Undoped Algaas/Gaas Quantum Dots with Thermally Robust Quantum Properties

THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF PHYSICS

Undoped AlGaAs/GaAs Quantum Dots with Thermally Robust Quantum Properties

Andrew Ming See

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy

September 2011

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Surname or Family name: SEE

First name: MING Other name/s: ANDREW

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Title: Undoped AlGaAs/GaAs Quantum Dots with Thermally Robust Quantum Properties

Abstract 350 words maximum: (PLEASE TYPE)

In a modulation-doped AlGaAs/GaAs heterostructure, electrons in the two-dimensional electron gas (2DEG) are provided by ionization of Si dopants in the AlGaAs layer. To reduce the effect of Coulomb scattering between ionized dopants and electrons in the 2DEG, an undoped AlGaAs spacer is grown between the doped AlGaAs and the undoped GaAs layers, which reduces large-angle scattering, thus increasing the mobility and electron mean free path. In a traditional semiclassical picture of an open quantum dot, if the electron mean free path exceeds the dot width, transport becomes ballistic and the corresponding magneto-conductance fluctuations (MCF) are considered as a Fourier sum of periods arising from all possible Aharonov-Bohm loops that intercept the quantum point contacts and are formed by scattering from the dot walls alone. As a result, these devices, known as semiconductor billiards; were seen as ideal for studies of dynamical chaos in the quantum mechanical limit. However, modulation-doped devices are not without their problems. For example, it has recently been demonstrated that small-angle disorder scattering causes unpredictable changes in the device’s electronic properties each time it is cooled for use. This finding forces a careful reconsideration of our notions of ballistic transport in these devices. Another problem associated with modulation-doped devices is the temporal instability due to rapid switching of the dopants between ionized and de-ionized states, hindering the development of ultrasensitive quantum devices.

This thesis reports the development of undoped quantum dots, where the ionized dopants are removed and the 2DEG is populated electrostatically by applying a positive bias to a degenerately doped cap. Our “induced” devices produce MCF that are reproducible with high fidelity after thermal cycling to 300 K. By performing a comparative analysis between nominally identical undoped and modulation-doped billiards, we conclude that small-angle scattering dominates transport in dots. Our work has important implications for studies of quantum chaos and ballistic transport. Additionally, measurements of our small undoped quantum dot operating in the Coulomb blockade regime showed features of excited state transport and spin- dependent transport blockade.

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3 Acknowledgements

Firstandforemost,IwouldliketothankmysupervisorsA/Prof.AdamMicolich andProf.AlexHamiltonfortheirunconditionalsupportandinspirationsthroughoutthe course of my PhD candidature. Adam and Alex have taught me a lot about semiconductor physics, research methods, experimental techniques, science communication skills as well as life in general. I am very grateful to have them as supervisors,mentorsandfriends.

Secondly,IwanttosaythankstomycollegesattheQuantumElectronicDevices (QED)groupformakingthisjourneysoenjoyable:toDr.OlehKlochanforshowingme howtorunthefridgeandsetupthemeasurementsproperly;toDr.JackCochranefor histechnicalsupport;toDr.WarrickClarkeforgettingmestartedinthecleanroom;to Daisy Wang for simulating the transport scattering times, and to Dr. Ted Martin, Dr. LasseTaskinen,Dr.LapHangHo,Dr.AdamBurke,Dr.ZacharyKeane,Sarah Macleod, Jason Chen, Sebastian Fricke, Patrick Scriven, LaReine Yeoh, Ashwin Srinivasan, Rifat Ullah,SpencerRussettfor,DavidWaddingtonandRoyLiforhelpfuldiscussions.

I would also like to acknowledge the Australian National Fabrication Facility (ANFF) for providing us with the cleanroom and measurement facilities. I want to especiallythankBobStarrettandDavidBarberfortheirassistanceincryogenics;Dr.Eric Gauja,Dr.FayHudson,Dr.JoannaSzymanska,Dr.AndreasFuhrer,Dr.AndreaMorello andFrankWrightforusefuladviceindevicefabrication.IwanttoalsothankProf.Saskia Fischer,Prof.MarkEriksson,Prof.AndrewSachrajda,Prof.KlausEnsslinandProf.Sven Roggeforinsightfuldiscussionsthatledtosomeoftheresultspresentedinthisthesis.

Additionally, I want to thank everyone in the School of Physics whom I have great pleasure working with: to the Head of School Prof. Richard Newbury for his support in writing this thesis; to David Jonas and Kristien Clayton for resolving our computerͲrelated issues; to Prof. Michael Gal, Prof. Michael Ashley, Sue Hagon, Ranji Balalla,AlbertMcMaster,JoselitoConducto,StephenLo,PatriciaFurst,DaveRyanand mygoodfriendPatrickMcMillian,fortheiradministrativesupport.

4 DuringAugustͲSeptember2010,IhadtheopportunitytocollaboratewithProf. RichardTaylorandhisgroupattheUniversityofOregonduringmyfourweeksofvisit.I wanttothankRichard,Dr.BillyScannell,IanPilgrim,RickMontgomeryandPeterMorse forawonderfulexperience(well,apartfromeatingthatBaconMapleAle!).

DuringthecourseofmyPhDcandidature,Ihadconcurrentlyparticipatedinthe Research Commercialization Training Scheme (CTS): I would like to thank Dr. Wallace BridgeformakingthisgraduatecourseavailableforPhDstudentsacrossanumberof universitiesinAustralia;Ihadawonderfultimelearningaboutthecommercialsideof research,aswellascompetinginthefinalsofthePeterFarrellCup2008withDr.Kai WeiChuandSidneyHsiong.

Lastbutnotleast,Iowemymostsinceregratitudetomyparents–Leonand Amy See, who have worked so hard in Hong Kong in order to support my studies in AustraliasinceYear9.Withouttheirunconditionalloveandsupport,Iwouldneverhave made it this far. I also want to thank my grandparents for always looking after me. Finally, I want to thank my beloved wife Selina for her patience and understanding duringthetoughtimesinthisjourney.

5 ListofPublications

Journalpublications x “ImpactofSmallͲAngleScatteringonBallisticTransportinQuantumDots”–A.M. See,I.Pilgrim,B.C.Scannell,R.Montgomery,O.Klochan,A.M.Burke,M. Aagesen,P.E.Lindelof,I.Farrer,D.A.Ritchie,R.P.Taylor,A.R.HamiltonandA. P.Micolich,Phys.Rev.Lett.,108,196807(2012). x “AlGaAs/GaAssingleelectrontransistorfabricatedwithoutmodulationdoping” –A.M.See,O.Klochan,A.R.Hamilton,A.P.Micolich,M.Aagesen,andP.E. Lindelof,Appl.Phys.Lett.,96,112104(2010). x “Isittheboundariesordisorderthatdominateselectrontransportin semiconductor`billiards'?”–A.P.Micolich,A.M.See,B.C.Scannell,C.A. Marlow,T.P.Martin,I.Pilgrim,A.R.Hamilton,H.Linke,R.P.Taylor, arXiv:1204.4882v1,(acceptedforpublicationinFortschrittederPhysik). x “ProbingtheSensitivityofElectronWaveInterferencetoDisorderͲInduced ScatteringinSolidͲStateDevices”–B.C.Scannell,I.Pilgrim,A.M.See,R.D. Montgomery,P.K.Morse,M.S.Fairbanks,C.A.Marlow,H.Linke,I.Farrer,D.A. Ritchie,A.R.Hamilton,A.P.Micolich,L.Eaves,andR.P.Taylor,–inPressfor Phys.Rev.B,arXiv:1106.5823. x “RoleofbackgroundimpuritiesinthesingleͲparticlerelaxationlifetimeofatwoͲ dimensionalelectrongas”–S.J.MacLeod,K.Chan,T.P.Martin,A.R.Hamilton, A.See,andA.P.Micolich,M.AagesenandP.E.Lindelof,Phys.Rev.B,80, 035310(2009). x “SpinͲdependenttransportblockadeobservedinundopedAlGaAs/GaAsSET”–A. M.See,O.Klochan,A.P.Micolich,A.R.Hamilton,M.AagesenandP.E.Lindelof, (manuscriptinpreparation). x “RadioͲfrequencyreflectometrymeasurementsofundopedAlGaAs/GaAsSET”– S.J.MacLeod,A.M.See,P.Scriven,Z.Keane,A.R.Hamilton,M.AagesenandP. E.Lindelof,(manuscriptinpreparation).

6 Conferenceproceedings x “FabricationofundopedAlGaAs/GaAselectronquantumdots”–A.M.See,O. Klochan,A.P.Micolich,A.R.Hamilton,M.AagesenandP.E.Lindelof,IEEE Xplore10.1109/ICONN.2010.6045253.InternationalConferenceonNanoscience andNanotechnology,2010. x “FabricationofundopedAlGaAs/GaAselectronquantumdot”–A.M.See,O. Klochan,A.P.Micolich,A.R.Hamilton,M.AagesenandP.E.Lindelof,AIPConf. Proc.,1399,343(2011).30thInternationalConferenceonthePhysicsof Semiconductors,2010.

7 TableofContents

Acknowledgements...... 4 ListofPublications...... 6 ThesisIntroduction...... 12 ThesisOutline...... 14 ChapterOne–Background...... 16 1.1–Introduction...... 16 1.2–TheAlGaAs/GaAsheterostructures&confinementtechniques...... 17 1.2.1–Modulationdoping...... 17 1.2.2–UndopedAlGaAs/GaAsheterostructure...... 18 1.2.3–l Latera confinementofthe2DEG:From2Dtolowerdimensions...... 20 1.3–Electricalcharacterizationofa2DEG...... 21 1.3.1–ShubnikovdeHaas&Hallmeasurements...... 21 1.3.2–Elasticmeanfreepathandballistictransport...... 23 1.3.2.1–Quantumpointcontacts...... 24 1.3.3–Scatteringmechanismsin2Dsystems...... 25 1.3.3.1–Remoteionizedimpurityscattering...... 26 1.3.3.2–Backgroundimpurityscattering...... 27 1.3.3.3–Interfaceroughness...... 27 1.3.3.4–Surfacestates...... 28 1.3.3.5–Alloydisorderscattering...... 29 1.4–Basicsofelectrontransportinquantumdots...... 30 1.4.1–Theconstantinteractionmodel...... 31 1.4.2–Coulombblockadeandsingleelectrontunneling...... 34 1.4.2.1–ThelineshapeoftheCoulombblockadeoscillations...... 36 1.4.3–HighbiasregimeandCoulombdiamonds...... 37 1.5–MagnetoͲtransportofopenquantumdots...... 39 1.5.1–TheAharonovͲBohmeffect...... 40 1.5.2–SemiconductorbilliardsandMagnetoͲConductanceFluctuations(MCF)...... 41 1.5.3–Thesemiclassicalapproach...... 43

8 1.5.4–Thequantummechanicalapproach...... 44 1.5.5–Billiarddynamicsandquantumchaos...... 47 1.5.5.1–Proposedsignaturesforquantumchaosinquantumdots...... 49 1.5.5.2–Modificationtothequantumpicture:dynamicaltunneling...... 50 1.6–Howdoesthisthesisfitwithinthe contextofexistingknowledgeofbilliards?...... 51 Bibliography...... 52 ChapterTwo–Devicefabricationandlowtemperaturemeasurementtechniques...... 62 2.1–Introduction...... 62 2.2–Heterostructureoverview...... 63 2.2.1–NBI30undopedAlGaAs/GaAsheterostructure...... 64 2.2.2–C2275modulationͲdopedAlGaAs/GaAsheterostructure...... 65 2.3–Devicefabrication...... 65 2.3.1–ProcessflowforNBI30undopedquantumdotdevices...... 66 2.3.2–Processflowformodulationdopedquantumdotdevices...... 70 2.4–Lowtemperaturemeasurementtechniques...... 72 2.4.1–4.2Kdipping...... 72 2.4.2–Pumped4HeVTIsystem...... 73 2.4.3–Pumped3HeHelioxsystem...... 74 2.4.4–3He/4Hedilutionrefrigerationsystem...... 75 2.5–Electricalsetupsanddataacquisition...... 77 2.5.1–Electricalsetupformeasuringsingleelectrontransistors...... 77 2.5.2–Electricalsetupformeasuringelectronbilliards...... 80 Bibliography...... 82 ChapterThree–CharacterizationmeasurementsofNBI30andC2275heterostructures.....84 3.1–Introduction...... 84 3.2–ShubnikovdeHaasandHallmeasurements...... 85 3.2.1–TopgatedependenceoftheelectrondensityinNBI30...... 87 3.3–TransportscatteringtimeandquantumlifetimeofNBI30andC2275...... 88 3.3.1–ExtractingthequantumlifetimefromShubnikovdeHaasoscillations...... 90 3.4–SurfacedepletionofNBI30...... 93 3.4.1–Thedepletionregionexperiment...... 94

9 3.5–Chaptersummary...... 97 Bibliography...... 98 ChapterFour–DevelopmentofundopedAlGaAs/GaAsquantumdotsoperatinginthe Coulombblockaderegime...... 100 4.1–Introduction...... 100 4.2–Measurementsofthelargerquantumdotdevice...... 102 4.2.1–MeasurementofCoulombblockadeoscillationsfor thelargedot...... 103 4.2.2–Estimationofthechargingenergyforthelargedot...... 106 4.2.3–TemperatureanalysisoftheCBoscillationsfromthelargedot...... 108 4.3–Measurementsofthesmallerquantumdot...... 109 4.3.1–Gatecharacterizationforthesmalldot...... 110 4.3.2–Coulombblockadeoscillationsforthesmallerdot...... 111 4.3.3–Biasspectroscopyforthesmalldot...... 113 4.3.4–LineshapeoftheCBpeaksinthequantumregime...... 114 4.3.5–Preliminarynoiseperformanceanalysis...... 116 4.4–SpinͲdependenttransportblockadeinundopedquantumdots...... 119 4.4.1–Groundstatesuppressionandnegativedifferentialconductance...... 119 4.4.1.1–Temperatureactivatedlineartransport...... 121 4.4.1.2–BiasspectroscopyofsuppressedCoulombblockadepeaks...... 122 4.4.2–Spinblockadeandothersuppressionmechanisms...... 124 4.4.2.1–Spinselection rules...... 124 4.4.2.1–Othergroundstatesuppressionmechanism...... 127 4.5–Chaptersummary...... 127 Bibliography...... 128 ChapterFive–ComparisonofthermalstabilityofMCFinundopedandmodulationͲdoped billiards...... 134 5.1–Introduction...... 134 5.2–MCFmeasurementsofandundope AlGaAs/GaAsbilliard...... 136 5.2.1–Deviceandmeasurementsetup...... 136 5.2.2–Thethermalcyclingexperiment...... 137 5.2.3–SensitivityofMCFtolocallyinduceddisorder...... 142 5.3–MCFmeasurementsofamodulationͲdopedAlGaAs/GaAsbilliard...... 145

10 5.3.1–Deviceandmeasurementsetup...... 146 5.3.2–Billiarddefinitionandsidegatevoltagecharacteristics...... 147 5.3.3–ThermalcyclingofmodulationͲdopedbilliarddevice...... 148 5.4–Chaptersummary...... 153 Bibliography...... 154 ChapterSix–StudiesofdisorderusingmagnetoͲconductancefluctuations...... 157 6.1–Introduction...... 157 6.1.1–Billiarddevicesusedinthethesis...... 158 6.2–ActivationenergyofdopantsintheC2275billiard...... 160 6.2.1–PreviousresultsfromaGaInAs/InPbilliarddevice...... 160 6.2.2–ThermalcyclinganalysisforMDͲL...... 161 6.2.3–ChargerelocationmechanismsinthedopantlayerofC2275...... 165 6.3–ExtendedcomparativeMCFstudiesbetweenUDͲMandMDͲM...... 166 6.3.1–Thermalstabilityofgatecharacteristics...... 167 6.3.2–ThermalstabilityofmagnetoͲconductancefluctuations...... 169 6.3.3–MCFcorrelationanalysisbetweenUDͲMandMDͲM...... 172 6.3.4–StatisticsofMCFfluctuations...... 173 6.4–Effectsofdisorderinanundopedbilliarddevice...... 175 6.4.1–EffectofsurfaceoxidationonMCFreproducibility...... 176 6.4.2–MCFreproducibilityofalargeundopedbilliard...... 180 6.5–Summeryanddiscussions...... 182 6.5.1–Implicationsofourfinding...... 182 6.5.2–Correlationbetweenthermalrobustnessandchargestability...... 184 6.5.3–Futurework...... 185 Bibliography...... 186 AppendixA–Deviceprocessingtechniques...... 192 A1–AssessmentofNBI30undopeddevices...... 192 A2–MethodsforremovingleakingOhmiccontacts...... 193

11 ThesisIntroduction

The traditional way of obtaining a twoͲdimensional electron system in an AlGaAs/GaAsheterostructureisviaatechniqueknownasmodulationdoping[1].Ina modulationͲdopedsystem,theionizeddopantsareseparatedfromtheheterojunction by an AlGaAs spacer layer in order to reduce the Coulomb interactions between the dopantsandthe2DEG.Whilethistechniquehasenabledelectronmobilitiesashighas 35millioncm2/Vstobeachieved[2],itisalsoknowntocausenoiseissuesinquantum dots[3,4],quantumpointcontacts[5Ͳ10]andmesahallbardevices[11Ͳ13],duetothe switchingeventsintheremoteionizedimpuritylayers.Inthiswork,wehavestudied quantum dots formed on a heterostructure where the modulation doping has been removed.Theseundopedheterostructureshavebeenpreviouslyusedforstudiesofthe metalͲinsulatortransitionatlowtemperatures[14Ͳ16]duetotheirlowdisorder.Wewill firstdemonstratetheoperationoftwoundopedquantumdotsproducedbyextending onamethoddevelopedbye Kan etal.[17,18].ThesetwodotsoperateintheCoulomb blockaderegime,andoneofthemwassufficientlysmallthatexcitedstateresonances [19]andfeaturesofspinͲdependenttransportblockadecouldberesolved.

Anotherconcernthathasrecentlybeenraised[20]regardingmodulationdoping is the validity of the ballistic transport assumption in open dots (billiards). In a semiconductor billiard it is normally taken that, when the electron mean free path exceeds the width of the billiard, the magnetoͲconductance fluctuations (MCF) representthemultitudeofAharonovͲBohmloopsthatarisefromelectrontrajectories determinedbythebilliardwalls[21Ͳ23].Itisalsocommonlyacceptedthattheremote ionizedimpuritiesonlycausesmallperturbationstothosetrajectories[21,24].Thusthe MCF was considered as an accurate and highly sensitive magnetoͲfingerprint of the billiardgeometry,whichenabledquantumdotstobeusedasamodelsystemforstudies of quantum chaos [21,22,25Ͳ31]. Recent studies using scanning gate microscopy to imageelectrontransportinaquantumpointcontacthavefound“branching”ofelectron paths, providing evidence that smallͲangle scattering significantly affects transport at lengthscalessmallerthanthetraditionalballisticmeanfreepath[32,33].Morerecently,

12 Scannelletal.[20]reportedthatafterthermalcyclingaGaInAs/InPmodulationͲdoped billiard to 130 K, the characteristics of the corresponding MCF changed significantly despite the billiard geometry remaining unchanged. As a result, charge redistribution amongsttheremoteionizedimpuritiesisbelievedtoberesponsiblefortheobserved decorrelation.

The above findings trigger serious concerns regarding the role that ionized impuritiesplayindeterminingelectrondynamicsinaquantumdotbilliard.Inthiswork, we demonstrate that small angle scattering plays a dominant role in determining the characteristics of the MCF. This is achieved by presenting comparative studies of the MCF produced by undoped and modulationͲdoped billiards with very similar electron densities, mobilities and geometries [34]. We have compared three sets of these billiardswithdifferentsizes.Wefoundthatinallthreecases,theMCFcharacteristicsof the undoped billiard remained mostly unchanged after thermal cycling to room temperature.Incontrast,thermalcyclingconsiderablyalteredtheMCFcharacteristicsof the modulationͲdoped billiards. As part of a collaborative effort with Prof. Richard Taylor’s group at the University of Oregon, we have used a decorrelation analysis approach to study the temperatureat which the MCF decorrelates [20].This analysis reveals that charge reconfiguration of the dopants occurs aṯ150 K [35], which is consistentwiththepropertiesoftheDXcenter,awellknownscatteringsitefoundin AlGaA/GaAsheterostructureswhereSiisusedasmodulationͲdopant[36].Ourfindings have significant implications for studies of quantum chaos based on semiconductor quantumdots,andmayforceareinterpretationofsomepreviousstudiesonthistopic.

13 ThesisOutline

Chapter1isanintroductoryreviewofvariousconceptsnecessarytounderstand the results presented in this thesis. First, the twoͲdimensional electron gas (2DEG) in modulationͲdopedandundopedAlGaAs/GaAsheterostructuresisintroducedinSection 1.2, this is followed by discussions of the 2DEG electrical characterization techniques and scattering mechanisms in Section 1.3. In Section 1.4, we discuss the basics of electron transport in a lateral semiconductor quantum dot using the constant interactionmodel.Lastly,theconceptsofmagnetoͲtransportinopenquantumdotsand billiarddynamicsareintroducedinSection1.5.

In Chapter 2, we introduce various fabrication and measurement techniques used to obtain the results presented in this thesis. An overview of the various heterostructuresusedisfirstpresentedinSection2.2,whileSection2.3discussesthe processingstepsforthefabricationofundoped(Section2.3.1)andmodulationͲdoped (Section2.3.2)devices.Lastly,inSections2.4&2.5,wes discus variouslowtemperature electricalmeasurementtechniquesusedinthiswork.

QuantumdotsformedontheNBI30undopedandtheC2275modulationͲdoped AlGaAs/GaAsheterostructuresarediscussedthroughoutthisthesis.Chapter3provides useful characterization information of these two heterostructures. In Section 3.2, we presentelectrondensityandmobilityvaluesofthesetwowafersobtainedfromHalland ShubnikovdeHaasmeasurementsofthecorrespondingHallbardevices.Comparisons ofthequantumandtransportlifetimesarediscussedinSection3.3.Lastly,inorderto determinethewidthofthedepletionregioninNBI30’sdopedcapasaresultofsurface states,whichlimitstheminimumfeaturesizeofstructuresachievablewhilemaintaining high conductivity gate structures at low temperature; we report conductivity measurements of a series of narrow wires defined using the doped cap of NBI30 in Section3.4.

In Chapter 4 we present the development of two undoped AlGaAs/GaAs quantum dots.InSection4.2,wefirstdemonstratetheoperationofalargerquantum

14 dotintheCoulombblockaderegimeasaproofofconcept.InSection4.3,wepresent moredetailedtransportmeasurementanalysisofasmallerquantumdotoperatingin the quantum regime. A preliminary noise analysis of the smaller dot is presented in Section4.3.5whereachargefluctuationof0.8%of anelectronwasmeasuredinthe period of 15 minutes. Lastly, we report ground state suppression and negative differential conductance observed in our small dot in Section 4.4, which we attribute theiroriginstospinͲdependenttransportblockade.

