure 2.3. Notice that theinsulating regions as measured by the SET do not necessarily correspond to thevanishing Rxx. This is more evident where the filling factor is slightly higher thanν = 4 where Rxx is zero, but the SET signal indicates a conductive state.In Figure 7.2.(d), the region near ν = 3 is blown up. What is striking is thatthe noisy region is divided into 3 visually distinct regions. The region at the centrelines up with the integer filling factor, sandwiched by the two regions around itthat look similar to each other. The same pattern is observed near all filling factors62Figure 7.2: (a) Longitudinal and Hall resistance between filling factors 3 and5. (b) The circuit diagram used for measuring the conductance throughthe SET as a function of a voltage applied to the 2DEG, Vg, and the mag-netic field, perpendicular to the plane. (c) The differential conductancethrough the SET as a function of the magnetic field and Vg on the samemagnetic field axis as (a). (d) The region around ν = 3 in (c) is zoomedin.from 2 to 11. Above filling factor 11, the insulating region becomes too narrowand distinguishing these different regions becomes impossible.Looking at the insulating regions more carefully, we noticed that the centralpart exhibits hysteretic behaviour. Figure 7.3.(a) and (b) show the insulating regionnear ν = 42 as the magnetic field is swept up and down, as indicated by the arrows.The entire insulating region as a whole has stayed roughly at the same field, but theregion at the centre appears to be leading the magnetic field.To confirm that the insulating region is in fact not responding to the voltage2The density is different because this is a different sample than the one used for other figures inthis Chapter.63200 300 400 500 600Vg (µV)2.852.902.953.003.05MagneticField(T)(a)200 300 400 500 600Vg (µV)(b)0 20 40 60 80 100 120 140Vg (µV)a.u.B = 2.85 TB = 2.91 TB = 2.96 T(c)0 20 40 60 80 100 120 140Time (s)a.u.B = 2.85 TB = 2.91 TB = 2.96 T(d)Figure 7.3: G as a function of the magnetic field and Vg while the magneticfield is increased (a) and decreased (b) as indicated by the arrows. Aforward hysteretic behaviour is observed for the region at the centre. (c)G vs. Vg, at fields that are specified in each box and correspond to thered dashed lines in (a). (d) G vs. Time at the same set of magnetic fieldvalues.64applied to the 2DEG, we compared multiple scans of Vg at the same magnetic fieldand found no correlation between them. We also compared Vg scans and time scansat constant Vg at fields specified by dashed red lines in Figure 7.3.(a). The resultsare shown in Figure 7.3.(c) for Vg and (d) for time scans. Qualitatively, the timescans are not distinguishable from Vg scans. The centre section is very clearlydistinct from the two side regions in both time and Vg scans. The side regionslook like white noise, although their precise characterization is limited by the timeconstant of the lock-in amplifier which was set to 100 ms in this picture. The centreregion, however, looks like a two-level fluctuator (TLF).TLF-type noise has been previously observed in SETs [93] with the same char-acteristics as the data shown here, namely 1) the duty-cycle, the time fraction spenton high (or low), is always around 50%, 2) the amplitude ∆G, the difference be-tween high and low, is roughly the size of the oscillations in G as a function of Vg,and 3) the signal is not a single clean TLF; there seems to be many layers, someclose to the top and some close to the bottom. Note that the Vg scan seems to os-cillate at a lower frequency, but scanning Vg over the range shown took about 30 s,which means the oscillation period the same in both figures.Even though it has been argued [94] that the source of TLF noise can only bethe defects inside the tunnelling junctions of the SET, the clear dependence of thenoise on the QHE state is only compatible with the source of noise being outside ofthe SET structure [95], the 2DEG in this case.A likely explanation for the source of TLF noise is the existence of deep po-tential wells. When the system is in a IQHE state, the bulk of the sample becomesincompressible, but charge can jump between a few potential traps3. If those trapsare close enough to the island of the SET, TLF noise is observed on the SET. As themagnetic field moves away from an integer state, the number of traps available tothe energy levels of adjacent electrons increases. In this region, perhaps the ran-dom potential creates shallow puddles in the 2DEG that can host charge carriers,and those charge carriers can jump from one puddle to another. Even though thebulk is still incompressible, that is, Rxx is still zero, there are more traps for elec-trons to hop around, and hence the noise is not TLF anymore and 1/ f power spectra3See Figure 5 in Reference [93] that shows how a few charge traps can give rise to TLF noise.65100 101 102 103 104Frequency (Hz)0.40.50.60.70.80.91.0MeasuredAmp./InputAmp.Figure 7.4: The frequency response of the sound card. The measured ampli-tude is divided by the known amplitude generated by the lock-in ampli-fier at various frequencies.is expected in this regime. Once the bulk is compressible, the electrochemical po-tential reaches equilibrium, and the SET signal becomes very quiet; no discretejumps occur near the SET’s island.7.4 Sound Card MeasurementsIn order to further investigate the noise characteristics of the incompressible re-gions, we used a standard computer sound card. A sound card is capable of measur-ing a voltage on its microphone jack at frequencies to which human ear is sensitive.This is generally between 10 Hz and 17 kHz [96].Figure Figure 7.4 shows the sound card is capable of reading the correct ampli-tude of an oscillating voltage in the range of 10 Hz to a few kHz. To produce thisdata, the output signal of a lock-in amplifier was fed into the sound card at constantamplitude and various frequencies. The plot shows the amplitude of the signal asmeasured by the sound card as a function of frequency.At Vg = 0, and DC bias of 20 µV, the drain of the SET, through a currentamplifier, was connected to the microphone jack of the sound card. Figure 7.5.(a)shows the log of the noise power spectra, log[FFT(Id)2] where Id is the output from66101 102 103Frequency (Hz)6.06.26.46.66.87.0MagneticField(T)471013 (b) B=6.54T471013 (c) B=6.41T471013 (d) B=6.33T101 102 103Frequency (Hz)0369 (e) B=5.99T19log(FFT(Id )2)Figure 7.5: (a) The log of the square of the Fourier-transformed current noise,log(FFT(Id)2), as a function of frequency and magnetic field. The cur-rent noise data is collected with a sound card at 11 kHz samping rate.(b)-(e) Cuts of data in (a) at magnetic fields specified, and shown in (a)by white dashed lines. The red dashed line in each plot shows the slopeof 1/ f for comparison.the current amplifier, as a function of the magnetic field. Data for selected magneticfields, indicated by the white dashed lines, are shown in plots (b)-(e). At 6.54 T (b)and 6.33 T (d), the sample is in the incompressible mode, and the power spectradetected is roughly proportional to 1/ f . At 6.41 T, the centre of ν = 2 state, thenoise is expected to be compatible TLF noise, following a Lorentzian curve [93],but the “knee frequency” of the Lorentzian is around a few Hz, which is lower thanwhat the sound card can detect. Below 6.2 T, the bulk is in a compressible state.Coulomb blockade oscillations are visible, and the noise power, as shown in (e) for5.99 T, is very low.67Even though the data collected from the sound card are in general agreementwith the arguments regarding the sources of the noise presented above, the methoddid not turn out to be very useful, since the two ranges of frequencies that are inter-esting for this problem were not available to the sound card. The knee frequencyof the Lorentzian that is supposed to characterize the TLF noise is below 10 Hz.The frequency of the pinning-modes of WCs is in the GHz range which is ordersof magnitude above the highest frequency a sound card can detect. To circumventthe first problem, any DC voltmeter with sampling rate higher than a few Hz wouldsuffice. Data similar to Figure 7.3.(c) and (d) over longer periods of time are suf-ficient to characterize the TLF, but were not available during the analysis of thedata. To solve the second problem, a high frequency voltmeter such as a networkanalyzer can be used.7.5 Temperature Dependence of The Localized StatesAnother method for probing localized states is to measure the width ∆B of the van-ishing Rxx in magnetic field as a function of temperature. Previously, ∆B in trans-port measurements was found to agree with SET measurements [92]. Figure 7.6.(a)and (b) show the SET measurements and transport measurements over the centre offilling factor 2 as a function of magnetic field. The red vertical lines in (a) are thebest attempts in visually locating the onset of the incompressible state. The blackdashed line in (b) is where Rxx crosses 100 Ω. Comparing these two sets of valuesin (c) as the temperature rises, it seems that the width ∆B is decreasing with risingtemperatures in both measurements with the same slope.In Figure 7.6.