Chapter5and6constitutethemainresultsofthisthesis.InChapter5,weuse thesamesmallquantumdotdeviceinChapter4,butoperatedintheopenregime,to perform a comparative study of magnetoͲconductance fluctuations (MCF) in an undoped(Section5.2)andamodulationͲdopedbilliarddevicewithnominallyidentical geometry (Section 5.3). The result in Chapter 5 suggests that the electron dynamics established by the billiard geometry are largely randomized due to the presence of remoteionizedimpurities.TheimplicationofthisresultisdiscussedfurtherinChapter6, whereaMCFdecorrelationanalysisofamodulationͲdopedbilliardispresented(Section 6.2). We attribute the decorrelation of the MCF to the reconfiguration of charge and ionizationstateoftheDXcentersintheAlGaAs.Section6.3presentsamoredetailed comparisonofMCFfromundopedanddopedbilliardsusingamoreconventionalsquare billiard design. Finally, using the characteristics of the MCF as a sensitive probe of disorder scattering in Section 6.4, we investigate the effect of disorders such as backgroundimpuritiesandsurfaceoxidationinourundopedbilliarddevices.

Chapter1

Background

1.1–Introduction

In this thesis we report investigations of closed and open quantum dots fabricated on undoped and modulationͲdoped heterostructures. In this chapter, we discussvariousconceptstoalevelsufficienttounderstandtheexperimentalresults.For clarity, we divide this chapter into four main sections. We will first introduce the concept of the twoͲdimensional electron gas (2DEG) and the various electron confinement techniques. This is followed by discussions of the 2DEG characterization methods and scattering mechanisms. We will then discuss the constant interaction modelofCoulombblockadeinquantumdots,whichcanbeusedtoexplainmostofthe results presented in Chapter 4. Finally, we discuss various important concepts in the magnetoͲtransport of open quantum dots, such as the semiclassical and the fully quantummechanicaltreatmentsofbilliards,aswellasthetraditionalviewsofbilliard dynamics.

1.2–TheAlGaAs/GaAsheterostructures&confinement techniques

ThetwoͲdimensionalelectrongas(2DEG)inanAlxGa1ͲxAs/GaAsheterostructure providesastartingpointforthedevicesstudiedinthisthesis.Suchaheterostructureis realized byepitaxial growth of a wider bandͲgap semiconductor (AlGaAs) on top of a narrower bandͲgap semiconductor (GaAs). When the bands of the two material are joined to form a straddling gap*[37], a quantum well is formed at the AlGaAs/GaAs interface, where theelectrons are strongly confined in the direction perpendicular to theinterface(zdirection).ThisconfinementwidthiscomparabletotheelectronFermi wavelength(̱50nm)leadingtostrongquantizationinthezdirection.Typicallyonly theloweststate,or2Dsubband,ispopulatedresultinginathinsheetofelectronswith motion constrained to the inͲplane directions (i.e. x and y). This sheet of electrons is commonlyknownasatwoͲdimensionalelectrongas(2DEG).

1.2.1–Modulationdoping

InthecaseofamodulationͲdopedheterostructure,theelectronsthatpopulate the 2DEG are provided by Si dopants in the AlGaAs region. In order to reduce the scattering between the ionized dopants and the electrons at the 2DEG, an undoped AlGaAs spacer layer is grown between the doped AlGaAs and undopedGaAs regions. Thistechniqueisknownasmodulationdoping,andwasfirstreportedbyDingleetal.in 1978[1].Figure1.1(a)showsatypicalmodulationͲdopedAlGaAs/GaAswaferstructure grownonthesurfaceofaGaAssubstratewithacorrespondingenergybanddiagram showninFig.1.1(b).Startingfromthesubstrate,anarrowGaAsbufferlayerisgrown first,followedbyalayerofundopedAlGaAs.ThepurposeoftheGaAslayeristoeven out any surface roughness in the substrate. Doping of the AlGaAs region is achieved typically by substituting some of the lattice positions with silicon (Si). Silicon donor

* ThebandͲgapsatܶ=0KforAl0.33Ga0.67AsandGaAsare1.836and1.424eVrespectively.Whenthese formaheterojunction,theGaAsconductionbandsits0.363eVbelowtheAlGaAsconductionbandandthe GaAsvalencebandsits0.049eVabovetheAlGaAsvalenceband[38].

Figure1.1:(a)TypicalwaferlayoutforamodulationͲdopedAlGaAs/GaAsheterostructure. (b) Energy band diagram showing the conduction band corresponding to the heterostructure in (a). The dashed line corresponds to the Fermi levelܧி . The level highlightedinblueisthe2DEG,whichisformedattheAlGaAs/GaAsheterojunction.

atomshaveanextravalenceelectroncomparedtoAlandGa.FinallyaGaAscapis grownontoptopreventtheoxidationofAlintheAlGaAslayer.

1.2.2–UndopedAlGaAs/GaAsheterostructure

Ratherthanrelyingondopantstoformthe2DEG,asinthecaseofmodulation dopingdiscussedabove,anotherwaytoproducea2DEGistouseabiasedtopgateto electrostaticallyinducecarriersattheAlGaAs/GaAsinterface.Adeviceoperatingonthis principle is the MetalͲOxideͲSemiconductorͲFieldͲEffectͲTransistor (MOSFET) in enhancementmode[39].Historically,theseundopedorinducedheterostructureswere fabricatedthrougheitherthestandardSemiconductorͲInsulatorͲSemiconductor(SIS)or invertedSIS(ISIS)structures.IntheSISstructuresadopedcaplayerisusedasthegate forelectrostaticallypopulatingthe2DEG.IntheISISstructure,abiasedsubstrateisused asthegate[40,41].Duetotheeliminationofremoteionizedimpurities,theundoped devices were found to have higher mobilities at low carrier densities݊compared to thosefabricatedonmodulationͲdopedstructures[42].Wewillreturntodiscussionsof

18 thisinSection1.3.3.1,whenweintroducevariousscatteringmechanismsin2Dsystems. Theabilitytoreachextremelylowcarrierdensities(̱0.2ൈ1010cmͲ2)hasenabledthe studyofstronglyinteractingsystemsandthemetalͲinsulatortransition[14Ͳ16].Other advantagesofundopedsystemsincludetheabilitytoeasilyvarytheelectron[43]and hole[44]densities,andtheformationofambipolardeviceswhereboth݊Ͳtypeand݌Ͳ typecarriersarepresent[45].

There are two common techniques of fabricating topͲgated undoped heterostructures.FromthemethodofKaneetal.[17,18],aGaAscapisdegenerately dopedtofunctionasametallicgatetoinducecarriers(Fig.1.2).Thiscapisseparated fromthe2DEGbyaGaAsspacerandanAlGaAsebarrier.Th Ohmiccontactsareformed by first etching down to near the 2DEG beforemetallization. This method of forming Ohmic contacts is known as the selfͲaligned technique. The main difficulty of this techniqueisthattheOhmiccontactscannotbeplacedtoocloseortoofarawayfrom theconductingcap,otherwisetheOhmicswilleitherleaktothegateornotcontactthe 2DEG properly. This also limits the density range that can be achieved using these structures.Furthermore,thereneedstobeanoverlappingregionbetweenthecapand the diffused Ohmic metal so that the 2DEG extends all the way to the Ohmics. A schematicofthisapproachisshowninFig.1.2.

Figure1.2:Anundoped heterostructureproposedby Kane etal. [17,18]. Adegenerately doped cap is used to induce carriers at the 2DEG (blue dashed line). The cap needs to overlaptheOhmiccontactsforthistowork.

InthemethodofHarrelletal.[42,46],twosetsofmetalgatesareusedrather thanadopedsemiconductorgate.Thefirstgateconsistsofametallayerdepositedon the heterostructure that covers the whole 2DEG region except for a narrow strip

19 surroundingtheOhmiccontacts.Thisgapisneededtopreventthegatefromshortingto theOhmiccontactsand2DEG.Theentirestructureisthencoveredbyathinlayerof polyimideandthesecondsetofgatesisdepositedontopofit.Thesegatescoverthe gapbetweentheesurfac gateandtheOhmiccontacts,ensuringcontinuityacrossthe wholesample.Inthisthesis,allundopeddeviceswerefabricatedbasedonthemethod ofKaneetal.[17,18]only.

1.2.3–Lateralconfinementofthe2DEG:From2Dtolower dimensions

The development of electron beam lithography (EBL) has allowed lower dimensionaldevicestobemade.StudiesofoneͲdimensional(1D)andzeroͲdimensional (0D)electronsystemssuchasquantumpointcontacts(QPCs)andquantumdots(QDs) rely on applying further lateral confinement to the 2DEG in an AlGaAs/GaAs heterostructureatscalesecomparabl totheelectronFermiwavelength.Thisistypically achievedusingEBLandoneorbothofthetwolateralconfinementtechniques:using metal split gates to deplete regions of 2DEG underneath or by etching parts of the semiconductoraway.Figure1.3illustratestheseideasoflateralconfinement.

Figure1.3:Theseschematicsillustratetheconceptoflateralconfinementusing(a)asplit gate, (b) deep etching and (c) electrostatically inducing electrons with a shallowͲetched, degeneratelydopedcap.Thedarkregionsinthemiddleofthedevicescorrespondtothe 2DEGattheAlGaAs/GaAsheterostructure.Theyellowregionin(a)correspondstothemetal splitͲgates, and the orange region in (c) corresponds to a patterned degeneratelyͲdoped semiconductorcap.

The first splitͲgate device on an AlGaAs/GaAs heterostructure was made by Thornton et al. in 1986 [47]; this technique operates by electrostatically repelling electronsatthe2DEGtoachievelateralconfinement.AreviewofthistechnologybyR.P. Taylorcanbefoundin[48].Etchingcanbeimplementedintwoways:deepandshallow. Deepetchingworksbydirectlyconfiningthe2DEGbylimitingthelateralextentofthe AlGaAs/GaAs interface. The surface states formed deplete the 2DEG at the edges leading to a 2DEG with a slightly smaller area than the area of the remaining AlGaAs/GaAs material. Alternatively, shallow etching involves terminating the etch beforereachingtheAlGaAs/GaAsinterface,andoftenafteretchingthroughmostorall ofthedopinglayersabovethe2DEG.The2DEGisdepletedintheseregionsbyamixture of a lack of doping to adequately populate the 2DEG and electrostatic depletion by surfacestates,dependingontheprecisedetailsoftheetchdepthandheterostructure. Fordopedcapheterostructuressuchasthosediscussedinthisthesis,shallowetchingis usedtodividethecapintoindividuallybiasablegatesfordefiningandcontrollingalowͲ dimensionaldevice.

1.3–Electricalcharacterizationofa2DEG

Inthissection,wediscusstechniquesforelectricalcharacterizationofa2DEG.In particular,wefirstdiscussmethodsforextractingtheelectrondensity݊andmobilityߤ fromShubnikovdeHaasandHallmeasurements.Thisisfollowedbydiscussionsofthe electronmeanfreepathandtheconceptofballistictransport.Finally,wewillbriefly discuss a few dominant scattering mechanisms in AlGaAs/GaAs heterostuctures that limitthemobility.Characterizationdetailsoftheheterostructuresusedinthisthesisare providedinChapter3.

1.3.1–ShubnikovdeHaas&Hallmeasurements

When a magnetic fieldܤis applied perpendicular to the plane of the 2DEG, electronsexperienceaLorentzforcethatcausesthemtomoveincircular“cyclotron” orbits in the plane of the 2DEG with radiusݎ௖௬௖=԰݇ிȀ݁ܤ. At sufficiently large fields,

21 harmonic oscillator quantization of the cyclotron orbit occurs, and as a result, the energyoftheelectronsܧ௝alsobecomesquantized.Theresultingdiscreteenergylevels areknownasLandaulevels,givenbyEqn.1.1:

ͳ ܧ ൌ൬݊൅ ൰԰߱ ሼͳǤͳሽ ௝ ʹ ௖

כ where߱௖ ൌ݁ܤȀ݉ is the cyclotron frequency. Note that for simplicity, we have omittedthespinͲsplittingterminEqn.1.1.IntheLandaulevelregime,the2DEGdensity of states ߩ൫ܧ௝൯ appears as a series of “broadened” delta functions, where the broadening߁௅௅=԰Ȁ߬௤measured at the full width half of maximum is determined by scatteringeventswithacharacteristictime߬௤.Thischaracteristicstimeisalsoknownas thequantumlifetime,whichwewilldiscussinSection3.3.Asܤisincreasedsuchthat

԰߱௖>߁௅௅,successiveLandaulevelsriseabovetheFermienergyanddepopulate,leading tooscillatorystructureinthelongitudinalresistanceܴ௫௫thatisperiodicinͳȀܤ.These oscillationsareknownasShubnikovdeHaasoscillations[49].Theelectrondensitycan beextractedviaEqn.1.2:

ͳ ݃ ݁ ο ൬ ൰ൌ ௅௅ ሼͳǤʹሽ ܤ ݄݊

where݃௅௅=2istheLandauleveldegeneracywhenspinͲsplittingisneglected.

Another way to extract the electron density is to use the classical Hall effect where݊=ܤȀܴ݁௫௬ (ܴ௫௬ is the transverse/Hall resistance). Note that Landau level quantizationisalsoresponsibleforthequantumHalleffect,whichproducestransverse resistance plateaus at resistances of݄/ߥ݁ଶ, whereߥis the filling factor. An inͲdepth discussion of the quantum Hall effect can be found in [50]. Based on the electron density,theelectronmobilityߤofa2DEGcanbeobtainedfromEqn.1.3:

ͳ ܮ ߤ ൌ ଴ ൬ ൰ሼͳǤ͵ሽ ܴ݁݊௫௫ ܹ

whereܮandܹarethelengthandwidthofthe2DEG–theratioܮȀܹisknownasthe ଴ aspectratioofthe2DEG,andܴ௫௫isthelongitudinalresistanceatzeromagneticfield.

1.3.2–Elasticmeanfreepathandballistictransport

The elastic mean free path݈௘௟(given by Eqn. 1.4 in terms of݊andߤ) is an important parameter in transport studies. It is defined as the average length that an electron in the 2DEG travels between largeͲangle elastic collisions with impurities.

԰ߤ ݈ ൌ൬ ൰ ξʹߨ݊ሼͳǤͶሽ ௘௟ ݁

The nature of electron transport in a laterally confined region is determined by the

relationship between the dimensions of the confined region and݈௘௟ . Figure 1.4 illustratesthisideaforthecasewheretransportoccursinarectangularchannel.InFig. 1.4(a),asanelectrontravelsfromlefttorightundertheinfluenceofanelectricfield, theaveragedistancebetweenelasticcollisionsbetweentheelectronandimpuritiesis

short.Inthisregime,݈௘௟isshorterthanboththewidthܹandlengthܮofthechannel, andwehavediffusivetransport,wherethecurrentisdominatedbytheelectrondrift

velocity.As݈௘௟increases,orܹandܮdecrease,electrontransportfirstentersthequasiͲ

ballisticregimewhereܹ ൏݈௘௟ ൏ ܮ.Here,transportisdependentonbothscattering

fromimpuritiesandthedeviceboundaries;thisisshowninFig.1.4(b).Lastly,when݈௘௟is increasedfurthersuchthatitislargerthanbothܹandܮ,thesystementersthefully ballistic regime, where on average, there are no impurities in the channel and the electrontrajectoriesareentirelydeterminedbyscatteringfromthechannelboundary. This is illustrated in ig. 1.4(c). Note that the classical concept of conductivityߪൌ

Figure1.4:Illustrationsof(a)diffusive,(b)quasiͲballisticand(c)ballistictransportregimes. ThereddotsrepresentimpuritiesthatcauselargeͲanglescattering.

݃ሺܮΤ ܹሻ,Fwhere݃istheconductance,asdefinedinDrude’smodelofconduction[51], isonlyvalidinthediffusiveregime.InthequasiͲballisticandballisticregimes,thelocal conductivityisnolongerdefinedanditisnecessarytoviewtheconductanceintermsof the transmission probabilityܶof electrons through the channel via the Landauer formula ݃ൌሺʹ݁ଶΤ݄ሻܶ [52]. Full details of the Landauer formula are discussed extensivelyin[47].TheexperimentalresultspresentedinthisthesisinChapter5and6 raisequestionsaboutthevalidityoftheballistictransportassumptioninmodulationͲ dopedbilliards.

1.3.2.1–Quantumpointcontacts

Thequantizationoftheelectricalconductanceinquantumpointcontactswas first observed by Wharam et al. [53] and van Wees et al. [54] in 1988, and is an importantexampleofaballistictransporteffectinnanoscaledevices.Aquantumpoint contact(QPC)isanarrowconstrictionbetweentwowideconductingregionswherethe widthandlengthoftheconstrictionarecomparabletotheelectronFermiwavelength

ߣி and much smaller than the elastic mean free path݈௘௟ . Figure 1.5(a) shows a schematicofaQPC.UsingthesplitͲgatemethod,aQPCisdefinedbyapplyinganegative biasܸ௚to the gates to electrostatically deplete the 2DEG in regions underneath. The widthoftheQPCcanbecontrolledusingܸ௚.Themeasuredconductance݃asafunction ofܸ௚for a QPC is shown in Fig. 1.5(b). Asܸ௚becomes more negative and the QPC

Fiigure 1.5:(a) A schematic of a quantum point contact, the yellow region on the top represents metal split gates which are biased atܸ௚to deplete regions underneath at the

2DEG (shown as black). (b) A typical QPC pinch off trace plotting݃vs.ܸ௚showing the quantizedconductanceinstepsof2݁ଶ/݄.Thedatain(b)wasobtainedfromoneofmytest structures fabricated on a modulationͲdoped AlGaAs/GaAs heterostruucture that is not discussedinthisthesis.

narrows, the conductance drops and shows a series of steps at integer multiples of 2݁ଶ/݄. The risers between the steps correspond to successive 1D subbands passing throughtheFermienergy,andeitherpopulatingordepopulatingdependingonwhether the QPC is being widened or narrowed. Each conductance step݃ ൌ ݌ሺʹ݁ଶȀ݄ሻ corresponds to an integer numberr݌of halfͲFermi wavelengthsߣி/2 fitting across the width of the QPC, reflecting the importance of the transverse quantum confinement imposed by the gates. The factor of 2 above reflects the spinͲdegeneracy at zero magnetic field. At high fields, the steps are observed at multiples of݁ଶȀ݄ due to Zeemansplittingofthe1Dsubbandss.

1.3.3–Scatteringmechanismsin2Dsystems

Over the last four decadees, there have been enormous efforts devoted to optimizing transport in twoͲdimensional electron gases (2DEGs) [55,,56]. Mobilities as highas35millioncm2/Vshavebeenrecentlybeenreported[2].Suchhhighmobilitiesare importantforstudiesofstripesanddbubblephasesinthequantumHallregime[57]and evenͲdenominator fractional Quantum Hall States [58] for topoological quantum computing applications. However, even at lower mobilities, scatteriing is of profound

25 importancetoaproperunderstandingoftransport,astheresultsinthistheesishighlight. InparticularthereismoretothepicturethansimplytheamountoflargeͲangleelastic scattering. We now look more closely at the physics of scattering in AlGaAs/GaAs2D systems.

1.3.3.1–Remoteionizedimpurityscattering

RemoteionizedimpurityscatteringismostlyduetothedopantsiintheAlGaAs layerofamodulationͲdopedsystem,whichprovidethechargecarriersinthe2DEG.The impuritiesarechargedionizeddopantsthatinteractwiththeelectronsinthe2DEGvia theCoulombinteraction.Theeffectof theremoteionizedimpuritieson mobilitywas demonstratedbyHarrelletal.in1999[442].Intheirpaper,theycomparedthemobility asafunctionofcarrierdensitybetweentwosamplesmadefromdifferentwafers:one modulationͲdoped and the other undoped. The two waferswere grownn in the same molecularbeamepitaxy(MBE)chamberandthereforehavesimilarbackgroundimpurity levels.Figure1.6showstheirexperimenntalresults;belowacarrierdensityof̱1x1011

Figure 1.6:Mobility vs. carrier density comparison between an induced 2DEG and a modulation doped 2DEG grown in the saame MBE chamber measured atܶ= 1.7 K. Figure reproducedfromHarrelletal.,Fig.3ofRef[42].

26 cmͲ2,themobilityoftheundoped(induced)sampleishigherthanthatofmodulation dopedsample.Harrelletal. thusconcludedthat remotelyionizedimpurityscattering dominatesinthelowdensitylimit[42].

1.3.3.2–Backgroundimpurityscattering

Background impurities arise due to the growing environment of the heterostructure as it is impossible to eliminate all unintended species from the MBE chamber and achieve perfect vacuum. Background impurities get randomly deposited on the surface of the heterostructure as it is being grown, and incorporated into the layers. These unintended species include semiconductor atoms (i.e. Al, Ga, As), stray dopants(i.e.Si),andatmosphericgasspeciessuchasN,O,etc.However,theonlyones that we are interested in here are impurities ending up at the interface that directly scatterelectronsinthe2DEG,andstrayionizedimpuritiesthat causescatteringviathe Coulombinteraction.

With more conventional semiconductor growth techniques, where the background impurity level is relatively high, it is possible to estimate the amount of background impurities that act as ionized dopants by growing a thick layer (a few microns)andextractingthearealcarrierconcentrationfromtransport measurements. However, for MBE grown heterostructures, where the background dopant levels are extremely low, this method is not suitable. Instead we rely on the fact that carriers scattered by background impurities are generally scattered at largeͲangle due to the strongshortͲrangerepulsionbetweenthecarriersandimpurities.Thiseffectisstronger atlowerdensitiesandthisbackgroundlevelalsodeterminesthethresholdbelowwhich the 2DEG would break up into inhomogeneous puddles. The impurity concentration levelofAlGaAswasfoundto2to3higherthanthatinGaAs[59,60].

1.3.3.3–Interfaceroughness

Interface roughness scattering is caused by the roughness of the interface betweentheGaAsandAlGaAslayersatwhichthe2DEGresides.Studieshaveindicated

that optimum mobilities are achieved at growth temperaturesܶ௚between 600 and

700°C [39,64], and drops rapidly forܶ௚outside this range. This range of optimumܶ௚ coincides with the range of temperature over which the surface roughness of the AlGaAs/GaAsheterojunctionwasfoundtobethesmallest[63].Theeffectofinterface roughnessscatteringisstrongerathigherelectrondensitywherethe2DEGispulled closertotheAlGaAs/GaAsinterface.Itiscurrentlyoneofthemainfactorsthatlimit themobility[59].RecentstudiesusingashortͲperiodsuperlattice(SPSL)dopinginstead of the more standard݊ͲAlGaAs allows the use of smaller AlAs mole fraction spacer, which leads to lower background impurity levels as well as better interface quality. Mobilitiesashighas35ൈ106cm2/Vswereobservedinsuchstructures[2].