(a), near the centre of ν = 2, there is a region where the noiselevel is low. We identify this region with the TLF noise discussed above. Noticethat with the rising temperature, this region remains unaffected. This point is moreobvious in Figure 7.7 where the sound card was used to measure the current noisearound ν = 3 at different temperatures. As the temperature increases, the width ofthe two sections with 1/ f noise shrinks, but the region sandwiched between themwith lower levels of noise remains unchanged up to 120 mK.685.8 6.2 6.6 7.0 7.4Magnetic Field (T)G(a.u.)(a)5.8 6.2 6.6 7.00.00.20.40.60.81.0Rxx(kΩ)(b) 20 mK40 mK60 mK80 mK100 mK120 mK5.8 6.2 6.6 7.0Magnetic Field (T)20406080100120Temperature(mK)(c)RxxSETν=2Figure 7.6: (a) Differential conductance of the SET as a function of the mag-netic field for various temperatures. Traces are shifted up for clarity. (b)Rxx as a function of the magnetic field for various temperatures. Thecolours correspond to the same temperatures as in (a). (c) The onset ofincompressible regions in (a) marked by red vertical lines, and cross-over 100 Ω in (b) are plotted to show that they follow roughly the sameslopes.7.6 Final RemarksEven though the data presented here does not conclusively characterize the sourceof the noise and the nature of localization in the sample under investigation, withthe right tools and techniques, answering those questions are not out of reach. Aunique opportunity has presented itself here, because the incompressible regiondemonstrates two very distinct behaviours that correspond to two different knownstates of the IQHE.A thorough low-frequency investigation of the TLF noise can reveal more de-tails about the deep potential puddles that are close to the SET’s island. If two SETs694.14.24.34.4T=20 mK4.14.24.34.4T=50 mK4.14.24.34.4MagneticField(T) T=60 mK4.14.24.34.4T=80 mK100 101 102 103Frequency (Hz)4.14.24.34.4T=120 mKFigure 7.7: The current noise measured by the sound card around ν = 3 atvarious temperatures. Even though the two noisy sections around thecentre of the integer filling factor fade slowly with increasing temper-ature, the centre that we identify with the TLF-noise state remains un-changed.70can be fabricated close to each other, investigating correlations between their sig-nals can reveal information about the size of charge puddles or the domain size ofthe WC. Smaller islands can also be more sensitive to individual localized states.A high-frequency investigation, in the range of 100s of MHz to 10s of GHz,of the current noise from the SET can reveal some characteristics of the pinningmodes, and more importantly on these samples, the transition between TLF and1/ f noise can be studied. Also a radio-frequency SET [97] might be able to resolvemore features in the 1/ f regions.There is evidence to believe that good quality of FQHE states can be achievedwithout low-temperature illumination. This was observed for a short time duringone of our cool-downs as well. If a reliable method for achieving high qualityFQHE states without illumination is found, all of the techniques described abovecan reveal interesting features of those states as well. At the same time, the break-down of the SET signal after illumination, Figure 6.4, has not been observed in thepast and can reveal important information about the movement of charge carriersbetween the 2DEG and the donor sites, and perhaps other unknown potential trapsin QW samples.71Chapter 8ConclusionAt the end of this work, the question of whether the quasiparticles at ν=5/2 possessnon-Abelian exchange statistics or not remains an open one. The answer, however,seems more within the reach than before.We showed that strong FQHE states are reproducibly and reliably obtainableon QW samples. Illumination alone at low temperature brings the sample to awell-defined low density state, and annealing to 2.2-6.5 K for more than 2 minutesincreases the density and optimizes the FQHE characteristics. After the maximumdensity and the optimum quality is obtained, annealing does not have any effect.The time and effort for making measurements are greatly reduced since after anelectrical shock, the sample can be easily reset in-situ in less than one hour withoutthe temperature having to go over 2.2-6.5 K. Resetting the sample was a processthat used to involve warming up the fridge to close to room temperature and coolingback down, taking several hours or even days. We also measured one of the highestenergy gaps at ν=5/2 observed in the community using this technique, which isessential for making successful measurements of ∂µ/∂T , since ∂T should be smallcompared to this energy gap.As we saw in Chapter 6, the illumination technique makes the SET signal ex-tremely noisy. The mechanism for this effect is, at this point, unknown, and in myview, is an interesting question. It is unclear, for example, whether the noise isproduced by the 2DEG, the donors layer, or the bulk of the semiconductor that sitsunderneath the SET. To investigate