1.3.3.4–Surfacestates

Atthetopsurfaceoftheheterostructurethelatticeisabruptlyinterrupted.Asa result there are a large number of dangling bonds. Some of these join with those of adjacentatomsforsurfacereconstructiontominimizethesurfaceenergy,whilesome bond to reactive atmospheric species such as oxygen, and some remain free.This surfacechemistryiscomplex,buttheresultisalargepeakinthedensityofstates(DOS) atthesurfaceinthemiddleofthebandͲgapforanAlGaAs/GaAsheterostructure.The surfaceFermilevel“pins”attheDOSpeak,whichleadstothecharacteristicSchottky barrierformetalgatesdepositedonthissurface.Asubstantialnetchargeisassociated withthesesurfacestates,anditcanhaveasignificantimpactonscatteringandmobility asrecentlyhighlightedbyMaketal.[59].Intheirstudies,undopedAlGaAs/GaAswafers with different AlGaAs layer thicknesses grown in the same MBE chamber were measured.Figure1.7showstheirexperimentalresults.SampleswheretheAlGaAslayer thickness݀wassmall(i.e.݀=20nmor30nm)werefoundtohavesignificantlylower mobilitythansampleswiththickerAlGaAslayers.Thiseffectsaturateswhen݀>100nm, andasimilarbehaviourwasalsoobservedforsamplesfromanotherMBEsystem.

Figure1.7:(a)Schematicoftheundopedheterostructureusedin[59].(b)Comparisonof mobility ʅ vs. carrier density n for the undoped 2DEGs with different AlGaAs layer thicknesses݀.AllthreeheterostructuresweregrowninthesameMBEchamber.Thesample with݀=20nmshowssubstantiallylowermobilitythanthosewithlarger݀.For݀>70nm, thiseffectsaturates.(c)SimilarbehaviourisobservedwithsamplesgrowninChamberV. FiguresreproducedfromMaketal.,Fig.1(a)andFig.3of[59].

1.3.3.5–Alloydisorderscattering

Gallium Arsenide is a binary crystalline semiconductor with an alternating (GaͲAs)motifsuchthatthefournearestneighboursforeachGaatomsareAsatoms, and vice versa. In AlGaAs, the structure is not as straightforward, the group III component is essentially an alloy AlxGa1Ͳx, and beyond maintaining this ratio, the allocationofanAlorGaatomtoeachgroupVsiteisrandom.TheGaͲAsandAlͲAs bondsareslightlydifferent,andthisleadstoarandomdistributionofdeviationsin anotherwiseuniformscatteringpotential.Thisisknownasalloydisorderanddoes notoccurforGaAs.Becausethe2DEGresidesontheGaAssideoftheinterface,the effect of alloy disordermay appear to be negligible. However, in thehigh density limitwherethe2DEGispulledclosertotheAlGaAs/GaAsinterfaceandtheelectron

29 wavefunction extends slightly into the AlGaAs layer, the effect of alloy disorder scatteringisnolongernegligible[39].

In this thesis, we focus on studies of remote ionized impurity scattering, but rather than considering their effect on the mobility, we look at their effects on the characteristicsofthemagnetoͲconductancefluctuations(MCF)ofanopenquantumdot (introducedinSection1.5).Ourstudiessuggestthatsignificantreconsiderationofthe interpretationofMCFinquantumdotsisrequired,particularlywithrespecttoballistic transportandquantumchaos,asdiscussedinSections1.5.3and1.5.5.1respectively.

1.4–Basicsofelectrontransportinquantumdots

Semiconductorquantumdotscanbeconsideredassmallislandsconfinedinthe 2DEGthatarecapacitivelycoupledtooneormoregates.Thosegatescanbeusedto tunetheelectrostaticpotentialofthedotwithrespecttothereservoirs,changingthe numberofelectronsontheisland.Whenthesize ofthequantumdotiscomparableto the wavelength of the electrons that occupy it, the system exhibits a discrete energy spectrum.Asaresult,quantumdotsareoftenreferredtoasartificialatoms[65].

Thecouplingsbetweenthedotandthesourceanddrainreservoirsareusually controlledviaapairofquantumpointcontacts[53,54].Dependingontheconductance ofthepointcontacts݃,aquantumdotcaneitherbeoperatedinaweaklyͲcoupled(݃<< 2݁ଶȀ݄) or open (݃> 2݁ଶȀ݄)) regimes. Figure 1.8 shows an AFM image of a lateral quantumdotdevicediscussedinChapter4.Adegeneratelydopedcapispatternedinto sevengatesofwhichthetopgate(TG)isusedtodefinethequantumdotatthe2DEG.

TheQPCgates(QPCLandQPCR)areusedtochangethecouplingtothesourceanddrain contacts,andaplungergate(PG)isusedtotunetheelectronoccupancy.Inthissection, wefocusonthebasicsofquantumdottransportintheweaklyͲcoupledregime.

Figure1.8:AnAFMimageofaquantumdotdevicesimilartotheonediscussedinChapter4. Apatterneddopedcapisshowndividedintosevenseparategates.Thetopgate(TG)isused todefineadotatthe2DEG220nmbelowthesurface.TheQPCgates(QPCL&QPCR)are usedtotunetheconductanceofthepointcontacts,whiletheplungergate(PG)isusedto changethenumberofelectronsinthedot.

1.4.1–Theconstantinteractionmodel

Here,wedescribeasimplifiedpictureofhowelectronsonthedotinteractwith each other and with the reservoirs. It follows Averin et al. [66], Meir et al. [67] and Beenakker[68],whogeneralizedthechargingtheoryformetalsystemstoincludezeroͲ dimensional (0D) states. Two assumptions are made here: first, the Coulomb interactionsamongelectronsinthedot,andbetweenelectronsinthedotandthosein the environment, are parameterized by a single, constant capacitance ܥఀ . This capacitanceismostlythesumofthecapacitancesbetweenthedotandthesourceܥௌ, the drainܥ஽, and the gatesܥ௚. Second, the single particle energy level spectrum is independentoftheseCoulombinteractionsandhenceindependentofthenumberof electronsinthedot.Figure1.9showsaschematicrepresentationofalateralquantum dot in (a) and its equivalent circuit diagram in (b). This model is referred to as the constantinteraction(CI)model,formoredetaileddiscussions,seeKouwenhovenetal. [69].

Figure1.9:(a)Schematicrepresentationofaquantumdotinthelateralconfiguration.The dotisconnectedtosourceanddrainviatunnelbarriers.Transportmeasurementscanbe

madebymeasuringthecurrentܫinresponsetoabiasvoltageܸௌ஽andagatevoltageܸ௚.(b) The corresponding circuit diagram assuming the CI model. In this model, all electronͲ

electron interactions are represented by a single parameterܥఀ ൌܥௌ ൅ܥ஽ ൅ߑܥ௚, where

ߑܥ௚isthesumofthegatecapacitances,asthereisoftenmorethanonegatecoupledtothe dot.

UnderthesetwoassumptionsintheCImodel,thetotalenergyܷሺܰሻofadot withܰelectrons in the ground state is given by Eqn. 1.5, which contains two terms:

ே ൅ܳሻଶ כሺെ݁݊ ܷሺܰሻ ൌ෍ܧ௜ ൅ ሼͳǤͷሽ ʹܥఀ ௜

ே Thefirsttermisadiscretesumσ௜ ܧ௜ofallthesingleparticleenergiesrelativetothe bottomoftheconductionband,fromthelowestelectronuptotheܰthelectron.The ଶ כ second termሺെ݁݊ ൅ܳሻ Ȁሺʹܥఀሻis the electrostatic contribution of the gates to the כ כ energy, where݊ is the number of added electrons defined as݊ ൌܰെܰ଴(ܰ଴ corresponds toܰat zero gate bias) andܳis the excess charge induced by the

gatesܳ ൌ ܥௌܸௌ ൅ܥ஽ܸ஽ ൅ߑܥ௚ܸ௚. From Eqn. 1.5, we can calculate the electrochemical

potentialenergyofthedotwithܰelectronsߤௗ௢௧ሺܰሻ,whichisdefinedastheminimum th .energy necessary to add theܰ electron to the dotߤௗ௢௧ሺܰሻ ؠܷሺܰሻ െܷሺܰെͳሻ

ThisisshowninEqn.1.6asthesumofthechemicalpotentialܧேandtheelectrostatic

ߤௗ௢௧ሺܰሻ ؠ ܷሺܰሻ െܷሺܰെͳሻ

ே ேିଵ െͳሻ൅ܳሻଶ כ൅ܳሻଶ ሺെ݁ሺ݊ כሺെ݁݊ ൌ ൝෍ ܧ௜ ൅ ൡെ൝෍ܧ௜ ൅ ൡ ʹܥఀ ʹܥఀ ௜ ௜

ଵ כ ଶ ሻ ݁ ሺ݊ െ ଶ ݁ܳ ൌ ሼܧேሽ൅ቊ െ ቋሼͳǤ͸ሽ ܥఀ ܥఀ

potential݁߮ே. Hereߤௗ௢௧ሺܰሻdenotes the transition between theܰͲ electron ground state GS(ܰ) and the (ܰͲ1)Ͳ electron ground state GS(NͲ1). The difference in the electrochemical potentials of the transitions between successive ground states are

spacedbytheadditionenergyοߤௗ௢௧:

οߤௗ௢௧ ؠߤௗ௢௧ሺܰ൅ͳሻെߤௗ௢௧ሺܰሻ

ଶ ଵ ଶ ଵ ሻ ሻ ݁ ሺ݊ ൅ ଶ ݁ܳ ݁ ሺ݊ െ ଶ ݁ܳ ൌ ቊܧேାଵ ൅ െ ቋെቊܧே ൅ െ ቋ ܥఀ ܥఀ ܥఀ ܥఀ

݁ଶ ൌ οܧ ൅ ሼͳǤ͹ሽ ܥఀ

ଶ theterm݁ Τܥఀiscommonlyknownasthechargingenergyܧ஼,whereasοܧisthesingle particle energy level spacing between two discrete quantum levels. This can be approximatedusing:

ʹߨ¾ଶ ؆ ሼͳǤͺሽ ܧο ܣכ݉

isthedot’sareaand԰isPlanck’sconstantdividedbyܣ,istheeffectivemassכwhere݉ ଶ .ఀܥ՜ Ͳ,οߤௗ௢௧ ؆݁ Τ ܧߨ.Inalargequantumdotwhereοʹ

1.4.2–Coulombblockadeandsingleelectrontunneling

TheadditionenergyshowninEqn.1.7canleadtotransportblockadeasshown schematicallyinFig.1.10(a).Inthiscase,noneoftheavailabledotlevelsߤௗsitswithin thebiaswindowߤ௅ െߤோ,thereforetransportisnotallowed.ThisisknownasCoulomb blockade(CB)andtransportisonlypossiblebythermalactivationortunnelingviavirtual states[70].However,asshowninFig.1.10(b),thisblockadecanbeliftedbyvaryingthe electrostatic potential with a gate bias ܸ௚ , such that ߤௗ௢௧ሺܰ ൅ ͳሻ lies within

Figure1.10:Thesediagramsexplain(a)theCoulombblockadeand(b)thesingleelectron tunneling in the low bias regime (ߤ௅ െߤோ <οߤௗ௢௧ ) atܶ= 0 K. In (a), there is no electrochemicalpotentiallevelofthedotߤௗthatlieswithinthebiaswindowߤ௅ െߤோ.Asa result,transportisnotallowedandthisisknownasCoulombblockade(CB).In(b),theߤௗ௢௧ ladderisvariedbythegatebiasܸ௚,suchthatߤ௅ ൐ߤௗሺܰ൅ͳሻ ൐ߤோandtransportblockade islifted.Inthiscase,anelectronfromtheleftleadcantunnelintothedot,increasingܰby1, and subsequently tunnel out of the dot to the right lead. In the process, the number of electronsoscillatesbetweenNandN+1.Thisisknownas singleelectrontunnelingbecause onlyoneelectronisallowedtoenterthedotatatime.In(c),thecurrentܫvs.ܸ௚isshown withasetofCoulombblockadeoscillationscorrespondingtoCoulombblockadeandsingle electrontunneling.Themagnitudeofthecurrentdependsonthetunnelingratesbetween theleadsandthedot.

34 thebiaswindow.Inthiscase,anelectronfromtheleftleadcannowtunnelintothedot, andsubsequentlytunneloutofthedotviatherightlead.Inthistransition,thenumber ofelectronsinthedotoscillatesbetweenܰandܰ+1,resultingincurrentflowthrough thedot.Thecorrespondingܸ௚valuesarecommonlyknownasdegeneracypoints,and coincidewithcurrentpeaksinaplotofܫvs.ܸ௚,asshowninFig.1.10(c).Inthelowbias regime,wherethebiaswindowissmallerthantheadditionenergyofthedotߤ௅ െߤோ,<

οߤௗ௢௧, only one electron can tunnel into the dot at a time, this is known as single electrontunneling[69].

Asthegatebiasiscontinuouslyvaried,thecurrentܫthroughthequantumdot oscillates between zero and nonͲzero due to Coulomb blockade. These oscillations, referredtoasCoulombblockadeoscillations,areshownschematicallyinFig.1.10(c).The numberofelectronsisfixedinbetweenthepeaks,andtheperiodofCBoscillationsis given by Eqn. 1.9. The quantityܥ௚Ȁܥఀ is commonly referred to as the

ଶ ͳ ܥఀ ݁ οܸ௚ ൌ൬ ൰ ቊοܧ ൅ ቋሼͳǤͻሽ ݁ ܥ௚ ܥఀ

lever arm. In the limit ofοܧ ՜ Ͳ, such as in a large dot,οܸ௚ ՜݁Ȁܥ௚and we have perfectlyperiodicoscillations.

There are two conditions that must be satisfied in order to observe Coulomb blockade.Firstly,thenumberofelectronsinthedotmustbewelldefined.Thisimplies thattheenergyuncertaintyofanelectronduetoitsdwelltimeonthedotshouldbe muchsmallerthanthechargingenergy(i.e.οௗ௢௧<<ܧ஼).Equivalently,itmeansthatthe barrier conductance should be much smaller than 2݁ଶȀ݄. Secondly, the thermal distributionofelectronsneartheFermilevelintheleadsmustbemuchsmallerthanthe chargingenergy(i.e.݇஻ܶ<<ܧ஼).

1.4.2.1–ThelineshapeoftheCoulombblockadeoscillations

ForCoulombblockadeoscillationsobservedinthetransportmeasurementsof quantum dots, two temperature regimes are particularly important: classical and quantum. The classical Coulomb blockade regime corresponds to the condition: ଶ اܧοاఀ , whereas the quantum regime corresponds to:݇஻ܶܥȀ ݁ا஻ܶ݇ ا ܧο ଶ ݁ Ȁܥఀ. According to “orthodox” Coulomb blockade theory [68], the lineshape of an individualconductancepeaktakesoneofthetwoformsbelow:

݃ ߜ݇Τ ܶ ൌ ஻ ݃ஶ ʹሺߜΤ ݇஻ܶሻ

ͳ ߜ ൎ ିଶ ൬ ൰ሼͳǤͳͲሽ ʹ ʹǤͷ݇஻ܶ

݃ ο ߜ ൌ ௗ௢௧ ିଶ ൬ ൰ሼͳǤͳͳሽ ݃ஶ Ͷ݇஻ܶ ʹ݇஻ܶ

Associatedwithbothformsareanumberofparametersthatneedintroduction.Firstly we have݃andܸ௥௘௦which correspond to the peak conductance and the gate voltage correspondingtothepeak.Theparameterߜൌ݁ሺܥீΤܥఀሻሺܸ௥௘௦ െܸ௚ሻisthehalfͲwidthat halfͲmaximumofthepeakinenergyunits(2ߜȀ݁istheFWHMinaplotof݃vs.ܸ௚).In thequantumcase(Eqn.1.11),οௗ௢௧representsthebroadeningofthedotlevelsdueto coupling with the leads. Finally݃ஶis a normalization factor such that 1/݃ஶ= 1/݃௅+

1/݃ோcorrespondstothemeasuredresistanceofthedotinthehightemperaturelimit ଶ (݇஻ܶ>>݁ Ȁܥఀ).

In the case of the classical CB regime (Eqn. 1.10), the maximum peak height

݃௠௔௫does not change with temperature and is given by݃ஶȀʹ. For the quantum CB regime, the height decreases linearly with increasing temperature, ݃௠௔௫ ൌ

݃ஶሺοௗ௢௧ΤͶ݇஻ܶሻ. Equations 1.10 and 1.11 assume that the barrier conductances are ଶ Τ݄.Thisassumptionimpliesthatοௗ௢௧ismuchsmallerthan݇஻ܶeven 2݁ا small݃௅ǡ݃ோ

36 atlowܶ.Thisisequivalenttotherequirementthatonlyfirstordertunnelingistaken intoaccountandhigherordertunnelingviavirtualintermediatestatescanbeneglected.

Atreatmentoftheregimewhere݇஻̱ܶοௗ௢௧isgivenby[71Ͳ73].

1.4.3–HighbiasregimeandCoulombdiamonds

AnotherwaytolifttheCoulombblockadeconditionistowidenthebiaswindow byraisingtheelectrochemicalpotentialofoneoftheleadsrelativetotheother.Inthe followingwewillconsiderthecasewhereonlytheleftleadisbiasedwhiletherightlead isheldatground (asymmetricbiasing).

The application ofܸௌ஽causes the degeneracy points along theܸ௚axis (i.e. currentmaxima)tosplit.Figure1.12explainsthedynamicsofthissplitting[74].First,we drawtheenergylevelsofܷሺܰሻandܷሺܰ ൅ ͳሻincludingtheexcitedstates,asshownin Fig.1.12(a),andthenwemapoutallthepossibletransitionasindicatedinFig.1.12(b). These include the ground state transition GS(ܰ)՞GS(ܰ+1) and the excited state transitions: GS(ܰ)՞ES(ܰ+1) and GS(ܰ+1)՞ES(ܰ).* When we plot the differential conductance݀ܫȀܸ݀ௌ஽asafunctionofܸௌ஽andܸ௚,thesplittingateachdegeneracypoint alongtheܸ௚axisformsa“V”shapeasshowninFig.1.12(c).Assumingtherightleadis grounded, the two slopes areെ݁ܥ௚Ȁሺܥఀ െܥௌሻandሺܥ௚Ȁܥௌሻ. Along the black line of positiveslope,ߤௗ௢௧isfixedandthedotlevelcorrespondingtothetransitionofGS(ܰ)՞

GS(ܰ+1)isalignedwiththeelectrochemicalpotentialoftherightleadߤ஽,whiletheleft leadmovesupinenergy.Conversely,ifwefollowtheblacklinewiththenegativeslope, thesameGStransitionlevelisnowalignedwiththeleftleadandmovesalongwithitas theleftleadmovesupinenergy.Attheintersectionbetweenthereddlinean theblack lineǡES(ܰ+1)openstotheleftleadandallowsthetransitionES(ܰ+1)՞GS(ܰ)tooccur. Similarly,attheintersectionbetweentheblueandtheblackline,ES(ܰ)opensuptothe rightleadandtransitionbetweenES(ܰ)՞GS(ܰ+1)isallowed.

*Althoughtransitionsbetweenexcitedstatescanoccur[75],theyhavebeenignoredheretomaintainthe simplicityoftheschematic.

Figure1.12:ThesediagramsexplaintheexcitedstatesseenintheCoulombdiamonds.(a) Boththegroundstate(GS)andtheexcitedstate(ES)ofthequantumdotwithܰandܰ൅ͳ electronsareshown. The possibletransitionsareindicatedwith colouredarrowsandthe transitionbetweenES(ܰ)andES(ܰ+1)isignored.(b)Theelectrochemicalpotentialofthe dotߤௗ௢௧withܰ+1electronscorrespondstothethreepossibletransitionsshownin(a).(c)A schematicofbiasspectroscopyshowinghowthethreetransitionsin(a)evolveasafunction ofthesourceͲdrainbiasenergyܸ݁ௌ஽.ForthetransitionbetweenGS(N)andGS(N+1),theline extendingtothemorepositivegatevoltageܸ௚correspondstothesituationwheretheright leadisconstantlyinlinewiththeGS(ܰ)՞GS(ܰ+1).Thelineonthemorenegativesideofܸ௚ correspondstotheleftleadtrackingtheGS(ܰ)՞GS(ܰ+1)transitionasitmovesupwardsin energy.FigurereproducedandmodifiedfromFig.5of[74].

Figure1.13showsaseriesofdiamondͲlikeblockaderegions(ingrey)observed when both the positive and negativeܸௌ஽sides are plotted. The pink regions indicate where single electron tunneling occurs. In regions where the bias window is large enoughformorethanonedotstatestooccupy,multipleelectronscantunnelintothe ଶ dotatthesametime.Theadditionenergyοߤௗ௢௧ ൌοܧ൅݁ Ȁܥఀcanbeextractedfrom thehalfdiamondheight,andοܧcanbeextractedbytakingtheverticalseparationfrom theܸ௚axistotheintersectionoftheexcitedstatelineandthediamondedgeasshown inthefigure.

Figure1.13:SchematicofbiasspectroscopyshowingCoulombdiamondsingrey.Thepink regionscorrespondtoregionsofsingleelectrontunneling.Theadditionenergyοܧ ൅ ܧ௖can be extracted from half diamond height, whereasοܧcan be obtained from the vertical separationoftheintersectionoftheexcitedstateline(showninred)andthediamondedge, fromtheܸ௚axis.

1.5–MagnetoǦtransportofopenquantumdots

Oneofthemainfocusesofthisthesisistostudydisorderinheterostructuresby usingthemagnetoͲconductancefluctuations(MCF)producedbyanopenquantumdot toactasasensitiveprobeoftheelectrondynamicswithinthedot.Inthissection,we introducethekeyconceptsofmagnetoͲtransportin“open”quantumdots,wherethe

39 barrierconductanceofthequantumdotislargerthan2݁ଶȀ݄.Beforeweproceed,we willfirstdiscussthebasicsofelectroninterferencephenomena.

1.5.1–TheAharonovǦBohmeffect

Advances in the fabrication of nanoscale electronic devices have allowed experimentsthatwereoriginallyplannedonlyforvacuumtobeperformedinasolidͲ stateenvironment.OnesuchexperimentistherealizationoftheAharonovͲBohm(AB) effect.FirstpredictedbyY.AharonovandD.Bohmin1959[76],theAharonovͲBohm effect demonstrates the fundamental interplay between quantum mechanics and electromagnetism.Theoriginalproposedexperimentwastosplitabeamofelectrons travelling in vacuum such that two halves of the beams travel around a region containingamagneticfluxandthenrecombineintoasinglebeamuponexit.Asaresult of quantum interference at the combination point, the intensity of the beam was expectedtooscillateasafunctionofappliedmagneticfield.TheAharonovͲBohmeffect wasfirstdemonstratedforasolidstatesystembyWebbetal.[77]whoperformedthe experimentusingananoscalegoldringconnectedonoppositesidestosourceanddrain contacts. A schematic for this structure is shown in Fig. 1.14. Electron partial waves

Figure1.14:SchematicrepresentationsoftheAharonovͲBohmeffectproducing(a)݄/݁and (b)݄/2݁oscillationsundertheinfluenceofasmallperpendicularmagneticfieldܤ.In(a), theelectronpartialwavestakeboththeleftandtherightpathsandexitatthetop.Upon recombination, etheir phas difference isο߮ ൌ ʹߨܤܣȀሺ݄Ȁ݁ሻ, which leads to a sinusoidal oscillationinconductance݃vs.ܤwithaperiodοܤ ൌ ݄Ȁ݁ܣ.In(b),theelectronpairstravel allthewayaroundtheringandrecombineatthestartingpoint.Theygiverisetooscillations withοܤ ൌ ݄Ȁʹ݁ܣ.FigurereproducedfromFig.2.5of[78].

40 enteringtheringcantakeeithertheleftorrightarmandinterfereattheothersideof thering.Inthepresenceofamagneticfieldܤperpendiculartotheplaneofthering,a ݄ phase difference ofο߮=ʹߨܤܣȀሺ ൗ݁ሻ, develops between the two interfering paths, whereܣistheareaenclosedbythering.Atlowtemperaturewherephasecoherence can be maintained over distances larger than the ring circumference, quantum interference leads to a sinusoidal oscillation in the magnetoͲconductance with period οܤ ൌ ݄Ȁ݁ܣǤ

NotethatitisalsopossiblefortheelectronpartialͲwavestotravelalltheway aroundtheringbeforerecombining,asshowninFig.1.14(b).Thisprocessconstitutes coherentbackscatteringattheentrance.Suchtrajectoriesenclosetwiceasmucharea, andhencetwiceasmuchmagneticflux.ThesetimeͲreversedpairsproduceoscillations withperiod߂ܤ ൌ ݄Ȁሺʹ݁ܣሻǡtheyareanalogoustotheweaklocalizationeffectin2D[79], andareoftenknownasAl’tshulerͲAronovͲSpivak(AAS)oscillations[80].

1.5.2–SemiconductorbilliardsandMagnetoǦConductance Fluctuations(MCF)

A semiconductor billiard is formed by confining the 2DEG in an AlGaAs/GaAs heterostructuretoaspecificplanargeometrythatismuchsmallerthantheelasticmean freepathoftheelectronsinthe2DEG.Thebilliardisconnectedtotherestofthe2DEG vianarrowentranceandexitopenings(i.e. QPCs)intheconfinementgeometry.Under thisdefinition,anopenquantumdotisconsideredtobeabilliarddevice.Historically, twotreatmentswereusedtounderstandthetransportinsemiconductorbilliarddevices: thesemiclassicaltreatmentandthefullquantummechanicaltreatment.Thefollowing discussions provide the basic concepts and implications of each approach. However, beforeweproceed,itisnecessarytodistinguishthedifferentmagneticfieldregimes.

Four magnetic field regimes for electron transport can be identified, they are shown schematically in Fig. 1.15 (assuming a hardͲwalled confinement potential) depending on the relationship between the fieldܤ, the width of the billiardܹ஻and

Figure 1.15:Schematics of four different magnetic field regimes: (a) Atܤ=Ͳ, electron travelsinastraightpathasthereisnomaagneticͲinducedcurvature.(b)Whenݎ௖௬௖>ܹ஻/2, electrontrajectoriesintersectingtheQPCsarecurvedandabletoaccessthewholeregionof thebilliard,bothregimesin(a)and(b)arereferredtoasthelowfieldregime.(c)Asݎ௖௬௖<

ܹ஻/2, the cyclotron orbit is such that electron trajectories intersecting the QPCs can no longeraccessthecentralregionofthebilliard,insteadthey“skip”alongthebiilliardwalls.(d)

For higher fields such thatݎ௖௬௖<ܹ௤/2, the radius of the orbitis so small such thatan electrontravellingalongthebilliardwallisguaranteedtoexitatthefirstQPCitencounters, and can only travel as far as the distance between the entrance and the exit. The semiclassicaltheoryholdsinthelowfieldregime.FigurereproducedfromFig..2.8of[78].A hardͲwalledconfinementpotentialisassumed.

thewidthoftheQPCܹ௤.Thefirstregimeshownin(a)occursatܤ=0T,wherethereis no cyclotron curvature in the trajectories and no fieldͲinduced phase difference betweentrajectories.Anyphasedifferencecomespurelyfromthedifferencceinthepath lengths. The second regime shown in Fig 1.15(b) corresponds to fields where the cyclotronradiusݎ௖௬௖=԰݇ிȀ݁ܤislargerthhanhalfܹ஻.Heretheelectronfolllowsacurved trajectory,butthatcurvatureisnotsolargethatelectrontrajectoriesinteerceptingthe QPC cannot access the central region oof the billiard. The regimes in (a)) and (b) are referred to as the low field regime, where the MCF can be meaningfullly considered within the semiclassical picture introduced in Section 1.5.3.At higher magnetic fields therearetwofurtherregimes:thefirstoccurswhenܹ௤/2<ݎ௖௬௖<ܹ஻/2andisknownas theskippingorbitregime(Fig.1.15c).Herethecyclotronradiusissuchthattrajectories interceptingtheQPCscannolongeraccessthecenterofthedot,butinsteadfollowthe walls uuntil they reachaQPC and escappe. The second regimeoccurs whenݎ௖௬௖<ܹ௤/2

42 andisknownastheedgestateregime(Fig.1.5d).Thedifferencebetweenthisandthe skippingorbitregimeisthatthecyclotronradiusissosmallthatanelectronenteringthe dotwillfollowthewallandisguaranteedtoescapeatthefirstQPCthatitencountersas itcannotbouncefarenoughineachorbittojumpacrossthegappresentedbytheQPC.

Notethatinpractice,duetothesmearinginܧி,theshapeofthedotwalls,etc.,these distinctions in the field regimes are by no means precise, but they enable sensible choicestobemadeintermsoftherangeinܤoverwhichtheMCFcanbeconsideredto representthesemiclassicaldynamicsofelectronswithinthebilliard.

1.5.3–Thesemiclassicalapproach

Under the semiclassical theory, ifܹ஻is much larger than the electron Fermi wavelengthߣி, then electrons can be pictured as moving along classical trajectories [25,47,81].Eachofthesetrajectorieshasaquantummechanicalamplitudeandphase associated with it. At sufficiently low temperature, the phase of electron waves is preserved over distances much larger thanܹ஻as they traverse the billiard. With the application of small magnetic fieldܤperpendicular to the plane of the billiard, the magneticfluxenclosedbypairsofinterferingelectronpartialwavescanbeconsidered asanAharonovͲBohmloop,whichgeneratesaperiodicoscillationinconductancewitha periodinverselyproportionaltotheareathatthetwopathsenclose[76].Aschematic representationofthissemiclassicaltreatmentinthecasewhereasmallܤisappliedis showninFig.1.16.Electrontransportfromthereservoirintothebilliardviathepoint contactisanalogoustothediffractionoflightthroughasmallpinhole;thisresultsinan incidentelectronbeamspreadingintothebilliarduponentry[82].Inthesemiclassical model,thisdiffractionoftheelectronwaveisaccountedforbytakingthesumofthe interference terms between all possible pairs ofclassical trajectories that an electron couldtakebetweentheentranceandexitQPCs.ThosepathsthatendupattheexitQPC willcontributetothetransmission,whereastheonesthatendupattheentranceQPC willcontributetobackscattering.Intermsofelectroninterference,theycorrespondto theABandAASeffectsrespectively(seeFig.1.14).eThes setsofpossiblepathsleadtoa distributionofdifferentloopareas,whichinturnproduceaseriesofdifferentperiodsin

Figure 1.16:A schematic of the semiclassical treatment of billiard transport is shown. A squarebilliardwithalignedQPCsisoutlinedinblack,whiletheelectronpartialwavesare showninredandblue.Thetrajectoriesshownarecurvedduetoasmallperpendicularfield ܤ. At the QPCs where the two paths intersect and interfere, their phase difference is a functionofܤaccordingtotheABeffect.ThecorrespondingMCFconsistingoftheFourier sumofalltheperiodsduetothepossibleABloopsintersectingtheQPCs,arecommonly consideredtobearepresentativeofbilliarddynamics.TheeffectofsmallͲanglescattering arecommonlyconsideredtobenegligibleinthetraditionalviewofballistictransport,when

݈௘௟>>ܹ஻.

ܤ.TheresultingmagnetoͲconductancefluctuations(MCF)consistoftheFouriersumof alltheseperiods,andcanthereforebeviewedasamagnetoͲfingerprintoftheclassical trajectoriesthatissensitivetotheprecisedistributionofloopareassupportedbythe billiard[21Ͳ23].

Inthetraditionalballistictransportregimewherethemeanfreepathismuch largerthanܹ஻ǡlargeͲanglescatteringprimarilyoccursatthebilliardwalls.Itistherefore widelyacceptedthatthecorrespondingMCFrepresentelectrondynamicsestablished primarilybythebilliardgeometryalone[83].Withinthissemiclassicaltreatment,smallͲ anglescatteringisconsideredtoonlycausesmallperturbationstotheoverallbilliard dynamics [21,24]. In this thesis, we will present experimental evidence to show that smallͲanglescatteringplaysasignificant,andperhapsdominantroleindeterminingthe magnetoͲconductancefluctuationsofthebilliard[34,35].

1.5.4–Thequantummechanicalapproach

The fully quantum mechanical approach abandons any notions of classical trajectoriescarryingaquantumphaseasinthesemiclassicalpicture,andreliesinstead

on numerically obtaining the electron waveͲfunction ߖሺݔǡ ݕሻ , and thereby the probabilityoftransmissionthroughthedot.Theprocedureinvolvesdefininga2Dlattice of points within the dot with the waveͲfunction at each point calculated using an iterativematrixapproach[84,85].Theconductanceߜ݃isobtainedusingtheLandauerͲ Büttikerformalismbasedonthetransmission/reflectionprobabilitiesobtainedfromthe resulting waveͲfunction. Simulation of the magnetoͲconductance reveals there are periodic features inߜ݃vs.ܤthat correspond to a very similar set of quantum waveͲ functionamplitudesinsidethedot.Figure1.17showsanexampleofߜ݃vs.ܤcalculated forasquarebilliard[84]similartotheoneshowninFig.1.16.Periodicfeaturesinߜ݃vs. ܤare highlighted by the arrows, and the corresponding waveͲ function amplitudes ߖሺݔǡ ݕሻȁare shown at the bottom of Fig. 1.17, where the darker regions represent| higher amplitude. Interestingly in all four |ߖሺݔǡ ݕሻȁsimulations, essentially the same diamondͲshapedpatternsareobserved,withtheamplitudesreachingmaximaalongan underlyingclassicalorbit.Experimentalevidencefortheseperiodicstateswereinitially providedbymeasurements[86]wherethesmallͲscalefluctuationsindotconductance

Figure1.17:Anexampleofusingfullyquantummechanicalsimulationsfora0.3ʅm×0.3 ʅmsquarebilliard.TheplotshowsthecalculatedmagnetoͲconductancefluctuationsߜ݃asa function ofܤ. Four resonance features that appear in the magnetoͲconductance are indicatedbythearrows,andthecorresponding waveͲfunctionamplitudesareshown.The darker regions correspond to regions of higher amplitude, plotted as a function of the billiardcoordinates.FigurereproducedfromFig.2of[84].

݃vs.ܸீ,(ܸீisthesplitgatebias)wereinvestigated.Figure1.18showsacomparisonof theFourierspectraofthemeasuredandcalculatedfluctuationsin݃vs.ܸீ,forasquare billiard shown in the upper inset. These Fourier peaks represent the frequencies at whichspecificsetsofscarͲlikewaveͲfunctionfeaturesrecurinthedots[86].Foursetsof differentrecurringscarͲlikewaveͲfunctionfeaturescorrespondingtothelabeledpeaks areshowninthelowerinsets.Thesimulationresultwasfoundtocloselyreproducethe frequenciesatwhichthepeaksoccur.

Figure 1.18:Results from [86] showing a comparison between calculated (dashed) and measured(solid)Fourierspectrumofoscillationsin݃vs.ܸீ.TheupperinsetshowsaSEM imageofthesquarebilliarddeviceused.ThelowerinsetsshowthreesetsofrecurringscarͲ likewaveͲfunctionamplitudesthatcorrespondtothepeakslabeledinthemainfigure.The simulationcloselyreproducedtheexperimentalresult,providingevidenceoftheexistence of periodic orbits proposed by the full quantum mechanical treatment of open quantum dots.

RecentadvancesinscanninggatemicroscopyusingabiasedAFMtiptoraster scan over an the area of interest while concurrently mapping the variations in conductancecausedbytheperturbationhavealloweddirectimagingoftheseperiodic

46 orbits[87Ͳ88].Figure1.19showsresultstakenfrom[88],wherea“diamond”periodic orbitwasimagedfromanetchedsquarebilliard.

Figure1.19:Resultsfrom[88]showingdirectevidenceofperiodicorbitsaspredictedbya fullyquantummechanicaltreatmentofbilliardtransport.(a)AnAFMimageofthebilliard usedisshownonthetopleftpanel.(b)Asimulationresultcorrespondingtothebilliardin(a) isshown.(c)Animagegeneratedfromscanninggatemicroscopyispresented,showinga diamondstructureingreatresemblancetothesimulatedresultin(b).Thedasheddiamond shapeisdrawnasaguidetotheeyefortheperiodicorbitwithinthedot.Thecolouraxesin (b) and (c) represent the changes in the dot conductance, which correspond to the amplitudeoftheelectronwavefunction.

1.5.5–Billiarddynamicsandquantumchaos

Given the central role of classical trajectories in the semiclassical theory, the natureofbilliarddynamicsbecomesimportant.InFig.1.20(a),acircularbilliardisshown with the particle motion constrained only by specular reflections from the boundary walls.Acircularbilliardisthereforeclassicallyregular,whichmeansthatalltrajectories resultinstableperiodicorbitsthatretracethemselves.The“diamond”orbitdepictedin Fig.1.20(a)isarepresentativeexample.Incontrast,thestadiumbilliardshowninFig. 1.20(b)isclassicallychaotic[89].Trajectoriesinsuchsystemshowextremesensitivityto

Figure1.20:Schematicrepresentationsoftwoclassicalclosedbilliardsystems.(a)Acircular billiardisshownwithaperiodicelectrontrajectory.Theelectronpath(red)retracesitself indefinitely.Thissystemisreferredtoasregular.In(b),astadiumbilliardexhibitschaotic behaviourasitselectrontrajectoryshowsexponentialsensitivitytoinitialconditions.Orbits inachaoticsystemarenonͲperiodic.(c)Aphasediagramcommonlyknownasa“Poincaré section”thatcorrespondstoasliceacrossthecircularbilliardindicatedbyadashedlinein (a), showing two discrete points. (d) The corresponding phase diagram for the chaotic billiardsystemin(b),showingpointsthatfillupthewholephasespacewhenthesystemis allowedtoevolveforasufficientlylongtime.

initial conditions. This means that pairs of trajectories that were initially very close to one another will diverge away from each other exponentially in time. OrbitsinsuchsystemarenotstableandreferredtoasnonͲperiodic.Aphasediagram

commonlyknownasaPoincarésectionshowingtheparticle’svelocityݒ௫asafunction ofitspositionݔisoftenusedtorepresentthedynamicsofasystem:ifwetakeaslice across the circular billiard (shown as dashed line) in Fig. 1.20(a), the corresponding phase diagram will have two points as indicated in Fig. 1.20(c). For more complex periodicorbits,wewillgetadditionalnumberofpoints.Incontrast,thephasediagram thatcorrespondstothedashedlineinthestadiumbilliardshowninFig.1.20(b)fillsthe wholespacewhenweletthesystemevolveforasufficientlylongtime.Thisisshown schematicallyinFig.1.20(d).Inamorecomplicateddynamicalsystemthatcontainsa mix of regular and chaotic dynamics, the corresponding phase space breaks up into

48 smallislandswithdiscretepointsinthemthatrepresentperiodicorbits,andaseaof pointsaroundtheoutsidewhichcorrespondstochaos.These“islands”ofstableorbits inachaotic“sea”aretypicallyreferredtoasKolmogorovͲArnoldͲMoser(KAM)islands [89].In1996,Ketzmerick[90]proposedthatthepresenceofsoftͲwallpotentialprofiles inmesoscopicsystemsingeneralwouldleadtoamixedphaseͲspace.

1.5.5.1–Proposedsignaturesforquantumchaosinquantumdots

Acentralquestioninquantumchaosiswhetherscatteringfromasystemwhich classicallyexhibitschaoticdynamicswillgiverisetoanobservablequantumsignature thatisdistinctfromthatofanonͲchaoticsystem.AsdiscussedinSection1.5.3,under thetraditionalviewofballistictransport,smallͲanglescatteringisnormallyassumedto be negligible in comparison to scattering from the dot walls imposed by the billiard geometry. This initiated the study of quantum chaos using semiconductor billiards [21,22,25Ͳ31].In1992,Marcusetal.[21]reportedanexperimentalstudycomparingthe propertiesoftheMCFproducedbyacircularbilliard(nominallyregular)andastadiumͲ shapedbilliard(nominallychaotic).Theircorrespondingpowerspectrawerecompared, anditwasfoundthatthecircularbilliardshowedanincreasedhighͲfrequencyspectral content compared to the stadium, and hence the spectral content of MCF was consideredtobeasignatureofquantumchaos.Followingthisexperiment,Baranger, JalabertandStoneperformedtheoreticalinvestigationsregardingthelineshapeofthe “weaklocalization”peakinthemagnetoͲresistanceofaquantumdot.Inparticular,they predictedthatachaoticsystemshouldleadtoaLorentzianlineshape,whilearegular system would lead to a linear lineshape [26]. In 1994, Chang et al. reported the observation of the predicted lineshapes in circular and stadium cavities [22]. Their experimentaldatawasobtainedbystudyingtwodevices,eachcontaining48nominally identical billiard structures connected in parallel between the source and drain reservoirs. This approach was taken to average out the fluctuations due to quantum interferenceineachindividualbilliardstructure.Anumberofsubsequentexperiments obtained results conflicting with those of Chang et al. For example, both lineshapes were obtained for a square billiard depending on the applied gate bias [91] or temperature [22]. Moreover, in some experiments involving circular cavities, where

49 linearbehaviourswereexpected,Lorentzianlineshapeswerefound[93,94].Finally,our main results here (see Chapter 5 & 6) demonstrate that the electron dynamics established by the billiard geometry is largely randomized due to the influence of remoteionizedimpurities[93,94].Undersuchascenariothelineshapeatܤ=0cannot beusedtodeterminethenatureofthebilliarddynamics.

1.5.5.2–Modificationtothequantumpicture:dynamicaltunneling

AninterestingquestionarisesintryingtounderstandthediamondorbitwaveͲ functionscarsinFig.1.17fromtheperspectiveofthesemiclassicalpicture,whereonly trajectories that intersect the QPCs can contribute to the billiard conductance. The questionishowdoesadiamondͲshapedperiodicorbitaffecttheconductancewhenit neverintersectstheQPCs?Underthesemiclassicalpicture,anelectronfollowingsucha pathcanneverenterorexitthebilliard.

AnanswertothisquestionwasproposedbydeMouraetal.[94],andinvolveda considerationoftheuncertaintyprincipleandthemixedphasespace.Figure1.20shows adynamicalsimulationbydeMouraetal.forasquarebilliard.Thecorrespondingphase space contains KAM islands, which correspond to stable periodic orbits such as the diamond scar, surrounded by a chaotic sea, which contains trajectories that intercept the QPCs, and can thus enter and leave the billiard. An important aspect of quantum chaos is the inclusion of quantum uncertainty to an otherwise precisely deterministic dynamical system. What this means is that a phase

space such asݒ௫ vs.ݔ is “blurry” at areas of order԰Ȁʹdue to the uncertainty

requirementο݌௫οݔ ൒ ԰Ȁʹ.IfaperiodicorbitonaKAMislandiswithinthisuncertainty regionofthechaoticsea,thenitispossiblefortheorbittoaccessatrajectorybywhich it can escape the dot and influence the conductance. This process is known as “dynamicaltunneling”,bydirectanalogywithquantummechanicaltunnelingacrossa potentialbarrier.

Figure1.20:PoincarésectionofelectronvelocityasaproportionoftheFermivelocityݒȀݒி vs.thehorizontalpositionݔofanelectronwhenitstrajectoryinterceptsthewhitedashed lineindicatedintheinset(upper).Thechaoticseasurroundingtheislandistheplotofone singletrajectorythatstaysneartheKAMislandforalongtimebeforeescapingfromthedot. TheinsetcorrespondstothecalculatedbilliardpotentialdefinedbyasplitͲgateatagate voltageof–0.6VwithFermienergy=14.3meV.FigureproducedfromFig.1of[94].

1.6–Howdoesthisthesisfitwithinthecontextofexisting knowledgeofbilliards?

Both the semiclassical and full quantum mechanical treatments of the billiard systemdiscussedabovetendtooverlooktheinfluenceofsmallͲanglescattering.Inthis thesis, we adopt the semiclassical view of electron transport in billiards with an importantdistinction:remoteionizedimpuritiesplayadominantroleindeterminingthe electrontrajectories.Toshowthis,weperformedcomparativeMCFstudiesonseveral separatecooldownsofthreepairsofbilliardswithnominallyidenticalgeometries.The billiards were fabricated from two different AlGaAs/GaAs systems: one from a modulationͲdoped heterostructure (C2275) and the other from an undoped heterostructure (NBI30). These results suggest that the MCF are predominately determinedbytheprecisespatialdistributionofremoteionizedimpurities(i.e.smallͲ angle scatterers) rather than by the geometry of the billiard walls. This argument is consistentwithpreviousstudiesshowingthatotherpropertiesoftheMCF,suchasthe fractaldimensionortrendsinthepowerspectrathataremoreorlessindependentof billiardgeometry[95,96].

Bibliography

[1]“ElectronmobilitiesinmodulationͲdopedsemiconductorheterojunctionsuperlattices” –R.Dingle,H.Störmer,A.C.GossardandW.Wiegmann,Appl.Phys.Lett.,33,665(1978).

[2]“MBEgrowthofultraͲlowdisorder2DEGwithmobilityexceeding35×106cm2/Vs”– V.Umansky,M.Heiblum,Y.Levinson,J.Smet,J.NüblerandM.Dolev,J.Cryst.Growth, 311,1658(2009).