this question, one could fabricate SETs on vari-72ous samples, such as pure GaAs sample, or GaAs with a metal back-gate, and thenilluminate the sample at low temperature and measure the noise characteristics ofthe SET. It would also be interesting if the SET can be removed from the surfaceat low temperatures and placed back on after illumination. See [92] for a possibleway of moving the SET around at low temperatures. It would also be interestingto place two SETs close to each other and measure the cross correlation betweenthe noise measured in them, which can tell us, depending how close the SETs are,where the source of noise is.It is also believed that placing an SET on a layer of boron-nitride enhances thenoise characteristics of the SET. We made several attempts to fabricate the SETson boron-nitride flakes, but we failed to make working devices. Similar kinds oftechnique where the SET is separated from the 2DEG can be used to minimize theeffect of noisy layers under the SET. We showed that in this noisy state, the char-acteristic timescale of a jump in the SET’s potential is ≈1 s. It is conceivable that ameasurement technique that is much faster than 1 s can be used after illuminationto obtain useful data from an SET. This makes the possibility of fast temperatureoscillations desirable.We established a reliable technique for oscillating the temperature of the sam-ple by 10 mK at 1-20 Hz. Frequencies of up to 500 Hz have been achieved byother members of our lab in a different refrigeration unit and different samples. Ifthe temperature oscillation can be done at higher frequencies, the lock-in measure-ments will be easier to perform. We were limited to 20 Hz and below during thefinal set of measurements.We showed that oscillating the temperature by 10 mK and measuring the chem-ical potential changes in the range of 0.1 µV in the bulk of the sample are both pos-sible. This should give us the measurement tools for conclusive results, howeverthe ∂µ/∂T signal did not reveal any signature of an enhanced entropy of ν=5/2quasiparticles, perhaps because they didn’t exist in the sample. The transport mea-surements certainly suggest this to be the case. Transport measurements of sampleswith much better FQHE characteristics, however, are possible without the need ofillumination. These traces were in fact observed on one of our earlier iterations ofthe sample in a different refrigeration unit. Trying to obtain a high quality frac-tional state without illumination would be the shortest path to obtaining conclusive73results for the measurement.We showed that the properties of the bulk of an insulating 2DEG that are inac-cessible to a standard transport measurement can be studied using a SET. The SETis sensitive to the noise characteristics of various insulating states within an Integerplateau. If high quality fractional states are obtained, those can also be studiedusing an SET. A transport measurement is very limited in distinguishing variousfractional states, since ρxx vanishes in all such states, and only the shape of theiredge can be investigated. A SET fabricated on a high quality QW opens a new doorto such studies at various fractional states.74Bibliography[1] Samani, M. et al. Phys. Rev. B 90, 121405 (2014). → pages 20, 21[2] Leggett, A. J. Rev. Mod. 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The surface tension of gallium holds the GaAs substrate in place, while thegood thermal conductivity of gallium makes it possible to heat the substrate to≈ 600◦C by heating the puck. Gallium melts at 30◦C, therefore in order to avoidliquid gallium from running all over the sample during wet-treatment steps of thefabrication, it is important to remove the gallium from the back of the substrate.We followed the following steps in order to remove the gallium:1. In order to protect the top surface of the wafer, we first covered it withdroplets of PolyMethyl MethAcrylate (PMMA).2. The sample was then placed upside-down on a microscope glass. The PMMAlayer spread and created a protective layer between the sample and the glass.3. The microscope glass was heated to 100◦C on a hot plate.814. Liquid gallium could then be removed using a piece of clean-room wiper.It usually takes several minutes of wiping, but most of the gallium can beremoved.The 2DEG layer is located 200 nm below the surface of the sample. In orderto make electrical contacts with the 2DEG layer, we first placed 100 µm×100 µmmetallic squares on the surface of the sample using e-beam lithography techniques,following the recipe below.First we prepared the sample for e-beam lithography:1. Cleaned the sample by putting it in hot acetone for 3 minutes, hot methanolfor 3 minutes, and IsoPropyl Alcohol (IPA) for 1 minutes.2. Blow-dried the surface using nitrogen gas.3. Baked the sample on hot plate for 3 minutes at 185◦C. Then placed it on aroom temperature metal surface to cool fast.4. Placed the sample on the spinner (Laurell Technologies Corp. Model WS650MZ-23NPP/LITE), added 2 drops of Methyl MethAcrylate (MMA) and spin at5000 revolutions per minute (RPM) for 45 s.5. Baked the sample for 2 minutes at 185◦C.6. Spun MMA again at 5000 RPM for 45 seconds.7. Baked for minutes at 185◦C.8. Spun PMMA at 4000 RPM for 45 seconds.9. Baked for 12 minutes at 185◦C.Using the SEM (Zeiss Sigma), we used the following parameters to write thepattern:• Area dose: 280µC/cm2• Magnification: 200x82• Line spacing: 43nm• Centre-to-centre distance: 22nmAfter the pattern was written:1. Developed using a 1 to 3 ratio solution of Methyl IsoButyl Ketone (MIBK)and IPA for 90 seconds.2. Cleaned in IPA for 90 seconds.3. Blow-dried with nitrogen gas.4. Cleaned the surface using oxygen plasma etching for 5 seconds at 120 W.The pressure in the chamber was 350 mTorr and flow rate of 7.5 standardcubic centimetres per minute (SCCM).5. Evaporated the following metals on the surface in the following order usinga thermal evaporator:Nickel: 4 nmGold: 169 nmGermanium: 108 nmNickel: 74.5 nmGold: 20 nmThe base pressure in the evaporator’s chamber, after pumping for 12 hoursor more, was around 2× 10−7 Torr, which was further reduced to roughly7×10−8 Torr using a liquid nitrogen cryogenic pump.6. Immersed the sample in acetone for lift-off until the metal was visually de-tached from the surface. Rinsed in IPA, and blow-dried with nitrogen gas.For the metal films to make electric contact with the 2DEG 180 nm below the sur-face, the sample was annealed using rapid thermal annealing (RTA). The samplewas placed on a flat copper sheet for uniform thermal distribution. The coppersheet was then placed inside a small vacuum chamber. The vacuum chamber wasattached to a vacuum pump and a source of forming gas, a mixture of hydrogen(5%) and nitrogen (95%). By adjusting the valves attached to the pump and the83forming gas source, a pressure of 200 mmHg was maintained throughout the pro-cess. The copper plate was heated using a tungsten halogen bulb, and its tempera-ture was measured by thermocouples and a handheld multimeter. Controlling thetemperature was done manually, by looking at the temperature reading and adjust-ing the power repeatedly.Figure A.1: (a) Before annealing and (b) after annealing the sample in an RTAwhich causes the metals in the ohmic contacts to diffuse roughly 200nm into the sample and make electric contact with the 2DEG.First the temperature was increased to 350◦C, a process which took about 1min, and the temperature was kept there for 1 min. Then the temperature wasraised to 460◦C relatively quickly, and it was maintained there for 1.5 min. Afterthat the sample was cooled as quickly as possible by turning off the heater abruptlyand increasing the flow of the forming gas. Figure A.1 shows optical microscopeimages of the ohmic contacts (a) before and (b) after the annealing process. Thesmooth surface of the contacts are covered with bubbles that were produced duringannealing. The edges are slightly rounded, but the general shape of the contacts,down to 10 µm wide saw-tooth edges, is maintained intact. The saw-tooth shapeof the contact is believed to enhance the adhesion of the contact to the substrate.84A.2 SET FabricationThe SETs used in the experiments presented here were made by Professor AndrewHouck’s group at Princeton University’s department of electrical engineering. Dur-ing the first two years of my work as a graduate student, I made several attempts atmaking SETs. The SETs worked as expected, but the yield was low. Here I presentthe last recipes I used, and discuss the reasons I think the yield was low.After the standard 3-solvent cleaning as before, the following resists were spunon the GaAs substrate:1. Spun MMA-EL12 at 5000 RPM for 45 s. Ramp up at 500 RPM/s. Total 55 s.2. Baked for 2 min on a hot plate at 168◦C. Fast-cooled on an aluminum flatsurface.3. Spun PMMA-A4 at 4000 RPM for 45 s. Ramp up at 500 RPM/s. Total 53 s.4. Baked for 20 min on a hot plate at 168◦C. Fast-cooled on an aluminum flatsurface.The lithography was performed with the following parameters:• E-beam acceleration 27 keV• Aperture size 15 µm• Undercut exposure 60 nC/cm2• Bilayer exposure 280 nC/cm2• Working distance 5.2 mm.Development was done in IPA:MIBK (3:1) mixture for 90 s.Then aluminum was thermally evaporated on the sample. Figure A.2.(a) showsa diagram of the extensions added to the evaporation chamber in order to allow oxy-gen in the chamber while controlling the pressure. The sample was mounted on astage, depicted in Figure A.2.