[3]“ChargenoiseanalysisofanAlGaAs/GaAsquantumdotusingtransmissionͲtype radioͲfrequencysingleͲelectrontransistortechnique”–T.FujisawaandY.Hirayama,Appl. Phys.Lett.,77,543(2000).

[4]“ModulationofNoiseinSubmicronGaAs/AlGaAsHallDevicesbyGating”–Yongqing Li,CongRen,P.Xiong,S.vonMolnár,Y.Ohno,andH.Ohno,Phys.Rev.Lett.,93,246602 (2004).

[5]“LowͲfrequencynoiseinquantumpointcontacts”–F.Liefrink,J.I.Dijkhuis,andH. vanHouten,Semicond.Sci.Technol.,9,2178(1994).

[6]“DistributionsofsingleͲcarriertrapsinGaAs/AlxGa1оxAsheterostructures”–T. Sakamoto,Y.Nakamura,andK.Nakamura,Appl.Phys.Lett.,67,2220(1995).

[7]“TimeͲirreversiblerandomtelegraphsignalduetocurrentalongasinglehopping chain”–D.H.Cobden,A.Savchenko,M.Pepper,N.K.Patel,D.A.Ritchie,J.E.F.Frost, andG.A.C.Jones,Phys.Rev.Lett.,69,502(1992).

[8]“Conductanceinstabilitiesinquantumpointcontacts”–J.രC.Smith,C.Berven,M.രN. Wybourne,andS.രM.Goodnick,Surf.Sci.,361/362,656(1996).

[9]“OriginofswitchingnoiseinGaAs෸AlxGa1оxAslateralgateddevices”–M.PioroͲ Ladrière,J.H.Davies,A.R.Long,A.S.Sachrajda,L.Gaudreau,P.Zawadzki,J.Lapointe,J. Gupta,Z.Wasilewski,andS.Studenikin,Phys.Rev.B,72,115331(2005).

[10]“InSituReductionofChargeNoiseinGaAs/AlxGa1ͲxAsSchottkyͲGatedDevices”–C. Buizert,F.H.L.Koppens,M.PioroͲLadrière,HͲP.Tranitz,I.T.Vink,S.Tarucha,W. Wegscheider,andL.M.K.Vandersypen,Phys.Rev.Lett.,101,226603(2008).

[11]“ResistancefluctuationsinGaAs/AlxGa1ͲxAsquantumpointcontactandHallbar structures”–C.Kurdak,C.ͲJ.Chen,D.C.Tsui,S.Parihar,S.LyonandG.W.Weimann, Phys.Rev.B,56,9813(1997).

[12]“ModulationofNoiseinSubmicronGaAs/AlGaAsHallDevicesbyGating”– YongqingLi,CongRen,P.Xiong,S.vonMolnár,Y.Ohno,andH.Ohno,Phys.Rev.Lett., 93,246602(2004).

[13]“1/fNoiseinanAlxGa1оxAs/GaAsheterostructurebetween77and300K”–L.Ren andM.രR.Leys,Physica(Amsterdam)192B,303(1993).

[14]“ResistivityofDilute2DElectronsinanUndopedGaAsHeterostructure”–M.P.Lilly, J.L.Reno,J.A.Simmons,I.B.Spielman,J.P.Eisenstein,L.N.Pfeiffer,K.W.West,E.H. Hwang,andS.DasSarma,Phys.Rev.Lett.,90,056806(2003).

[15]“TwoͲDimensionalMetalͲInsulatorTransitionasaPercolationTransitioninaHighͲ MobilityElectronSystem”–S.DasSarma,M.P.Lilly,E.H.Hwang,L.N.Pfeiffer,K.W. West,andJ.L.Reno,Phys.Rev.Lett.,94,136401(2005).

[16]“DisappearanceofMetalͲLikeBehaviorinGaAsTwoͲDimensionalHolesbelow30 mK”–JianHuang,J.S.Xia,D.C.Tsui,L.N.Pfeiffer,andK.W.West,Phys.Rev.Lett.,98, 226801(2007).

[17]“VariabledensityhighmobilitytwoͲdimensionalelectronandholegasesinagated

GaAs/AlxGa1ͲxAsheterostructure”–B.E.Kane,L.N.Pfeiffer,K.W.WestandC.K. Harnett,Appl.Phys.Lett.,63,2132(1993).

[18]“HighmobilityGaAsheterostructurefieldeffecttransistorfornanofabricationin whichdopantͲinduceddisorderiseliminated”–B.E.Kane,L.N.Pfeiffer,andK.W.West, Appl.Phys.Lett.,67,1262(1995).

[19]“AlGaAs/GaAssingleelectrontransistorfabricatedwithoutmodulationdoping”–A. M.See,O.Klochan,A.R.Hamilton,A.P.Micolich,M.Aagesen,andP.E.Lindelof,Appl. Phys.Lett.,96,112104(2010).

[20]“Chaoticelectrontransportinsemiconductordevices”–B.Scannell,PhDthesis, UniversityofOregon(2010).

[21]“ConductanceFluctuationsandChaoticScatteringinBallisticMicrostructures”–C. M.Marcus,A.J.Rimberg,R.M.Westervelt,P.F.Hopkins,andA.C.Gossard,Phys.Rev. Lett.,69,506(1992).

[22]“WeakLocalizationinChaoticversusNonchaoticCavities:AStrikingDifferencein theLineShape”–A.M.Chang,U.H.Baranger,L.N.PfeifferandK.W.West,Phys.Rev. Lett.,73,2111(1994).

[23]“UniversalconductancefluctuationsinthemagnetoresistanceofsubmicronͲsizen+Ͳ GaAswiresandlaterallyconfinednоͲGaAs/(AlGa)Asheterostructures”–R.P.Taylor,M. L.Leadbeater,G.P.Whittington,P.C.Main,L.Eaves,S.P.Beaumont,I.McIntyre,S. ThomsandC.D.W.Wilkinson,Surf.Sci.,196,52(1988).

[24]“FractalConductanceFluctuationsinaSoftͲWallStadiumandaSinaiBilliard”–A.S. Sachrajda,R.Ketzmerick,C.Gould,Y.Feng,P.J.Kelly,A.Delage,andZ.Wasilewski,Phys. Rev.Lett.,80,1948(1998).

[25]“WeakLocalizationandInstabilityinBallisticCavities”–H.U.Baranger,R.A. Jalabert,andA.D.Stone,Phys.Rev.Lett.,70,3876(1993).

[26]“ConductanceFluctuationsintheBallisticRegime:aProbeofQuantumChaos?”–R. A.Jalabert,H.U.Baranger,andA.D.Stone,Phys.Rev.Lett.,65,2442(1990).

[27]“ClassicalandquantumballisticͲtransportanomaliesinmicrojunctions”–H.U. BarangerD.P.DiVincenzo,R.A.JalabertandA.D.Stone,Phys.Rev.B,44,10637(1991).

[28]“QuantumͲchaoticscatteringeffectsinsemiconductormicrostructures”–H.U. Baranger,R.A.Jalabert,andA.D.Stone,Chaos,3,665(1993).

[29]“Phasebreakinginballisticquantumdots:Experimentandanalysisbasedon chaoticscattering”–C.M.MarcusandR.M.Westervelt,P.F.HopkinsandA.C.Gossard, Phys.Rev.B,48,2460(1993)

[30]“Conductancefluctuationsinaquantumdotinthetunnelingregime:Crossover fromaperiodictoregularbehavior”–C.M.Marcus,R.M.Westervelt,P.F.Hopkins,A.C. Gossard,Surf.Sci.,305,480(1994).

[31]“Conductancefluctuationsandquantumchaoticscatteringinsemiconductor microstructures”–C.M.Marcus,R.M.Westervelt,P.F.Hopkins,andA.C.Gossard, Chaos,3,643(1993).

[32]“CoherentbranchedflowinatwoͲdimensionalelectrongas”–M.A.Topinka,B.J. LeRoy,R.M.Westervelt,S.E.J.Shaw,R.Fleischmann,E. J.Heller,K.D.Maranowskik andA.C.Gossard,Nature(London),410,183(2001),

[33]“UnexpectedfeaturesofbranchedflowthroughhighͲmobilitytwoͲdimensional electrongases”–M.P.Jura,M.A.Topinka,L.Urban,A.Yazdani,H.Shtrikman,L.N. Pfeiffer,K.W.WestandD.GoldhaberͲGordon,Nat.Phys.,3,841(2007).

[34]“ImpactofSmallͲAngleScatteringonBallisticTransportinQuantumDots”–A.M. See,I.Pilgrim,B.C.Scannell,R.Montgomery,O.Klochan,A.M.Burke,M.Aagesen,P.E. Lindelof,I.Farrer,D.A.Ritchie,R.P.Taylor,A.R.HamiltonandA.P.Micolich,Phys.Rev. Lett.,108,196807(2012).

[35]“ProbingtheSensitivityofElectronWaveInterferencetoDisorderͲInduced ScatteringinSolidͲStateDevices”–B.C.Scannell,I.Pilgrim,A.M.See,R.D.Montgomery, P.K.Morse,M.S.Fairbanks,C.A.Marlow,H.Linke,I.Farrer,D.A.Ritchie,A.R. Hamilton,A.P.Micolich,L.Eaves,andR.P.Taylor,–inPressforPhys.Rev.B, arXiv:1106.5823.

[36]“Deepdonorlevels(DXcenters)inIIIͲVsemiconductors”–P.M.Mooney,J.Appl. Phys.,67,R1(1990).

[37]“Quasielectricfieldsandbandoffsets:teachingelectronsnewtricks”–H.Kroemer, Rev.Mod.Phys.,73,783(2001). [38]“GaAs,AlAs,andAlxGa1оxAs:Materialparametersforuseinresearchanddevice applications”–S.Adachi,J.Appl.Phys.,58,R1(1985).

[39]“SelfͲConsistentResultsforaGaAs/AlxGa1ͲxAsHeterojunciton.II.LowTemperature Mobility”–T.Ando,J.Phys.Soc.Jpn.,51,3900(1982).

[40]“UseofsuperlatticestorealizeinvertedGaAs/AlGaAsheterojunctionswithlowͲ temperaturemobilityof2×106cm2/Vs”–T.Sajoto,M.Santos,J.J.Heremans,M. Shayegan,M.Heiblum,M.V.Weckwerth,andU.Meirav,Appl.Phys.Lett.,54,840 (1989).

[41]“HighͲmobilityvariableͲdensitytwoͲdimensionalelectrongasininvertedGaAsͲ AlGaAsheterojunctions”–U.Meirav,M.Heiblum,andF.Stern,Appl.Phys.Lett.,52, 1268(1988).

[42]“FabricationofhighͲqualityoneͲandtwoͲdimensionalelectrongasesinundoped GaAs/AlGaAsheterostructures”–R.H.Harrell,K.S.Pyshkin,M.Y.Simmons,D.A.Ritchie, C.J.B.Ford,G.A.C.Jones,andM.Pepper,Appl.Phys.Lett.,74,2328(1999).

[43]“SimpleͲlayeredhighmobilityfieldeffectheterostructuredtwoͲdimensional electrondevice”–R.L.Willett,L.N.Pfeiffer,andK.W.West,Appl.Phys.Lett.,89, 242107(2006).

[44]“UndopedhighmobilitytwoͲdimensionalholeͲchannelGaAs/AlxGa1оxAs heterostructurefieldͲeffecttransistorswithatomicͲlayerͲdepositeddielectric”–T.M.Lu, D.R.Luhman,K.Lai,D.C.Tsui,L.N.Pfeiffer,andK.W.West,Appl.Phys.Lett.,90, 112113(2007).

[45]“Fabricationandcharacterizationofambipolardevicesonanundoped AlGaAs/GaAsheterostructure”–J.C.H.Chen,D.Q.Wang,O.Klochan,A.P.Micolich,K. DasGupta,F.Sfigakis,D.A.Ritchie,D.Reuter,A.D.Wieck,andA.R.Hamilton,Appl. Phys.Lett.,100,052101(2012).

[46] “Veryhighquality2DEGSformedwithoutdopantinGaAs/AlGaAsheterostructures” –R.H.Harrell,J.H.Thompson,D.A.Ritchie,M.Y.Simmons,G.A.C.JonesandM. Pepper,J.ofCrystalGrowth,201,159(1999).

[47]“QuantumTransportinSemiconductorNanostructures”–C.W.J.BeenakkerandH. vanHouten,SolidStatePhysics,44,1Ͳ228(1991).

[48]“TheroleofsurfaceͲgatetechnologyforAlGaAs/GaAsnanostructures”–R.P. Taylor,Nanotechnology,5,183(1994).

[49]“ThePhysicsofLowͲDimensionalSemiconductors”–J.H.Davies,Cambridge UniversityPress(1998).

[50]“TheQuantumHallEffect–2ndEd.”–(e.d.)R.E.PrangeandS.M.Girvin,SpringerͲ Verlag,NewYork(1990).

[51]“PhysicsofSemiconductorDevices”–S.M.Sze,NewYork:Wiley,2nded.(1981).

[52]“SpatialVariationofCurrentsandFieldsDuetoLocalizedScatterersinMetallic Conduction”–R.Landauer,IBMJ.Res.Dev.,1,223,(1957);ibid,32,306,(1988).

[53]“OneͲdimensionaltransportandthequantisationoftheballisticresistance”–D.A. Wharam,T.J.Thornton,R.Newbury,M.Pepper,H.Ahmed,J.E.F.Frost,D.G.Hasko,D. C.Peacock,D.A.RitchieandG.A.C.Jones,J.Phys.C,21,L209(1988).

[54]“QuantizedconductanceofpointcontactsinatwoͲdimensionalelectrongas”–B.J. vanWees,H.vanHouten,C.W.J.Beenakker,J.G.Williamson,L.P.Kouwenhoven,D. vanderMarelandC.T.Foxon,Phys.Rev.Lett.,60,848(1988).

[55]“ElectronicpropertiesoftwoͲdimensionalsystems”–T.Ando,A.B.Fowler,andF. Stern,Rev.Mod.Phys.,54,437(1982).

[56]“TheroleofMBEinrecentquantumHalleffectphysicsdiscoveries”–L.Pfeifferand K.W.West,PhysicaE,20,57(2003).

[57]“EvidenceforanAnisotropicStateofTwoͲDimensionalElectronsinHighLandau Levels”–M.P.Lilly,K.B.Cooper,J.P.Eisenstein,L.N.Pfeiffer,andK.W.West,Phys. Rev.Lett.,82,394(1999).

[58]“ObservationofanevenͲdenominatorquantumnumberinthefractionalquantum Halleffect”–R.Willett,J.P.Eisenstein,H.L.Störmer,D.C.Tsui,A.C.Gossard,andJ.H. English,Phys.Rev.Lett.,59,1776(1987).

[59]“DistinguishingimpurityconcentrationsinGaAsandAlGaAsusingveryshallow undopedheterostructures”–W.Y.Mak,K.DasGupta,H.E.Beere,I.Farrer,F.Sfigakis, andD.A.Ritchie,Appl.Phys.Lett.,97,242107(2010).

[60]“RoleofbackgroundimpuritiesinthesingleͲparticlerelaxationlifetimeofatwoͲ dimensionalelectrongas”–S.J.MacLeod,K.Chan,T.P.Martin,A.R.Hamilton,A.See,A. P.Micolich,M.Aagesen,andP.E.Lindelof,Phys.Rev.B,80,035310(2009).

[61]“Effectofbackgrounddopingontheelectronmobility of(Al,Ga)As/GaAs heterostructures”–T.J.Drummond,W.Kopp,H.Morkoç,K.Hess,A.Y.Cho,andB.G. Streetman,J.Appl.Phys.,52,5689(1981).

[62]“TheEffectofGrowthTemperatureontheMobilityofTwoͲDimensionalElectron GasinSelectivelyDopedGaAs/NͲAlGaAsHeterostructuresGrownbyMBE”–S.Hiyamizu, T.Fujii,T.Mimura,K.Nanbu,J.SaitoandH.Hashimoto,Jpn.J.Appl.Phys.,20,(1981).

[63]“DependenceofthestructuralandopticalpropertiesofGaAsͲGa1оxAlxAs multiquantumͲwellstructuresongrowthtemperature”–C.Weisbuch,R.Dingle,P.M. Petroff,A.C.Gossard,andW.Wiegmann,Appl.Phys.Lett.,38,840(1981).

[64]“MBEgrowthofultraͲlowdisorder2DEGwithmobilityexceeding35x106cm2/Vs” –V.Umansky,M.Heiblum,Y.Levinson,J.Smet,J.Nübler,andM.Dolev,J.Cryst. Growth,311,1658(2009).

[65]“ArtificialAtoms”–M.A.Kastner,Phys.Today,46,24(1993).

[66]“TheoryofsingleͲelectronchargingofquantumwellsanddots”–D.V.Averin,A.N. Korotkov,andK.K.Likharev,Phys.Rev.B,44,6199(1991).

[67]“Transportthroughastronglyinteractingelectronsystem:Theoryofperiodic conductanceoscillations”–Y.Meir,N.S.Wingreen,and P.A.Lee,Phys.Rev.Lett.,66, 3048(1991).

[68]“TheoryofCoulombͲblockadeoscillationsintheconductanceofaquantumdot”– C.W.J.Beenakker,Phys.Rev.B,44,1646(1991).

[69]”Electrontransportinquantumdots”–L.P.Kouwenhoven,C.M.Marcus,P.L. McEuen,S.Tarucha,R.M.Westervelt,andN.S.Wingreen,inMesoscopicElectron Transport,NATOASISer.EVol.345,editedbyL.L.Sohn,L.P.KouwenhovenandG. Schön(Kluwer,Dordrecht,1997),pp.105–214.

[70]“Virtualelectrondiffusionduringquantumtunnelingoftheelectriccharge”–D.V. AverinandYu.V.Nazarov,Phys.Rev.Lett.,65,2446(1990).

[71]“MeasurementofdiscretechargeinthesystemsofultraͲsmalltunneljunctions”– Yu.V.Nazarov,J.LowTemp.Phys.,90,77(1993).

[72]“PeriodicconductanceoscillationsinthesingleͲelectrontunnelingtransistor”–D.V. Averin,PhysicaB,194,979(1994).

[73]“Mesoscopicquantumtransport:Resonanttunnelinginthepresenceofastrong Coulombinteraction”–H.SchoellerandG.Schön,Phys.Rev.B,50,18436(1994).

[74]“SpinsinfewͲelectronquantumdots”–R.Hanson,L.P.Kouwenhoven,J.R.Petta, S.TaruchaandL.M.K.Vandersypen,Rev.Mod.Phys.,79,1217(2007).

[75]“CompetingchannelsinsingleͲelectrontunnelingthroughaquantumdot”–J.Weis, R.J.Haug,K.v.Klitzing,andK.Ploog,PhysRev.Lett.,71,4019(1993).

[76]“SignificanceofElectromagneticPotentialsintheQuantumTheory”–Y.Aharonov andD.Bohm,Phys.Rev.,115,485(1959).

[77]“Observationofh/eAharonovͲBohmOscillationsinNormalͲMetalRings”–R.A. Webb,S.Washburn,C.P.Umbach,andR.B.Laibowitz,Phys.Rev.Lett.,54,2696(1985).

[78]“FractalMagnetoͲconductanceFluctuationsinMesoscopicSemiconductorBilliards” –A.P.Micolich,PhDthesis,UniversityofNewSouthWales(2000).

[79]“Weaklocalizationinthinfilms:atimeͲofͲflightexperimentwithconduction electrons”–G.Bergmann,Phys.Rep.,107,1(1984).

[80]“TheAaronovͲBohm effectindisorderedconductors”–B.L.Al'tshuler,A.G. Aronov,B.Z.Spivak,JETPLett.,33,94(1981).

[81]“ConductanceFluctuationsintheBallisticRegime:aProbeofQuantumChaos?”– R.A.Jalabert,H.U.Baranger,andA.D.Stone,Phys.Rev.Lett.,63,2442(1990).

[82]“QuantumTransportinOpenMesoscopicCavities”–J.P.Bird,K.Ishibashi,Y. Aoyagi,T.Sugano,R.Akis,D.K.Ferry,D.P.PivinJr,K.M.Connolly,R.P.Taylorc,R. Newbury,D.M.Olatona,A.Micolich,R.Wirtz,Y.OchiaiandY.Okubo,Chaos.Solitons& Fractals.8,1299(1997).

[83]“TheSignatureofChaos”–R.V.Jensen,Nature,373,16(1995).

[84]“Magnetotransportfluctuationsinregularsemiconductorballisticquantumdots”– R.Akis,D.K.FerryandJ.P.Bird,Phys.Rev.B,54,17705(1996).

[85]“Numericalanalysisof ballisticͲelectrontransportinmagneticfieldsbyusinga quantumpointcontactandaquantumwire”–T.Usuki,M.Saito,M.Takatsu,R.A.Kiehl, andN.Yokoyama,Phys.Rev.B,52,8244(1995).

[86]“LeadͲOrientationͲDependentWaveFunctionScarringinOpenQuantumDots”–J. P.Bird,R.Akis,D.K.Ferry,D.Vasileska,J.Cooper,Y.AoyagiandT.Sugano,Phys.Rev. Lett.,82,4691(1999).

[87]“Imagingscarredstatesinquantumdots”–A.M.Burke,RAkis,T.Day,G.Speyer, D.KFerryandB.R.Bennett,J.Phys.Condens.Matter,21,212201(2009).

[88]“PeriodicScarredStatesinOpenQuantumDotsasEvidenceofQuantumDarwinism” –A.M.Burke,R.Akis,T.E.Day,GilSpeyer,D.K.Ferry,andB.R.Bennett,Phys.Rev.Lett., 104,176801(2010).

[89]“ChaosinDynamicalSystems”–E.Ott,CambridgeUniversityPress(1993).

[90]“Fractalconductancefluctuationsingenericchaoticcavities”–R.Ketzmerick,Phys. Rev.B,54,10841(1996).

[91]“LeadͲinducedtransitiontochaosinballisticmesoscopicbilliards”–J.P.Bird,D.M. Olatona,R.Newbury,andR.P.Taylor,K.Ishibashi,M.Stopa,Y.Aoyagi,T.SuganoandY. Ochiai,Phys.Rev.B,52,R14336(1995).

[92]“Influenceofshapeonelectrontransportinballisticquantumdots”–M.J.Berry,J. A.Katine,R.M.WesterveltandA.C.Gossard,Phys.Rev. B,50,17721(1994).

[93]“Quantumtransportinchaoticandintegrableballisticcavitieswithtunableshape” –Y.Lee,G.Faini,andD.Mailly,Phys.Rev.B,56,9805(1997).

[94]“TunnelingandNonhyperbolicityinQuantumDots”–P.S.deMoura,Y.C.Lai,R. Akis, J.P.Bird,andD.K.Ferry,Phys.Rev.Lett.,88,236804(2002).