(b). A magnet was attached to one side of the see-saw,which made that side heavier, and with the help of the screw, the stage’s normal85Figure A.2: (a) The diagram of valves and pumps and the vacuum chamberfor leaking oxygen between the two Al evaporation steps. (b) Al evap-oration at 0◦. (c) Using a magnet from outside the vacuum chamber,the sample is oriented at 14◦ relative to the direction of the evaporation.vector stayed at 0◦ relative to the direction of the incoming aluminum vapour. After35 nm of Al evaporation, various valves were used to allow oxygen in the cham-ber. The oxygen was introduced either from an oxygen-argon cylinder, or fromthe air through a cold-trap which filtered water and other particles from the airand presumably only allowed oxygen and nitrogen in. Both techniques producedreasonable results.After a layer of Al2O3 is formed, another magnet from outside the vacuum86chamber was used to rotate the stage to 14◦, as shown in Figure A.2.(c). Then 35nm of Al was evaporated again. The following is a more detailed procedure thatwas followed:1. Evaporated 35 nm of aluminium at 0◦.2. Turned off the turbo pump and waited for 10 minutes until the sample cooleddown.3. Opened valve B then C.4. Waited for a few minutes.5. Closed valve B.6. Waited for a few minutes.7. Closed valve P, then C.8. Opened valve A, then B.9. Started the timer and waited until the pressure was 330 mTorr.10. After 5 to 6 minutes the pressure should be roughly 300 mTorr. Opened valveP a little. Tried keeping the pressure constant at this value for 10 minutes.11. Closed valves A, B, and C.12. Turned on the turbo pump and waited until the pressure was 1× 10−7 Torragain.13. Evaporated 35 nm of aluminum.14. Turned off the pumps and waited for 10 minutes until the sample cooled off.15. Opened the chamber and took out the sample.Because the dimensions of SET was small, the grain size of the evaporatedaluminum could be relevant. In order to obtain smaller grains, we tried cooling the87Figure A.3: SEM monographs of the SET trials without cooling the substrate(a), and while cooling the substrate with liquid nitrogen (b). The grain-sizes on the cooled substrate are smaller.sample to 77 K with liquid nitrogen during both stages of aluminum evaporation.The results are shown in Figure A.3.Cold evaporation produced smaller grains, but rendered the lithography recipesuseless. We gave up on cold evaporation, since optimizing the correct e-beamexposure was very time consuming.A.3 Fridge WiringThe measurements presented in this thesis were carried out on two different dilu-tion refrigeration [98, Chapter 7] systems: an OxfordTM“wet-fridge”, named so be-cause it is immersed in liquid helium during its operation, and a BlueForsTM“dry-fridge”, since there is no liquid helium bath.88Because electric wires need to connect the measurement apparatus, such asamplifiers, multimeters, and so on, that operate at room temperature, to the sampleunder investigation, often at around 10 mK, great care needs to be taken to avoidoverheating the sample. Both systems were provided with wires that go from roomtemperature down to the mixing chamber, with total electric resistance of around100 Ω. Although this resistance reduces the heat flow, there are other sources thatcan heat up the sample.An important source of heat in a measurement that involves sweeping the mag-netic field is the magnetocaloric effect [99]. This effect is due to the coupling ofthe magnetic ions in a material and the external magnetic field [100]. Using non-magnetic materials in places that are exposed to large external magnetic fields is,therefore, necessary. Alloys that include Cr, Mn, Fe, Ni, or Co tend to becomemagnetic at low temperatures. Pure metals such as Cu, Au, Ag, and Al are veryuseful for good thermal and electrical conductivity in high magnetic fields.Materials with good electrical conductivity also cause heating if they are ex-posed to large changes in magnetic fields. This type of heating is caused by Eddycurrents, and can be minimized with careful geometrical considerations. See [98,Section 10.5.2] for a thorough discussion of Eddy currents heating.Beside general considerations, an important part of the wiring in both fridges isthe low-pass filters. Since all measurements are performed at frequencies below afew hundred Hz, any other signal above that frequency is considered noise, causesoverheating of the sample, and should be filtered out before it reaches the sample.In the wet-fridge, each signal wire enters a resistor-capacitor (RC) filter board.The diagram of the filters is shown in Figure A.4.Figure A.4: Adapted from Reference [101]. RC filter board and the mi-crowave attenuator diagram for the dry-fridge.89The surface-mount components were soldered on a circuit board. The filterstart attenuating at 1 MHz. At 100 MHz, the attenuation is 10−3. The level ofattenuation, however, gets worse at higher frequencies, due to parasitic inductance.At 10 GHz, no significant filtering is provided by this board. The boards were fixednext to the mixing chamber, but because they were not exposed to high magneticfield, heat-sinking them to the mixing chamber was essential [101].To get rid of the GHz noise, long high-resistance constantan wires were sand-wiched between copper tapes. At radio frequencies, the transport of electric signalis limited to the outer boundaries of the wires. The skin-depth δ shrinks as√f ,which makes the effective surface area of the conductor smaller. Even though theconstantan wire only adds 100Ω to the total resistance of the signal wires, it provedto be a very effective radio-frequency filter.The dry-fridge follows a different strategy for filtering out noise. A set of metalthin-film meanders were evaporated on a heavily p-doped silicon substrate, coveredby 1 µm of SiO2. Each meandering line is made of 10 nm of Cr and 60 nm of Au.The length of the line is 53.5 mm, the width is 150 µm and the total resistanceis 305 Ω. The thin-films capacitively couple with the p-doped Si substrate, whichacts as the ground, with total capacitance of 297 pF. Silver was evaporated on theother side of the silicon wafer, connecting the p-doped silicon to the ground of thefridge.The idea here is, instead of having a 3-stage RC filter, as in the dry fridge, wehave a continuous filter, which produces a much steeper drop in gain as a functionof frequency, while keeping the total DC resistance of the filter small.In this set-up, the attenuation of -3 dBV is achieved at 4.3 MHz. No radio-frequency attenuation strategy is adopted in the dry-fridge. The thin Cr-Au filmsshould, in principle, attenuate radio-frequencies with skin-depths higher than 70nm, or 10 GHz, but anything above 10 GHz is not filtered.Unfortunately low electronic temperatures have not been achieved on this fridgeyet. My best estimate of the base electronic temperature of the sample is around25 mK, while the mixing chamber is at 12 mK. Recent more careful measurementsof the electronic temperature using graphene on a silicon substrate suggests highertemperatures at high magnetic fields.90Appendix BMaxwell RelationsFollowing reference [102, Appendix E], the Maxwell relation(∂µ∂T)n=(∂ s∂n)T(B.1)can be derived as follows.B.1 The Triple Product Rule of Partial DerivativesConsider a real analytic function1 of two variables u = u(x,y). The differential ofu isdu =(∂u∂x)ydx+(∂u∂y)xdy (B.2)Now assume a contour of u in the x-y plane, Figure B.1, so that as we move on thecurve, the value of u remains constant. So when we move from the point (x,y) to(x+dx,y+dy), then du= 0. We can break this move into two steps; first we movealong the x axis with constant y, and then we move along the y axis, with constant1A function is analytic if and only if its Taylor series about any point in its domain converges tothe function in some neighbourhood of the point.91Figure B.1: Contour of constant u on the x-y plane.x. (∂u∂x)ydx+(∂u∂y)xdy = 0 (B.3)The slope of the contour line is (∂y/∂x)u, so dy = (∂y/∂x)udx. Replacing dy inEquation B.3, gives us(∂u∂x)ydx+(∂u∂y)x(∂y∂x)udx = 0 (B.4)Therefore (∂x∂u)y(∂u∂y)x(∂y∂x)u=−1 (B.5)B.2 State FunctionsNow consider the following partial differential equation, or a “state equation” assuch equations are usually referred to in thermodynamicsdu = f (x,y)dx+g(x,y)dy (B.6)92where f (x,y) and g(x,y) are given functions. Comparing Equation B.6 with Equa-tion B.2, if u exists, thenf =(∂u∂x)yg =(∂u∂y)x(B.7)Taking derivatives of f and g with respect to y and x respectively, we obtain(∂ f∂y)x=∂ 2u∂x∂y(∂g∂x)y=∂ 2u∂y∂x(B.8)But for any analytic function u(x,y), we have∂ 2u∂x∂y=∂ 2u∂y∂x(B.9)Therefore (∂ f∂y)x=(∂g∂x)y(B.10)Given a thermodynamic state function, Equation B.10 can be used to derive variousMaxwell relations.B.3 Free EnthalpyIn a system with constant pressure (isobaric), the quantity that is minimized is thefree enthalpy G, defined asG =U + pV −T S (B.11)where U is the internal energy, p is pressure, V is the volume of the system, T isthe temperature, and S is the entropy. The differential of G isdG = dU +V ·d p+ p ·dV −T ·dS−S ·dT (B.12)93For a system with variable number of particles N, the differential of the internalenergy becomesdU = T ·dS− p ·dV +µ ·dN (B.13)where µ is the chemical potential, defined as the quantity that stays constant be-tween two systems that are in diffusive equilibrium. ThereforedG = µ ·dN+V ·d p−S ·dT (B.14)Using the same argument as we did