[95]“EvolutionofFractalPatternsduringaClassicalͲQuantumTransition”–A.P. Micolich,R.P.Taylor,A.G.Davies,J.P.Bird,R.Newbury,T.M.Fromhold,A.Ehlert,H. Linke,L.D.Macks,W.R.Tribe,E.H.Linfield,D.A.Ritchie,J.Cooper,Y.Aoyagi,andP.B. Wilkinson,Phys.Rev.Lett.,87,036802(2001).

[96]“ExperimentalInvestigationoftheBreakdownoftheOnsagerͲCasimirRelations”– C.A.Marlow,R.P.Taylor,M.Fairbanks,I.Shorubalko,andH.Linke,Phys.Rev.Lett.,96, 116801(2006).

Chapter2

Devicefabricationandlowtemperature measurementtechniques

2.1–Introduction

Inthisthesiswepresentexperimentalresultsfromsevendevices:fourofwhich werefabricatedontheNBI30undopedAlGaAs/GaAsheterostructure,andthreeonthe C2275modulationͲdopedAlGaAs/GaAsheterostructure.Anoverviewofthesedevicesis presentedinTable2.1.TheNBI30waferwasgrownattheNielsBohrInstitute,Denmark byM.AagesenandP.E.Lindelof,whiletheC2275waferwasgrownattheCavendish Laboratory, England by I. Farrer and D. A. Ritchie. All samples were fabricated at the SemiconductorNanofabricationFacility(SNF),andtheUniversityofNewSouthWales nodeoftheAustralianNationalFabricationFacility(ANFF).Allmeasurementspresented inthisthesiswereobtainedatUNSW,withtheexceptionofdeviceAS57NandC2275ͲA, whichweremeasuredattheUniversityofOregon(UO),UnitedStates,asacollaborative effortwithProf.RichardTaylorandhisgroup,duringmyfourweeksofvisitbetween AugustandSeptember 2010.

Device Wafer Doping Origin Chapter Measured Studies

NielsBohr SET, AS08N NBI30 induced 3,4 UNSW Institute scattering NielsBohr AS43N NBI30 induced 4,5 UNSW SET,MCF Institute NielsBohr AS57N NBI30 induced 6 UO MCF Institute NielsBohr AS61N NBI30 induced 6 UNSW MCF Institute

C2275ͲA C2275 doped Cavendish 6 UO MCF

C2275ͲC C2275 doped Cavendish 5 UNSW MCF

C2275ͲF C2275 doped Cavendish 6 UNSW MCF

Table2.1:Anoverviewofallthedevicesdiscussedinthisthesis.

2.2–Heterostructureoverview

The focus of this thesis is to study quantum dots fabricated on the NBI30 undopedAlGaAs/GaAsheterostructure.Theseundopeddevicesareinterestingdueto theabsenceofmodulationdopants.InChapter4,wedemonstratetwoofsuchdevices operating in the Coulomb blockade regime; while in Chapter 5 and 6, we provide comparativemagnetoͲconductancefluctuations(MCF)studiesbetweenbilliards(open dots) basedon the NBI30, and the C2275 modulationͲdoped wafers with comparable electron densities݊and mobilitiesߤ. In the following, we will discuss the layouts of theseheterostructures,whiletheircharacterizationdetailsareprovidedinChapter3.

2.2.1–NBI30undopedAlGaAs/GaAsheterostructure

Allofthe“induced”devicespresentedinthisthesiswerefabricatedbasedon this NBI30 undoped Al0.33Ga0.67As/GaAs heterostructure. After layers of GaAs/AlGa superlattice(SL)weregrownontopofa(100)GaAssubstrate,startingfromamicron thickundopedGaAsbufferandmovingupwards,theheterostructureconsistsof:a160 nmundopedAlGaAsbarrier,a25nmundopedGaAsspacer,anda35nm݊ାGaAscap. This cap is degenerately doped (doping density = 3.3ൈ1024mͲ3) to have a metallic conductivityatlowtemperature,providingasufficientlyhighelectrondensitytoscreen the 2DEG from ionizeddonors in the cap [1]. The heterostructure waswetͲetchedto defineaHallbarmesa,andNiGeAuOhmiccontactswerethenproducedusingaselfͲ aligned process [2]. As discussed in Section 1.2.2, this process ensures that the gate overlapsthecontactsbutremainselectricallyisolatedfromthem.Dependingonthetop gate bias,݊of 1.6 – 2.7ൈ1011cmͲ2,ߤof 274 – 400ൈ103cmͲ2/Vs (at 1.2 K) and a correspondingelasticmeanfreepath݈௘௟of1.8–3.4ʅmcanbeachieved.Figure2.1(a) showsaschematicofthewaferlayout,while(b)and(c)showschematicsofpatterned quantumdotandbilliarddevicesusedinChapter4and5&6respectively.

Figure 2.1:(a) Wafer layout for NBI30 undoped AlGaAs/GaAs heterostructure. In this heterostructure,a2DEG(bluedashedline)isformedbyapplyingapositivebiasonthe݊ା capstoinduceelectronsattheAlGaAs/GaAsinterface.Theyellowobjectnexttothe2DEG correspondstoadiffusedOhmiccontact.(b)Schematicofaquantumdotdevicediscussed inChapter4.(c)SchematicofabilliarddevicediscussedinChapter5&6.In(c/d),the݊ା caps(orange)arepatternedtoformthedevicestructuresatthe2DEGs(darkregions).

2.2.2–C2275modulationǦdopedAlGaAs/GaAsheterostructure

The C2275 modulationͲdoped wafer has a standard Al0.33Ga0.67As/GaAs 70 nm HighElectronMobilityTransistor(HEMT)structure.AftermultiplelayersofGaAs/AlAs andAlGaAs/GaAsSLsweregrownontopofa(100)GaAssubstrate,startingfroma500 nmundopedGaAsbufferandmovingupwards,theheterostructureconsistsof:a20nm undopedAlGaAsspacerlayer,a40nmSidopedAlGaAslayerwithadopingdensityof 1.2ൈ1018cmͲ3,anda10nmundopedGaAscap.Theheterostructurehasameasured electron density of 2.34ൈ1011cmͲ2, mobility of 333ൈ103cmͲ2/Vs (at 1.2 K) and a corresponding݈௘௟=2.66ʅminthedark.Figure2.2(a)showsaschematicofthewafer layout.ThesplitͲgatetechniquewasusedtofabricatesubmicrondevices.Schematicsof twobilliarddevicesusedinChapter5and6areshowninFig.2.2(b)and(c)respectively.

Figure2.2:(a)WaferlayoutforC2275modulationͲdopedAlGaAs/GaAsheterostructure.The yellow object shown represents a diffused Ohmic contact, which makes electrical connectiontothe2DEG(bluedashline).(b/c)Schematicsoftwobilliarddevicesdiscussedin Chapter5and6respectively.TheyellowgatescorrespondtotheTi/Ausplitgatesthatare usedtodepleteelectronsatthe2DEGanddefinethedevicestructures.

2.3–Devicefabrication

Inthissection,wedescribetheprocessingstepsforthefabricationofundoped, andmodulationͲdopeddevicespresentedinthisthesis.ForinͲdepthdiscussionsofGaAs processingmethods,acomprehensiveguidecanbefoundin[3].

2.3.1–ProcessflowforNBI30undopedquantumdotdevices

For a single device, a 3 mmൈ4 mm chip was cleaved from the NBI30 wafer usingaKarlͲSussHR100waferscriber(Fig.2.3a).Thechipwasthenplacedintoabeaker withacetoneandleftinanultrasonicbathfor15minstocleanthesurface.Inorderto avoiddryingstains,isopropanolandnitrogengaswereusedtorinseanddrythechip whereveracetonewasused.Whenthechipwasready,itwasspinͲcoated(usuallyat 5000rpmfor60s)withalayerofMicropositS1813photoresistof̱1.2μminthickness; whichcanbemeasuredusingaDektak3030surfaceprofiler.ThechipwasthenpreͲ bakedonahotplateat95°Ctodegasthesolventsfromtheresist.Atthebeginningofall spinningprocesses,thechipwasfirstspinͲcleanedunderajetofacetonetomakesure thatthesurfacewasascleanaspossible.

AQuintel600alignerwasthenusedtoexposethemesapatternonachromiumͲ onͲquartz mask to the chip. Ultraviolet light from a mercury vapor source was used, running in constant intensity mode at 10 mW/cm2 calibrated at 345 nm, usually an exposure time of 5 s was used (Fig. 2.3b). After the chip had been postͲbaked on a hotplate,thepatternwasdevelopedinAZ300MetalͲIonͲFreeTetramethylammonium hydroxide(TMAH)developerforaminute,followedbyarinseindeionized(D.I.)water (Fig.2.3c).AphotographofadevelopedmesapatternisshowninFig.2.4(a).

Inordertodefinethemesapattern,thechipwasetchedusingapremixedH2O:

HCl:H2O2(100:1:4)etchant.Theetchdepthneedstobelargerthanthethicknessof the GaAs doped cap, which is 35 nm for NBI30. After the chip was etched for the requiredtime(usually60–90s),itwasrinsedinD.I.waterandthenblowͲdriedwithN2 gas.Theleftoverresistcoveringthemesawasremovedusinganacetonesoak(Fig.2.4d).

To begin the Ohmic contact processing procedure; a new layer of photoresist wasfirstspunontothechip;theOhmicpatternwasthentransferredacrosstothechip usingthesameexposureanddevelopmentstepsasthemesa(Fig.2.3e).Afterthechip hadbeendevelopedandrinsed,itwasetchedinHCletchanttoobtainanetchdepthof 210nm,10nmabovetheAlGaAs/GaAsinterface.ThisselfͲalignedstepiscrucialandthe etch depth needs to be precise: if the etch was too shallow, the current leakage

Figure 2.3:Process flow of NBI30 undoped quantum dot devices. (a) Start, (b) mesa, exposure, (c) mesa development, (d) mesa etch, (e) Ohmic exposure, (f) Ohmic angled evaporation, (g) Ohmic liftͲoff (h) EBL, (i) etching for EBL defined gates, (j) interconnect exposure,(k)Ti/Audepositionforinterconnect,(l)interconnectliftͲoff,(m)packagingand bonding.

Figure2.4:Photographsof(a)developedmesapattern,(b)OhmiccontactliftͲoff,(c)etched quantum dot pattern showing alignment markers at the corners, and (d) gates/interconnectsdepositedonthedevicetomakeelectricalcontactstothemesa,Ohmic contactsandnanostructures.Theblackscalebarshown)in(a is500ʅmlong.

betweenthedopedcapandtheOhmicswouldincrease.Ontheotherhandiftheetch depthwastoodeep,diffusedOhmicmetalcannotmakeproperlycontacttothe2DEG, andthedevicedoesnotturnon.Inordertopreciselydeterminethisetchdepth,we performed multiple etch tests onea separat sacrificial chip prior to etching the real sample.Afteretching,thechipwasimmediatelyplacedintoaKurtJ.Leskerevaporator to avoid oxidation. The chamber was first evacuated to below 5ൈ10Ͳ6 Torr before metallization of nickel, germanium and Au separately (Fig. 2.4f). In order to prevent directleakagesbetweentheOhmicsandtopgate,metalswereevaporatedat35°with respectivetothe2DEGunderconstantrotation(thisangledependssensitivelyonthe resist profile, Ohmic etch profile, Ohmic etch depth and Ohmic metal thickness). The OhmicliftͲoffprocesswasperformedusingajetofacetoneaftersoakingthechipinNͲ methylͲ2Ͳpyrrolidone(NMP)onahotplateat80ƱCfor10mins.Aphotographofadevice aftertheOhmicliftͲoffprocessisshowninFig.2.4(b).

Thenextstepofprocessingistouseelectronbeamlithography(EBL)techniques to transfer patterns of the nanostructure device onto the mesa surface. A thin layer ሺ̱100 nm) of polyͲmethylͲmethacrylate (PMMA) A3 (950K, 3% in anisole) was first spun (at 5000 rpm for 60 s) onto the chip; followed by a long bake of 10 mins on a hotplateat180ƱCtohardentheresist.Adropofcolloidalgoldsuspensioninwaterfrom ESPIwasthenplacedatthecornerofthechipandallowedtodrybeforethechipwas placedintheelectronbeammicroscope.Twoelectronbeamsystemsavailableinthe SNFatUNSWaretheFEI/PhilipsXL30andFEISirion.Onceinitialbeamadjustmentssuch asfocusandastigmatismcorrectionsweremade,thecolloidalgolddropplacedearlier was used to ensure the focal point was optimized at the resist surface. Alignments markersplacedjustoutsidetheHallbarmesawereusedasreferencepointsfortheEBL process.DesignedpatternswerethenwrittenontothePMMAusingtheNPGSsoftware withatypicalareadoseof200ʅC/cm2at18keV(Fig.2.3h).Resistdevelopmentwas doneusingamixtureofmethylͲisobutylͲketone(MIBK)andisopropanolin1to3ratio.A postbakewasusuallyperformedtostrengthentheresistpriortoetching,whileasmall amountofsurfactantsuchasTritonXͲ100at0.5%wasaddedtotheH2O:H2SO4:H2O2 (160:1:8)sulfuricetchanttoincreaseetchuniformity(Fig.2.3i).Ouraimwastoetch throughthedopedcaptodefineseparategates,butnotsofarthattheetchpenetrates intotheAlGaAslayer,whereoxidationoccurs.Thisresultsatypicaletchdepthof45to 55nm,whichwasmeasuredusingaDimension3100atomicforcemicroscope(AFM).A photographofanetchedquantumdotpatternwithalignmentmarkersneartheedgeof the mesa is shown in Fig. 2.4(c). It was found that etching cannot be achieved consistently if the chip had been annealed; therefore, EBL steps were performed straightaftertheOhmicliftͲoffstep.

Thefinalstageofprocessingwastodepositgates/interconnectsthatallowthe Ohmics and the EBLͲdefined gates to be connected to a package for electrical measurements.A new layer of S1813 photoresist was first spun onto the surface, patternofthegates/interconnectwasthenexposed(Fig.2.3j)anddevelopedusingthe sameproceduresasdiscussedpreviously.Thesewerefollowedbymetallizationof15 nmofTiand85nmofAuusingtheEdwardsAuto306evaporatoratapressurebelow2 ൈ10Ͳ6mbar(Fig.2.3k).ToaidthegateliftͲoffprocess,wehardenedtheresistbysoaking the sample in chlorobenzene (C6H5Cl) for 5 mins before development [3]. LiftͲoff was

69 then easily achieved using a jet of acetone after 10 mins of soaking the device in acetone(Fig.2.3l).Aphotographofdepositedgates/interconnectsonadeviceisshown inFig.2.4(d).Finally,theprocessedchipwasstuckontoa20Ͳpinleadlesschipcarrier (LCC20)usingasmallamountofPMMA,followedbya180°Cbakeonahotplatefor10 mins.Bondingwasachievedusing50μmAuwireonaballbonder(Fig.2.3m).

2.3.2–Processflowformodulationdopedquantumdotdevices

ModulationͲdoped quantum dot samples were fabricated using standard photolithographyandEBLtechniques.Inthefollowing,onlyprocessingstepsthatare differentfromthatoftheundopeddevicesarediscussed.

Theprocessflowformakingmodulationdopedquantumdotdevicesisshown schematically in Fig. 2.5. The processing steps for the Hall bar mesa (Fig. 2.5aͲd) are largely the same as the undoped device, except here, the mesa needs to be etched throughtotheAlGaAs/GaAsinterfacefor2DEGisolation.FortheprocessingofOhmic contacts (Fig. 2.5eͲg), selfͲaligned step was not required, and metallization was performedwithouttheneedofangledevaporationorstagerotation.Additionally,since nanostructures in the modulationͲdoped samples were defined using the splitͲgate method [4] rather than etching, the Ohmic contacts were annealed prior to the EBL steps;andtheprolongedPMMAprebaketimerequiredintheundopedprocessingwas notnecessary(Fig.2.5h).MetallizationofthesplitͲgateswereperformedwith10nmof Tiand30nmofAu(Fig.2.5j).AftersoakingthedeviceinNMPat80°Conahotplatefor 20minutes,liftͲoffwasachievedusingajetofacetonewiththeaidofultrasonicbath treatment.Finally,thesamephotolithographyandpackagingstepswereusedtodeposit themetalinterconnects,whichprovidecontactstotheOhmicsandEBLͲdefinedgates forelectricalmeasurements(Fig,2.5jͲm).

Figure 2.5:Process flow of modulationͲdoped quantum dot devices. (a) Start, (b) mesa, exposure,(c)mesadevelopment,(d)mesaetch,(e)Ohmicexposure,(f)Ohmicdeposition, (g) Ohmic annealing, (h) EBL, (i) Ti/Au deposition for EBL defined gates, (j) interconnect exposure,(k)Ti/Audepositionforinterconnect,(l)interconnectliftͲoff,(m)packagingand bonding.

2.4–Lowtemperaturemeasurementtechniques

For experimental signatures of quantum effects, such as the excited state spectrum (Section 4.3.3) and the spin transport blockade features (Section 4.4) to be observed in single electron transistor quantum dots, electrical measurements were obtainedatlowtemperaturesܶ,wherethethermalenergy݇஻ܶneedstobelessthan thechargingenergyaswellasthesingleparticlelevelspacing(bothdependonthedot size) [5]. Closed dot transport measurements were thus obtained using a 3He/4He dilutionrefrigeratorwithabasetemperatureof̱40mK.Inthecaseofmeasuringthe magnetoͲconductancefluctuationsMCFinbilliards,inelasticscatteringevents,suchas electronͲphononscatteringneedstobesuppressedinordertomaintainasufficiently large phase coherence length for interference to occur [6]. This is not so stringent a requirement,soMCFmeasurementspresentedherewereperformedusingapumped 3HeHelioxsystemwithabasetemperatureof̱230mK.TheadvantageoftheHeliox systemisthatithasarelativelyquickturnaroundtime(̱5hours),idealforperforming thethermalcyclingmeasurementswereportinChapter5&6,whilestillprovidinga sufficientlylowtemperatureenvironmentfortheexperiment.Inthefollowing,wewill discussthevariouscryogenicsystemsusedatUNSW.

2.4.1–4.2Kdipping

Basic device assessments at liquid Helium temperature ሺ̱ 4.2 K) were performedbyattachingthepackagedsampletoadipͲprobe,andimmersingitinto a dewarofliquidHelium.ThedipprobeisfittedwithaCharnteksampleholder(seeFig. 2.6), which is also known as a zeroͲinsertion force (ZIF) socket. It contains beryllium copper pins to avoid magnetics, and makes electrical contacts to the leadlessͲchipͲ carrierLCC20package(withoutNiplatingonthepins)usingaspringcontactmechanism. The4HedipͲprobemethodisbyfarthesimplestwithaturnaroundtimeof̱2hours.

Figure2.6:AphotographoftheCharnteksampleholderattachedadipͲprobeatUNSW.

2.4.2–Pumped4HeVTIsystem

Temperatures down to 1.2 K are achieved using the Oxford Instruments VTI (variabletemperatureinsert)system.Aschematicdiagramofsuchsysteminacryostat isshowninFig.2.7.Deviceishousedinasampleholderattheendoftheinsert;the samplespaceisfilledwithliquid4Hefromthemainbathviaaneedlevalue;pumpingon thesamplespacereducesthetemperaturefrom4.3Kto̱1.2Kviaevaporativecooling [7,8].

Figure2.7:SchematicdiagramoftheVTIsystemusedinUNSW.

2.4.3–Pumped3HeHelioxsystem

FortheMCFexperimentspresentedinChapter5and6ofthisthesis,twoOxford Instruments Heliox VL (VL stands for vertical loading) 3He systems were used: one at UNSW,andoneattheUniversityofOregon.ThetwoHelioxsystemsareessentiallythe same,providingmeasurementtemperaturesdownto230mKwithatypicalholdtimeof 4Ͳ5 days. They have relatively short turnaround times of̱5 hrs (without swapping samples), whichisidealforperformingmultiplethermalcyclingexperiments.TheHeliox system at UNSW uses a 120Ͳlitre liquid 4He dewar as the main bath, and has a superconductingmagnetcapableofprovidingamagneticfieldupto2T(perpendicular tothesample)attachedtothebottomoftheinsert.TheHelioxsystematUOhowever, uses a nitrogenͲshielded 4He cryostat with a superconducting magnet capable of providingamagneticfieldupto8T.Withthissetup,heliumcanalsobeeasilyrefilled into the cryostat while keeping the insert cold, allowing experiments to be run continuously. In contrast, the Heliox setup at UNSW has a maximum run time of̱2

Figure2.8:(a)AschematicrepresentationoftheHelioxsystem.(b)Aphotographofthe HelioxinsertatUNSWisshown.

74 weeks,limitedbythecapacityofthedewar.AphotographoftheHelioxinsertatUNSW anditsschematicdiagramisshowninFig.2.8.

ThethreemainspartsoftheHelioxsystemare:the3Hesorptionpump,1Kplate and3Hepot,theyarehighlightedinFig.2.8.Tobegintheoperation,asmallamountof 4Heexchangegasisinjectedintoinnervacuumcan(IVC),slowingcoolingthesystemto ̱4.2 K via the liquid He in the main bath. A 4He cooled activated charcoal sorption pumplocatedat1Kplatewillthenphysisorbtheexchangegas(forܶ<̱30K),isolating thethermallinkbetweenthemainbathandtheinsert.The1Kplateisconnectedtothe mainbathviaavacuumshieldedpickͲuptubeandamechanicalpumpisusedtoachieve evaporativecoolingofliquidHe,reducingthetemperaturetoabout1.4K(hencethe name1Kpot).TheinflowofliquidHeiscontrolledwithaneedlevaluetopreventthe pot from emptying or overfilling. The 1 K plate and coil wraps around the channel between the 3He sorption pump (same principle as 4He sorb but slightly bigger as it needstophysisorbmoregas)andthe3Hepot,sothatwhenheatisappliedtothe3He sorb,allofthe3Hegaswillbereleased,condensedbythe1Kplateandthencollectedat the3Hepot.Oncetheheatiscutoff,the3Hesorbwillstarttopumpontheliquid3Hein thepotwherethesampleholderisattachedviaacoldfinger.Since3HehashigherzeroͲ point energy compared to 4He, evaporative cooling of 3He results a lower minimum temperatureof̱300mK.