for Equation B.8, we obtainµ =(∂G∂N)p,TS =(∂G∂T)p,N(B.15)and the Maxwell relation Equation B.10 takes the form(∂µ∂T)p,N=(∂S∂N)p,T(B.16)To obtain Equation B.1, we divide the numerator and denominator of the left handside by volume of the system, and make the assumption that the pressure is alwaysconstant implicit. Other Maxwell relations can be derived easily by the applicationof the Equation B.5.B.4 Chemical And Electrochemical PotentialSince thermodynamics is being discussed, it is useful to look at the meaning ofchemical potential and electrochemical potential more carefully, and understandwhy the SET is measuring the chemical potential of the bulk of the 2DEG.Consider two systems, Σa and Σb, having Na and Nb identical particles respec-tively, that are in thermal and diffusive equilibrium. Diffusive equilibrium is thecondition where the transfer of matter between two system is possible but the nettransfer of matter is zero [102, Section 7.2]. Both systems are in thermal, but notdiffusive equilibrium with a heat bath. If dN particles and energy dU is transferred94from Σa to Σb, the total change in entropy isdSt =[(∂Sb∂Nb)U−(∂Sa∂Na)U]dN+[(∂Sb∂Ub)N−(∂Sa∂Ua)N]dU (B.17)Assuming the heat bath being much larger than both systems, the temperature Tmust be constant across both systems and the heat bath. Therefore1Ta=(∂Sa∂Ua)N=(∂Sb∂Ub)N=1Tb(B.18)The chemical potential µ is then defined asµ =−T(∂S∂N)U,V(B.19)Two systems that are in thermal and diffusive equilibrium have the same chem-ical potential. Therefore chemical potential plays the same role for two systemsin diffusive equilibrium as temperature does for two systems in thermal equilib-rium. With this definition, it can also be shown that the particles flow from thesystem with higher chemical potential to the system with low chemical potential,just like heat flows from the system with higher temperature to the system withlower temperature.Now assume there is a conservative force acting on the particles. This could bethe electric force or the force of gravity. The electrochemical potential is definedas the sum of chemical potential and all other potentials. For electrons with chargee in an electric potential V , the electrochemical potential µe isµe = µ+ eV (B.20)In thermal and diffusive equilibrium, the electrochemical potential is the same be-tween two system. This is the basis for the operation of a battery. A chemicalreaction creates a difference between the two electrodes of a battery. Electronsmove in order to counter-balance this difference, creating a electrostatic potentialdifference between the two electrodes of the battery, which is the open circuit volt-age measured between the two electrodes.95The work function W is defined as the difference between the chemical po-tential of an electron outside and inside a solid. When two solids A and B, withdifferent work functions WA and WB make contact, the electrons move from thesolid with higher chemical potential, say A, to the one with the lower chemicalpotential BeVA+µA = eVB+µB (B.21)This process continues until the electrochemical potential, also referred to as theFermi level, is the same across both solids. A voltage difference Vc, called thecontact voltage, develops at the interface between the twoVc =VB−VA = µA−µBe =WA−WBe(B.22)Now consider the circuit diagram for a typical SET measurement in Figure 4.2. Theisland of the SET couples electrostatically with the bulk of the 2DEG [7]. Changingthe electrostatic potential of the 2DEG, for instance by applying an external voltageVg to it, or by affecting the density of electrons, shifts the electrostatic potential ofthe island, which in turn affects the conductance from the source to the drain.Applying a magnetic field, for instance, affects the chemical potential inside asolid, both in the 2DEG and the SET. The imbalance in chemical potentials causes acurrent to flow, since the two are connected to the ground of the system which actsas a reservoir of electrons. This flow of electrons creates a contact voltage betweenthe SET’s island and the 2DEGe · dVcdB=dµ2DEGdB− dµSETdB(B.23)since electrochemical potential in both systems are at equilibrium after everythingsettles down. Here µ2DEG is the chemical potential of the 2DEG and µSET is thechemical potential of the SET’s island. We argue that dµSET/dB is small, becausethe SET is made of bulk aluminum in its normal state, and can be neglected. There-fore the contact potential Vc gives us a direct measurement of the change in thechemical potential of the 2DEG. Vc is obtained by measuring the conductance fromthe source to the drain. Vg is used to tune the SET’s island into a resonance for96current flow.97

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