2.4.4–3He/4Hedilutionrefrigerationsystem

AphotographoftheKelvinoxK100dilutionrefrigeratoratUNSWisshowninFig. 2.9(a), the dilution unit, cryostat and external gasͲhandling system are shown schematicallyinFig.2.9(b).Thecryostatismadeupofthreeconcentricmetalcans:the outer vacuum can (OVC), the liquid Nitrogen (LN2) jacket and the main bath. A superconducting solenoid is immersed in liquid 4He at all times at the bottom of the mainbathandprovidesmagnetfieldupto10Tperpendiculartothe2DEGplane(not shownintheschematic).Thedilutionunitiscontainedinsidetheinnervacuumcan(IVC) andismadeupoffivemainparts:the1Kpot,condenser,still,heatexchangerandthe

Figure2.9:(a)AphotographoftheKelvinoxK100dilutionunitatUNSW.(b)Aschematic diagram of the refrigeration system showing the dilution unit, cryostat and external gasͲ handlingsystem.

mechanical pumps: one to circulate the 3He/4He mixture, one for the 1 K pot and anotheronefortheIVC.

Whenamixtureof3Heand4Heiscooledbelowacriticaltemperature̱ܶ0.87K, itwillseparateintoa3Herichphase,anda3Hedilutephase.Inthedilutionrefrigerator theconcentrationandvolumeofthismixturearechosensothatthelocationoftherichͲ dilutephaseboundaryoccursatthemixingchamber,andthesurfaceofthedilutephase isatachamberknownasthestill.Thestilliskeptatܶ=0.6to0.7Kwherethevapour pressureof3Heismuchhigherthanthatof4Heatthesetemperatures.Asaresult,when the still is pumped on, the concentration of 3He reduces and this creates an osmotic pressure difference between the mixing chamber and the still, causing 3He to flow across the richͲdilute phase boundary. Since the specific heat of 3He is larger in the

76 dilutephasethanitisintherichphase,coolingisachieved.Duringtheoperation,the rotarypumpcirculatesevaporated3Hefromthestilltopassthroughaseriesoffilters, nitrogen and helium cold traps (not shown) to remove any impurities, before being condensedbythe1KpotandsuppliedtothecondenserlineforreͲliquification.Theheat exchangerallowsthemixtureinthestillsidetocoolthe reͲliquifiedmixture,ensuring thatitisclosetobasetemperaturepriortoreͲenteringthemixingchamber.Aheater situatedinthestillisusedtocontrolthevapourpressureandthusthe 3Hecirculation rate;whilethetemperatureofthedevicemountedonthecoldfingercanbecontrolled withasecondheaterlocatedatthemixingchamber.

WhilethedilutionfridgeandtheHelioxsystemscanprovidemillikelvinlattice temperatures,amaterialcanhaveanumberoftemperaturesrelatingtotheentropies ofthevariouspartsofthematerial(i.e.thecrystallattice,electrons,nuclearspins,etc) [7,8].Atܶ>1K,thevariouspartsofthedevicearethermallywellcoupledanditmakes sensetotalkaboutasingletemperaturefortheexperiment.Howeveratܶ<1K,the couplingsbreakdown;inthiscase,whatcountsisthatelectronͲphonondecouples.The most important temperature in transport measurement is the electron temperature. Onewaytoassesstheelectrontemperatureistouseaquantumdotandmeasurethe temperaturedependenceofitsCoulombblockadepeaks[9].AsdiscussedinSection4, we found in our experiment that, although the dilution fridge system has a base temperature of̱40 mK according to thermometry, the electron temperature in the quantumdotdeviceremainedatabout140mK[10].

2.5–Electricalsetupsanddataacquisition

In this section, we describe the electrical setups used for measuring dots as singleelectrontransistors(SETs)inChapter4,andasbilliarddevicesinChapter5&6.

2.5.1–Electricalsetupformeasuringsingleelectrontransistors

BothstandardlowfrequencyaclockͲintechniques,aswellasdcelectricalsetupswere usedtoobtaintransportmeasurementsofSETsusingtheK100dilutionrefrigerator.A

77 schematicoftheacsetupfordeviceAS43N(seeSection4.3)isshowninFig.2.10. A constantexcitationvoltagewassuppliedtothesampleviathesourcecontactfromthe oscillator of a Stanford Research SR830 lockͲin amplifier using a voltage divider of typically 1 by 10000 ratio. The response currentܫfrom the drain contact was first detectedbyaStanfordResearchSR570currentpreͲamplifier,beforeitwasmeasured withtheSR830,andrecordedusingtheLabVIEWsoftware.Forgatebiasing,weeither usedthedigitaltoanalogue(DAC)outputsofaSR830lockͲinamplifier,aKeithleyK2400 sourceͲmeasure unit, or a Yokogawa 7652 voltage source. In the dc configuration, voltage excitation was provided by a Yokogawa 7652 voltage source, whileܫwas measured using a Keithley K2000 multimeter through a Femto LCAͲ200Ͳ10G current preamplifier. Quantum dot devices AS08N and AS43N presented in Chapter 4 were measured using this setup. As discussed in Section 1.4, in order to observe charging effectsinSET,boththeresistanceoftheleftandrightbarriersneedtobemuchlarger than13kё. SincethedotresistanceismuchlargerthantheresistanceoftheOhmic contacts (typically between 100ёto 1 kё) and the input impedance of the lockͲin amplifier (̱1 kё), most of the applied excitation dropped across the dot area and thereforeonlytwoͲterminalmeasurementswererequired.

Inordertominimizetheelectricalnoiseofthedilutionfridgesetup,alldcgate lineswerefilteredusinghomeͲmadeRCRfilterswithacutͲofffrequencyof0.1Hz.To reduce the high frequency noise, we connected the ac lines such as the Ohmic connections, to MiniͲCircuits BLPͲ1.9 commercial lowͲpass filters with a cutͲoff frequency of 1.9 MHz. Additionally, a set of passive internal RCR cold filters for all electrical lines going into the fridge were placed at the 1 K pot stage of the dilution fridge.Another source of noise often found in the electrical setup are caused by competing grounds in the circuit; in order to minimize this effect, all the electronics wereconnectedtothesamegroundpointandthecorrespondingGPIBinterfaceswere isolated from the ground of the computer via an optical isolator bus. Moreover, the SR570 current amplifier used was powered by a 12 V dc car batteries instead of the mains,andplasticKFclampsandOͲringswereusedtoelectricallyisolatethegroundsof thefridgepumpinglinesfromtheelectronics.

Figure:2.10:Schematicdiagramoftheacmeasurementsetupusedforquantumdotdevice (AS43N) discussed in Section 4.3. The circuit diagrams for the RCR filters (enclosed by dashedrectangles)areshownnearthebottom.

2.5.2–Electricalsetupformeasuringelectronbilliards

ForthemagnetoͲconductancemeasurementsofopendotsdiscussedinChapter 5and6,standardlowfrequencyaclockͲintechniqueswereused.Sincetheresistances oftheopendotsarecomparabletotheresistanceoftherestofthecomponentsinthe circuit, a fourͲterminal measurement configuration was used. Using a constant excitationvoltageoftypically100ʅV,thedotconductancewasobtainedfromdividing thepotentialdifferencebetweenthevoltageprobeslocatedoneachsideofthedot, measuredwithanEG&GInstruments5210lockͲinamplifier;bythecurrentacrossthe wholesamplemeasuredwithaSR830lockͲinamplifier.Figure2.11showsaschematic diagram of the measurement setup used for the billiard devices AS43N and AS61N discussedinChapter5&6respectively.

Figure2.11:Schematicdiagramofthemeasurementsetupfortheundopedbilliarddevice (AS61N)discussedinChapter6.

Bibliography

[1]“Newphenomenaincoupledtransportbetween2Dand3DelectronͲgaslayers”–P. M.Solomon,P.J.Price,D.J.Frank,andD.C.LaTulipe,Phys.Rev.Lett.,63,2508(1989).

[2]“HighmobilityGaAsheterostructurefieldeffecttransistorfornanofabricationin whichdopantͲinduced disorderiseliminated”–B.E.Kane,L.N.Pfeiffer,andK.W.West, Appl.Phys.Lett.,67,1262(1995).

[3]“ModernGaAsprocessingmethods”–RalphWilliams,ArtechHouseMicrowave Library,2nded.(1990).

[4]“TheroleofsurfaceͲgatetechnologyforAlGaAs/GaAsnanostructures”–R.P.Taylor, Nanotechnology,5,183(1994).

[5]“Electrontransportinquantumdots”–L.P.Kouwenhoven,C.M.Marcus,P.L. McEuen,S.Tarucha,R.M.Westervelt,andN.S.Wingreen,inMesoscopicElectron Transport,NATOASISer.EVol.345,editedbyL.L.Sohn,L.P.Kouwenhoven,andG. Schön,Kluwer,Dordrecht,(1997),pp.105–214.

[6]“IntroductiontoMesoscopicPhysics”–Y.Imry,OxfordUniversityPress,NewYork, (1997).

[7]“MatterandMethodsatLowTemperatures”–F.Pobell,SpringerͲVerlag,Berlin,2nd ed.(1996).

[8]“ExperimentalPrinciplesandMethodsBelow1K”–O.V.Lounasmaa,AcademicPress, London,(1974).

[9]“StatisticsandParametricCorrelationsofCoulombBlockadePeakFluctuationsin QuantumDots”–J.A.Folk,S.R.Patel,S.F.Godijn,A.G.Huibers,S.M.Cronenwett,C. M.Marcus,K.Campman,andA.C.Gossard,Phys.Rev.Lett.,76,1699(1996).

[10]“AlGaAs/GaAssingleelectrontransistorfabricatedwithoutmodulationdoping”–A. M.See,O.Klochan,A.R.Hamilton,A.P.Micolich,M.Aagesen,andP.E.Lindelof,Appl. Phys.Lett.,96,112104(2010).

[11]“OxfordInstruments,Kelvinox300DilutionRefrigeratorManual”,Oxford Instruments,Oxford,(1988).

Chapter3

CharacterizationmeasurementsofNBI30and C2275heterostructures

3.1–Introduction

Open and closed quantum dots formed on the undoped AlGaAs/GaAs heterostructure NBI30 are discussed throughout this thesis. In Chapters 5 and 6, we presentcomparativestudiesbetweendevicesformedontheNBI30anditsmodulationͲ doped counterpart C2275. These wafers have comparable electron density݊and mobilityߤ. This chapter aims to provide useful characterization information of these two heterostructures. We will first present Shubnikov de Haas (SdH) and Hall measurements performed on Hall bar mesas formed on these two systems, which allowed us to extract݊andߤ. Furthermore, in order to determine the dominant scatteringmechanismthatlimitsthemobilityoftheNBI30andC2275heterostructures, we discuss two scattering lifetimes: the transport scattering time߬௧and the quantum lifetime߬௤.Theratioof߬௧Ȁ߬௤canbeusedtocharacterizethescattering[1]forthetwo systems.Finally,duetothepresenceofsurfacestatesinNBI30,adepletionregionis formed underneath the surface as a result of Fermi level pinning. The size of this depletionregiondeterminestheminimumfeaturesizeofstructuresthatcanbemade

84 from the cap, ensuring that it remains conductive at low temperature. In order to determine the size of this depletion region, we report an experiment where the conductivities of narrow wires defined in NBI30’s degenerately doped cap were measured.

3.2–ShubnikovdeHaasandHallmeasurements

A photograph of the AS08N Hall bar device fabricated on the NBI30 heterostructure is shown in Fig. 3.1(a), while a schematic representation of the measurementsetupisshownin(b).Measurementswereperformedusingastandard fourͲterminal ac lockͲin technique in a constant voltage configuration. An Oxford InstrumentVarioxpumped4HesystematUNSWwithabasetemperatureof1.4Kwas used.DetaileddiscussionsoftheVarioxsystemcanbefoundinSection2.4.2.Forthe C2275modulationͲdopedheterostructure,aHallbardeviceC2275ͲCsimilartoAS08N wasmade.MeasurementswereperformedontheHelioxpumped3HesystematUNSW at1.1K.

Figure3.1:(a)AphotographofdeviceAS08NfabricatedonNBI30.Itwasusedtomeasure the electron density݊and mobilityߤat different top gate voltages்ܸீatܶ= 1.4 K. The scalebarshownis0.5mmlong,theyellowregionscorrespondtoTi/Augatesdepositedto make electrical contacts to the degenerately doped top gate and to the Ohmics. (b) A schematic representation of the standard ac lockͲin setup for measuring the longitudinal

(ܴ௫௫=ܸ௫௫Ȁܫ) and transverse (ܴ௫௬=ܸ௫௬Ȁܫ) resistances. The yellow TͲgate represents an electricalcontacttothedopedtopgate;asdiscussedinSection3.4,asmallSchottkybarrier isformedbetweenthetwo.

Figure3.2showsresultsoftheShubnikovdeHaas(SdH)andHallmeasurements for AS08N in (a), and C2275ͲC in (b). The corresponding longitudinal ܴ௫௫ and transverse/Hallܴ௫௬resistancesaredisplayedinblueandredrespectively.Asdiscussedin Section 1.3.1, the electron density݊can either be extracted from the SdH data using

Eqn. 1.2, or from the Hall data using the relation݊ൌܤȀܴ݁௫௬. All density values presented in this chapter were obtained using the Hall method, and correspond to measurements performed in the dark (no LED flash). Once we have obtained݊, the correspondingelectronmobilityߤ,canbecalculatedusingEqn.1.3.

Figure3.2:ShubnikovdeHaas(blue,leftaxis)andHall(red,rightaxis)measurementsfor(a) the undoped AS08N, and (b) modulationͲdoped C2275ͲC devices, showing longitudinal

ܴ௫௫and transverse/Hall resistancesܴ௫௬with a field resolution of 0.5 mT. The similarities between(a)and(b)suggestthat,theNBI30andC2275heterostructureshaveverysimilar electrondensitiesandmobilities.Bothmeasurementswereperformedinthedark,andthe datain(a)wereobtainedatatopgatevoltage்ܸீ=1V.

BasedonthedatainFig.3.2(b),weobtained݊=2.34×1011cmͲ2andߤ=3.33× 105cm2VͲ1sͲ1fortheC2275heterostructure.InordertogetanideaofhowmanySi dopantatomsareionizedintheC2275sample,weuseda1DSchrodingerͲPoissonsolver כ tosimulatetheelectrondensityatthe2DEGwithagivendopingdensityܰௗusedasa fittingparameter.Thisdopingdensitywasadjusteduntilthecalculatedelectrondensity matched what we extracted from the Hall measurements. The fraction of ionized כ dopantscanbethencalculatedbycomparingܰௗwiththeܰௗspecifiedfromthegrowth 18 Ͳ3 notes. Atܰௗ= 1.2ൈ10 cm and assuming a quantum well width of 25 nm, we כ calculatedthatܰௗȀܰௗ̱0.6,i.e.only60%ofthesilicondopantatomswereionizedin C2275ͲC.UnͲionizedSidopantinAlGaAscanalsoexistinasecondstateknownasDX center[2].WewilldiscusstheimplicationsofthisinSection6.2.3.

3.2.1–TopgatedependenceoftheelectrondensityinNBI30

BasedonthedatainFig.3.2(a),wheredeviceAS08Nwasmeasuredatatopgate 11 Ͳ2 5 2 Ͳ1 Ͳ1 bias்ܸீ=1V,weobtain݊=2.33×10 cm andߤ=3.67×10 cm V s .Inorderto obtain݊ሺ்ܸீሻ,weperformedSdHandHallmeasurementsbetween்ܸீ=0.8and1.1V instepsof50mV,andplotthecorrespondingdensitiesasafunctionas்ܸீinFig.3.3(a). The corresponding density dependence of mobilityߤሺ݊ሻis shown in Fig. 3.3(b). From 11 Ͳ2 the linear fit of the data in (a),݊(்ܸீ)ൌെ1.09൅3.42்ܸீin units of 10 cm was extracted.Thisequationimpliesthatthedensityreacheszeroat்ܸீ=0.32Vandnotat 0Vasonewouldexpectforasimpleparallelplatecapacitormodel.Thisfinitethreshold can be understood as the voltage required to overcome the effect of disorder (e.g. populated traps and surface states) on electrons at low densities [3]. As a result, we wouldexpectthethresholdvoltagetovaryslightlysampletosample.Incontrast,the slopein݊ሺ்ܸீሻrelatestothecapacitancebetweenthetopgateandthe2DEG,which shouldbeuniformacrossallNBI30samples.Notehowever,the“turnͲon”voltageofthe Ohmiccontacts,whichwedefineastheminimumvoltagerequiredforacurrenttoflow betweenapairofOhmiccontacts,dependsentirelyonthefabricationprocess.Wecan interpretthisasvariationsinthephysicallocationsoftheOhmicswithrespecttothe 2DEGandtopgate,whichresultvariationsintheminimumelectrondensityrequiredto

Figure 3.3:(a) Electron density݊vs. top gate voltage்ܸீ for AS08N. The blue line correspondstothelinearfitofthedata(redcircles),andhastheform݊(்ܸீ)ൌെ1.09൅ 11 Ͳ2 3.42்ܸீin units of 10 cm . (b) The corresponding mobilityߤplotted as a function of݊. MeasurementswereobtainedontheVarioxpumped4Hesystematܶ=1.4K.

trigger the conduction. For our devices, the Ohmics typically turn on at்ܸீbetween 0.35and0.8V,andstartleakingtothecapataround1.1V.

3.3–TransportscatteringtimeandquantumlifetimeofNBI30 andC2275

We discussed the various scattering mechanisms that limit the mobility in 2D AlGaAs/GaAssystemsinSection1.3.3.Inpractice,determiningthefactorsthatlimitthe mobilityinaparticularsampleisdifficultsincedifferentscatteringeventsactcollectively

accordingtoMatthiessen’srule[4],whichstatesthatthetotalscatteringrate1/߬ఀis the sum of all scattering rates due to different mechanismsሺͳȀ߬ఀ ൌͳȀ߬ଵ ൅ͳȀ߬ଶ ൅

ͳȀ߬ଷ ൅ǥሻ.

There are two experimentally accessible parameters that have been used to separate different scattering mechanisms: the transport scattering time߬௧, obtained from conductivity, and the quantum lifetime (also known as single particle relaxation time)߬௤ , obtained from the temperature dependence of the Shubnikov de Haas oscillations [5].߬௧is weighted towards largeͲangle scattering, which is scattering that causesasignificantchangeinmomentumandwhichcontributestothemobilityinthe

Drudemodel[5].Thequantumlifetime߬௤,however,takesallscatteringevents(i.e.both largeͲ and smallͲangle) into account. As pointed out by Harrang et al. [6],߬௧is only accurate for systems such as silicon metalͲoxideͲsemiconductorͲfieldͲeffectͲtransistors (MOSFETs), where scattering is mostly shortͲrange, therefore mostly largeͲangle in nature.FormodulationͲdopedsystems,largeͲanglescatterersareeffectivelyconverted intosmallͲanglescatterersduetothespacerlayerseparatingtheionizeddopantsand

2DEG;asaresult,߬௧and߬௤canbecomesignificantlydifferent.

However, the ratio of߬௧Ȁ߬௤ can be used to determine the nature of the predominantscatteringmechanisminthe2DEGatlowtemperatures[1,6Ͳ9].Togeta feelingfortheratio߬௧Ȁ߬௤,let’sassumethetotalscatteringtime(߬௤)consistsofonlythe sum of largeͲangle and smallͲangle scattering timesͳȀ߬௤ ൎͳȀ߬௟௔௥௚௘ ൅ͳȀ߬௦௠௔௟௟. In a system where the dominant scattering mechanism is largeͲangle scattering,ͳȀ߬௤ ൎ

ͳȀ߬௟௔௥௚௘,then߬௧Ȁ߬௤ ൎ߬௧Ȁ߬௟௔௥௚௘andtheratioiscloseto1.Incontrast,forasystem where the dominant scattering mechanism is smallͲangle,ͳȀ߬௤ ൎͳȀ߬௦௠௔௟௟ , then

߬௧Ȁ߬௤ ൎ߬௧Ȁ߬௦௠௔௟௟andtheratioismuchlargerthan1.AccordingtoMacLeodetal.[1], the“ruleofthumb”isthat:߬௧Ȁ߬௤൑10forsystemswherebackgroundimpurities(largeͲ angle) is the dominant scattering mechanism and߬௧Ȁ߬௤ ൒10 for systems where remoteionizedimpurities(smallͲangle)arethedominantscatteringmechanism.

In the following, we will present calculations of߬௧Ȁ߬௤for AS08N and C2275ͲC based on the experimental data from the previous section. Theoretical treatments of

߬௧Ȁ߬௤canbefoundin[1]andreferencestherein.

3.3.1–ExtractingthequantumlifetimefromShubnikovdeHaas oscillations

Asdiscussedearlier,thetransportscatteringtime߬௧canbeobtainedfromthe Drudemodelofconductivity[10],andisrelatedtothemobilityviaEqn.3.1.

ߤכ݉ ߬ ൌ ሼ͵Ǥͳሽ ௧ ݁

Experimentally,߬௤ can be extracted from the SdH oscillations using the LifshitzͲ Kosevitchanalysis[5].Itcanbeshownthat:

߂ߩ௫௫ሺܤሻ െɎ ଴ ൌʹሼ ሽሼ͵Ǥʹሽ ʹܦሺܺሻߩ௫௫ ߱௖߬௤

଴ כ ଶ whereܦሺܺሻൌ ܺȀሺܺሻ,ܺ=ʹߨ ݇஻ܶȀ¾߱௖ǡ߱௖ =݉ Ȁ݁ܤ, ߩ௫௫ is the longitudinal resistivity atܤ= 0 and߂ߩ௫௫is the amplitude of the SdH oscillations. If we take the naturallogarithmofbothsides,Eqn.3.2reducestothefollowinglinearequation:

כ οߩ௫௫ሺܤሻ ߨ݉ ͳ ൬ ଴ ൰ൌʹെ ൬ ൰ሼ͵Ǥ͵ሽ ʹܦሺܺሻߩ௫௫ ߬௤݁ ܤ

௱ఘೣೣ ଵ Thequantumlifetime߬௤canthusbedeterminedbytheslopeofሺ బ ሻvs. .The ଶ஽ሺ௑ሻఘೣೣ ஻

௱ఘೣೣ ௱ఘೣೣ ଵ quantity బ isknownasthereducedamplitudeandaplotofሺ బ ሻvs. is ଶ஽ሺ௑ሻఘೣೣ ଶ஽ሺ௑ሻఘೣೣ ஻ referred to as a Dingle plot. The amplitudes of the SdH oscillations߂ߩ௫௫can be ଵ ଵ calculated using the following definition, ߂ߩ ൫ܤ ൯ൌ ȁሼߩ൫ܤ ൯െ ሾߩ൫ܤ ൯൅ ௫௫ ௝ ଶ ௝ ଶ ௝ିଵ

ߩ൫ܤ௝ାଵ൯ሽ|[11],where݆isanintegerandߩ൫ܤ௝൯correspondstoSdHmaxima/minima. Figure3.4illustratesthisdefinition.

ଵ ଵ Figure 3.4:A diagram illustrating߂ߩ ൫ܤ ൯ൌ ȁሼߩ൫ܤ ൯െ ሾߩ൫ܤ ൯൅ߩ൫ܤ ൯ሽ[11]. ௫௫ ௝ ଶ ௝ ଶ ௝ିଵ ௝ାଵ Theboldarrowinthediagramcorrespondsto2߂ߩ௫௫,calculatedbytakingtwominimaand onemaximumpointoftheSdHpeak(inred).

11 Ͳ2 The transport lifetime߬௧ for NBI30 at݊= 2.33ൈ10 cm ,்ܸீ = 1V was calculatedtobe13.8psusingtheDruderesultfromEqn.3.1.Figure3.5(a)showsafew

SdH oscillations from AS08N obtained at்ܸீ= 1 V. The black markers represent the maximaandminimausedtocalculate߂ߩ௫௫.ThecorrespondingDingleplotisshownin Fig. 3.5(b) with a vertical intercept close to ݈݊ 2 ̱ 0.69, which indicates the

Figure3.5:(a)ThesameSdHoscillationsfromAS08N(undoped)at்ܸீ=1VinFig.3.2(a), showingpointsofmaximaandminima(asblackmarkers)usedtoextract߂ߩ௫௫ሺܤ௡ሻ.(b)The ௱ఘೣೣ correspondingDingleplotwhereሺ బ ሻvs.ͳȀܤisshown.Theverticalinterceptwas ଶ஽ሺ௑ሻఘೣೣ foundtobecloseto2,whichisanindicationofagoodDingleplot.Thetopgatevoltage waschosentohaveanelectrondensityof2.33ൈ1011cmͲ2,similartotheC2275ͲCdevice.

Theextracted߬௧and߬௤werefoundtobe13.8and1.61psrespectively.Thisgivesaratioof

߬௧/߬௤̱8.57.Usingtherelation߱௖߬௤=1,ܤ=0.24Tisobtained,whichagreesreasonably wellwithonsetofoscillationsin(a).

appropriatenessofusingEqn.3.3tofitthedata.Thequantumlifetime߬௤wasfoundto be1.61ps,thisgivesaratioof߬௧Ȁ߬௤̱8.57.

The scattering lifetimes߬௧ and߬௤ for the modulationͲdoped C2275ͲC device have also been extracted based on the SdH oscillations from Fig. 3.2(b). The correspondingDingleplotisshowninFig.3.6(b).Theextractedlifetimesare13.3psfor

߬௧and1.33psfor߬௤,resultingaratioof߬௧Ȁ߬௤̱10.Thisratioisonlyslightlylargerthan thatoftheundopedsample,whichqualitativelysuggeststhatbackground(largeͲangle) scattering has a dominant effect over the smallͲangle scattering, (since ͳȀ߬௤ ൎ

ͳȀ߬௟௔௥௚௘ ൅ͳȀ߬௦௠௔௟௟).Wehavetriedusingthescatteringformalismdiscussedin[1]to model߬௧and߬௤over a range of electron densities.* However, our calculation shows that in order to match the extracted scattering lifetimes, an assumption of 1.2% Si dopantionizationisrequired.Thisassumptionisclearlyunreasonable,andthesituation atpresentisanopenquestion.Asasuggestionforfuturework,itwouldbeinteresting tomeasuretheSdHoscillationsasfunctionoftheelectrondensity(usingagatedC2275

Figure 3.6:(a) The same SdH oscillations of C2275ͲC from Fig. 3.3(b), showing points of maxima and minima (as black markers) used to extract߂ߩ௫௫ሺܤ௡ሻ. (b) The corresponding ௱ఘೣೣ Dingleplotshowingሺ బ ሻvs.ͳȀܤ.Theverticalinterceptwasfoundtobecloseto ଶ஽ሺ௑ሻఘೣೣ 2,whichisanindicationofagoodDingleplot.Theextracted߬௧and߬௤were13.3and1.33 psrespectively.Thisgivesaratioof߬௧Ȁ߬௤ ൌ10.Usingtherelation߱௖߬௤=1,ܤ=0.29Tis obtained,whichagreesreasonablywellwithonsetofoscillations.

*Modellingof߬௧and߬௤wasperformedbyD.Wang.

device)andextractthecorrespondingdensitydependenceof߬௧and߬௤,andseeifthat canbeusedtoanswerthisquestion.

Althoughourdingleanalysisresultscannotbeusedtoshowthatthedominant scattering mechanism in C2275 is of smallͲangle type, in Chapter 5 and 6, we will demonstratethesignificanceoftheseremoteionizedimpuritiesinthecharacteristicsof magnetoͲconductancefluctuations(MCF),measuredoverdifferentroomtemperature thermalcyclesusingquantumdotdevice[12,13].

3.4–SurfacedepletionofNBI30

IntheNBI30device,electronsareinducedatthe2DEGbypositivelybiasingthe degenerately doped݊ାGaAs cap.In order to form nanostructures, electron beam lithographyisusedtotransferthedevicepatternontothePMMAresist,followedbya chemicalwetetch(fabricationdetailscanbefoundinSection2.3.1).Attheboundaries ofasemiconductorgatedefinedbyetching,adepletionregionܦisformedduetothe presenceofsurfacestates.Theexactformofthesesurfacestatesiscomplicated;there areoxidesofbothGaandAs[14],elementalAs[15],aswellasothersurfacedefects.All of those introduce extra electronic states that are collectively referred to as surface states. Figure 3.7(a) shows a schematic representation of a semiconductor gate

Figure3.7:(a)Aschematicdiagramofasemiconductorgateformedbyetchingthecapof NBI30.Thebluecirclesrepresentthesurfacestatesontheedges(statesontopofthegate arenotshown)(b)Anenergybanddiagramacrosstheregionindicatedbyadashedlinein (a).AdepletionwidthܦisformedduetotheFermilevelpinningatmidͲgapattheGaAs boundary.

93 formedbyetchingthecapofNBI30.Duetothehighdensityofsurfacestatespresent neartheedges,theFermilevelatthesurfaceispinnedatmidͲgap.Theenergyband diagramcorrespondingtothedashedͲlinein(a)isshowninFig.3.7(b).Thisdepletion regionܦcanbeestimatedusingthesametreatmentforcalculatingdepletionregionin anabruptpͲnjunctionwhereonesideismuchmoreheavilydopedthantheother.This isshowninEqn.3.4:

ʹߝܸ ܦ ൌ ඨ ௕௜ ሼ͵ǤͶሽ ݁ܰௗ

whereܸ௕௜isthebuiltͲinvoltage,ߝisthedielectricconstantandܰௗisthedopingdensity.

Substitutingܸ௕௜=0.76V,ߝ=12.9ߝ௢(߳଴istheelectricpermeabilityoffreespace̱8.85 Ͳ12 24 Ͳ3 ൈ10 F/m)andܰௗ=3.3ൈ10 m forNBI30,ܦwascalculatedtobeabout20nm.

3.4.1–Thedepletionregionexperiment

Thesizeofthisdepletionregioniscriticalindesigningdevicestructures:ifthe widthofthegatesissmallerthan2ൈ ܦ,theydonotconductatlowtemperaturesand willnotproperlydefineananoscaledevicesuchasaQPCorquantumdot.Inorderto experimentallydetermineܦ, wemeasuredtheconductivitiesoffivemicronͲlongwires formed on the cap of NBI30. The measurements were obtained at 10 K in the cryoͲ cooleratUNSW.AcolouredSEMimageofthreeofthewiresisshowninFig.3.8.The Ti/Au metal gates were deposited onto the wires to make electrical contacts. To measuretheresistanceofawire,weconnectedoneendofthewiretoaK2400source meter,whiletheotherendwasconnectedtoground.Thecurrentܫthroughthewire wasmeasuredasafunctionoftheappliedvoltageܸ.

Figure3.9(a)showstheܫͲܸcharacteristicsforthe100nm,130nm,180nmand 230 nm wires. The wires with widths larger than 100 nm exhibited Ohmic behaviour withacommonresistanceܴof̱10Mё.Thefactthatallthesewireshaveacommon resistance value suggests thatܴis dominated by the contact resistance due to a

Figure3.8:(a)AcolouredSEMimageofaNBI30devicewithpatternedwires.Threewires areshownhere(inthecentralgreenregion)withelectricalcontactsmadebythedeposited Ti/Au interconnects (shown in yellow). A schematic of the electrical setup is also shown, where the current responseܫis measured as a function of applied biasܸ. Measurements wereobtainedinacryoͲcooleratܶ=10K.Thegreenregionsrepresentareasoftheintact caplayer,whereasthegreyregionscorrespondtotheetchedregions.(b)Amagnifiedview oftheregioninsidetheredcirclein(a),showingonethewires.Thescalebarsshownare50 ʅmin(a),and1ʅmin(b)respectively.

SchottkybarrierformedbetweentheTi/Aufingergatesandthesurfaceofthedoped cap. The 100 nm wire however, did not turn on untilܸ= 20 mV, and theܫͲܸ characteristics became linear only forܸ> 80 mV. Qualitatively, this behaviour can be understood by considering the schematic in Fig. 3.7(b). The depletion edgeܦat each exposed surface has a fixed width. If the wire widthܹ൐ʹܦ, then there is a narrow conducting strip of widthܹെʹܦthat gives the Ohmic conductivity at higher wire widths.Asܹapproachesʹܦ,theconductingstripbecomesverynarrow,andthisdrives thefreeelectrondensity,andhencetheFermienergyinsidethewire,downrelativeto the conduction band. This leads to an effective “turn on” voltage to overcome the barrier and drive current through the wire. Eventually whenܹ̱ʹܦ, the Fermi level will pin at midͲgap with a much higher turnͲon voltage. Figure 3.9(b) shows theܫͲܸ characteristicsofthe55nmwire.Asexpected,theturnͲonvoltagewasevenlargeraṯ 2.2 V and theܫͲܸcharacteristics is diodic. Moreover, charging and discharging events wereobserved,asindicated bythecurrentfluctuationsintheinsetofFig.3.9(b).Finally, to determine whether these wires behaved as metals or insulators, we performed a

Figure3.9:(a)ܫͲܸcharacteristicsofthe100nm,130nm,180nmand230nmwires.The 230,180and130nmwiresexhibitOhmicbehaviour,whereasthe100nmisdiodicandwire does not turn on untiḻ20 mV. (b)ܫͲܸcharacteristics of the 55nm wire showing a large turnonvoltageof2.2V.Thecorrespondingtimetraceatܸ=2.5Visshownasaninset.The observedtimefluctuationssuggestcharginganddischargingofnearbytraps.BasedontheܫͲ ܸcharacteristics of the 130 nm and 100 nm wires, the minimum gate width that can be defined using the cap of NBI30 is̱120 nm wide, and a depletion region of̱60 nm is obtained.(c)Temperaturedependenceoftheresistancesܴofforthe130,180and230nm wires. Betweenܶ= 10 and 70 K,ܴremains roughly constant, which indicates metallic behaviour. At higher temperature, the Schottky barrier former between the Ti/Au finger gatesandthesurfaceofthecapisliftedbythermalactivation,ܴdrops.

temperature dependence analysis of ܴ for the conducting wires of width 130 nm,180nmand230nm.Betweentemperatureܶ=10and70K,ܴstayedroughlyat10 Mё,whichsuggestmetallicbehaviour.Forܶbetween70and250K,theirresistances drop rapidly (by a factor of 10 at 250 K) asܶis increased, which suggests thermal excitationovertheSchottkybarrierformedbetweentheTi/Aufingergatesandthecap.

BasedonthedatainFig.3.9,weconcludedthattheminimumwidthofgates definedusingtheNBI30capisabout120nm.Thedepletionregionܦthereforeequals

96 halfofthis,i.e.60nm(ܦappearsonbothsidesina2Dwire).Notethatthisvalueof depletionwidthdoesnotagreewiththeearlierapproximationof20nmverywell.We attribute this discrepancy to the limitations of the approximation method used. The assumptionofFermilevelpinnedatagivenlevelinthebandgapwastypicallymadefor treating states at the interface of metalͲsemiconductor contacts [10], where the experimentalܫͲܸcharacteristicoftheresultingSchottkydiodeisusedtodeterminethe “pinned”level.Whenthisapproachisappliedtoexposedsurfaceof heterostructures,it has limitations such as the discontinuity of the potential energy at the edges of the metalgatesontheexposedsurfaces[16].

3.5–Chaptersummary

In this chapter, we have extracted the density and mobility of two heterostructures used throughout the thesis: the undoped NBI30 and modulationͲ doped C2275. We found that they have comparable݊andߤǡsuitable for comparative transportstudies,withthepresenceofremoteionizedimpuritiesinC2275beingthekey 11 Ͳ difference.Basedontheratios߬௧Ȁ߬௤extractedfromtheirSdHdataat݊=2.33ൈ10 cm 2,wefoundthatthedominantscatteringmechanismfortheNBI30heterostructureisof largeͲangle, whereas the situation for C2275 is unclear. Finally, by looking at theܫͲܸ characteristicsofnarrowwiresthatwereformedontheNBI30݊ାcap,wededucedthe minimum feature size with conduction properties suitable for use as an electrostatic gateis120nm.

Bibliography

[1]“RoleofbackgroundimpuritiesinthesingleͲparticlerelaxationlifetimeofatwoͲ dimensionalelectrongas”–S.J.MacLeod,K.Chan,T.P.Martin,A.R.Hamilton,A.See,A. P.Micolich,M.Aagesen,andP.E.Lindelof,Phys.Rev.B,80,035310(2009).

[2]“Deep donorlevels(DXcenters)inIIIͲVsemiconductors”–P.M.Mooney,J.Appl. Phys.,67,R1(1990).

[3]“ConductioninGlassesContainingTransitionMetalIons”–N.F.Mott,J.NonͲCryst. Solids,1,1(1968).

[4]“ValidityofMatthiessen’sruleforcalculatingelectronmobilityinGa1оxAlxAsalloys”– A.K.SaxenaandM.A.L.Mudares,J.Appl.Phys.,58,2795(1985).

[5]“SmallͲanglescatteringintwoͲdimensionalelectrongases”–P.T.Coleridge,Phys. Rev.B,44,3793(1991).

[6]“Quantumandclassicalmobilitydeterminationofthedominantscattering mechanismin thetwoͲdimensionalelectrongasofanAlGaAs/GaAsheterojunction”–J. P.Harrang,R.J.Higgins,R.K.Goodall,P.R.Jay,M.Laviron,andP.Delescluse,Phys.Rev. B,32,8126(1985).

[7]“SingleͲparticlerelaxationtimeversusscatteringtimeinanimpureelectrongas”–S. DasSarmaandF.Stern,Phys.Rev.B,32,8442(1985).

[8]“LowͲfieldtransportcoefficientsinGaAs/Ga1оxAlxAsheterostructures”–P.T. Coleridge,R.Stoner,andR.Fletcher,Phys.Rev.B,39,1120(1989).

[9]“ThetransportͲtimetostateͲlifetimeratioinsemiconductorquantumͲwellalloys:a multiplescatteringanalysis“–M.J.Kearney,A.I.Horrell,andV.M.Dwyer,Semicond. Sci.Technol.,15,24(2000).

[10]“PhysicsofSemiconductorDevices”–S.M.Sze.NewYork:Wiley,2nded.,1981, ISBN0Ͳ47105661Ͳ8.

[11]"TransportPropertiesofGaAs/AlGaAsDoubleQuantumWells"–HolgerRubel, PhDthesis,UniversityofCambridge,CavendishLaboratory,1994.

[12]“ProbingtheSensitivityofElectronWaveInterferencetoDisorderͲInduced ScatteringinSolidͲStateDevices”–B.C.Scannell,I.Pilgrim,A.M.See,R.D.Montgomery, P.K.Morse,M.S.Fairbanks,C.A.Marlow,H.Linke,I.Farrer,D.A.Ritchie,A.R. Hamilton,A.P.Micolich,L.Eaves,andR.P.Taylor,–inPressforPhys.Rev.B, arXiv:1106.5823.

[13]“ImpactofSmallͲAngleScatteringonBallisticTransportinQuantumDots”–A.M. See,I.Pilgrim,B.C.Scannell,R.Montgomery,O.Klochan,A.M.Burke,M.Aagesen,P.E. Lindelof,I.Farrer,D.A.Ritchie,R.P.Taylor,A.R.HamiltonandA.P.Micolich,Phys.Rev. Lett.,108,196807(2012).

[14]“StorageconditionsforhighͲaccuracycompositionstandardsofAlGaAs”–K.A. Bertness,A.Roshko,S.E.AsherandC.L.Perkins,J.Vac.Sci.Technol.B,23,1267(2005).

[15]“NondestructivecharacterizationofinterfacelayersbetweenSiorGaAsandtheir oxidesbyspectroscopicellipsometry”–D.E.Aspnes,J.B.TheetenandR.P.H.Change,J. VacuumSci.Technol.,16,1374(1979).

[16]“Extractionofparametersofsurfacestatesfromexperimentalteststructures”–M. G.Pala,G.Iannaccone,S.Kaiser,A.Schliemann,L.WorschechandA.Forchel, Nanotechnology,13,373,(2002).

Chapter4

DevelopmentofundopedAlGaAs/GaAsquantum dotsoperatingintheCoulombblockaderegime

4.1–Introduction

AsdiscussedinSection1.2.2,thedevelopmentofundopedheterostructures[1Ͳ5] have enabled high quality 2D systems to be made without dopantͲinduced disorder present in the conventional modulationͲdoped systems. For example, the ability to obtain high electron mobilities at low carrier densities [4] has made undoped heterostructures ideal for studies of metalͲinsulator behaviour [6Ͳ8].Undoped heterostructureshavealsobeenusedtostudy1Dinteractingsystemssuchasquantum wires[9]andQPCs.Inthiswork,weextendtheselfͲalignedmethoddevelopedbyKane etal.[2,3]tofabricateundopedquantumdotsthatevolvedfromtheHeterostructure Insulated Gate Field Effect Transistor (HIGFET) conceived by Solomon et al. [1], for studiesinquasiͲ0Delectrontransport[10].

Semiconductor quantum dots have been used to realize singleͲelectron transistors (SETs) [11], artificial atoms [12], ultrasensitive electrometers [13], 2DEG

100 refrigeration and thermometry [14], and may ultimately be used as elements for quantum information applications [15,16]. Semiconductor quantum dots are typically definedusingnegativelybiasedsplitͲgatestodepleteregionsofthe2DEGformedina modulationͲdoped AlGaAs/GaAs heterostructure [17]. Although modulationͲdoping results in high electron mobilities [18], it can also cause significant charge noise and temporal instability due to rapid switching of the dopants between ionized and deͲ ionizedstates[19–21].MethodssuchasbiasͲcooling[20]ordepositinggatesonathin insulatinglayer[21]canreducechargenoisebutdonoteliminateitentirely,hindering thedevelopmentofultraͲsensitivequantumdevices.

Somereadersmaybefamiliarwiththefirstelectricalstudyofasemiconductor quantum dot by Meirav et al., where the dot device was fabricated on an inverted heterostructure [11]. However, the ISIS heterostructure usedhad a deltaͲdoped layer betweentheSchottkygatesandthe2DEGtocountertheGaAssurfacestates,leadingto similar disorder to that in modulationͲdoped heterostructures [22]. In contrast, our devices use a patterned top gate to form a quantum dot at the 2DEG. The gate is degeneratelydopedtohaveahighconductivityatlowtemperature,whichprovidesa sufficientlyhighelectrondensitytoscreenthe2DEGfromanyionizeddonorsinthecap [23].

Inthischapter,wepresentmeasurementsoftwoundopedquantumdotdevices.

ThemainfocushereistodemonstratequasiͲ0Dtransportinthequantumregime(݇஻ܶ

<<οܧ<<ܧ஼,seeSection1.4.2.1)withoursmallerdevice,sincemostofthequantum dotapplicationsdiscussedaboverelyonthepresenceofdiscreteenergylevelswithin the dot [12Ͳ16]. The larger quantum dot measured first was used primarily as a demonstration of the fabrication process and the experimental methods. Before we beginourdiscussions,IwouldliketoacknowledgethatwhileIfabricatedbothquantum dotdevicesandperformedallthedataanalysisbymyself,measurementsofthelarger dotdevicewereassistedbyDr.OlehKlochanwhileIwasabroad,duetoatightbooking scheduleofourKelvinoxK100fridgesystematthetime.

101

4.2–Measurementsofthelargerquantumdotdevice

AcolouredSEMimageofthelargerquantumdotdeviceAS08NisshowninFig. 4.1.Thelithographicdimensionsofthedotareapproximately0.84ʅmby0.76ʅm.The dopedcapisdividedintosevengates,asshowninFig.4.1.Thetopgate(TG)wasused toinduceelectrons attheheterojunctionanddefinethedot.TheQPCgates(QPCL&

QPCR) were used to form tunnel barriers between the dot and the source and drain reservoirs,andtheplungergates(PG)wereusedtotunetheelectronoccupancyofthe dot.DetailsofthefabricationprocesscanbefoundinSection2.3.1.AlsoshowninFig. 4.1isaschematicofthemeasurementsetup.StandardtwoͲterminala.c.measurement techniqueswereusedwiththedevicecooledtomillikelvintemperaturesbyaKelvinox K100 dilution refrigerator with a base temperature of 40 mK. Note that due to a leakageprobleminthebreakoutboxatthetime,onlytheupperpartoftheQPC gates were used, with the lower gates grounded, while the plunger gates were biasedsymmetrically.

Figure4.1:AcolouredSEMimageofalargequantumdotdevicesimilartoAS08Nisshown withaschematicofthemeasurementcircuit.Thedegeneratelydopedcapispatternedinto sevenseparategates.Thetopgate(TG,tintedinyellow)isusedtodefinethequantumdot aswellas toinduceelectronsatthe2DEG.TheQPCgates(QPCLinblue&QPCRingreen)are usedformthetunnelbarriers,andtheplungergates(PG,inred)wereusedtocontrolthe electron occupancy of the dot. The rectangles with crosses on either side of the device representtheOhmiccontacts.Thescalebarshownis1ʅmlong.

102

4.2.1–MeasurementofCoulombblockadeoscillationsforthe largedot

AsdiscussedinSection1.4.2,theobservationofCoulombblockadeoscillations requires sufficiently opaque tunnel barriersሺ݃<< 2݁ଶȀ݄) connecting the dot to the source and drain reservoirs. This condition can be realized by applying a sufficiently largenegativebiastotheQPCgates.Toinvestigatehowthesegatesbehave,wefirst pinched off the conduction channel using QPCL, QPCR and PG independently, and measuredthecorrespondingcurrentasafunctionofgatevoltageܸ௚.Theresultshave been converted into twoͲterminal conductance݃ଶ், and are shown in Fig. 4.2. The observedQPCpinchoffcharacteristicsisexpectedtobequalitativelydifferentfromtheone shown in Fig. 1.5(b). This is because in this kind of device, there are parts of device structures(thedotandtheotherQPC)thatarepermanentlypresent,unlikethesplitͲgate

Figure4.2:TwoͲterminalconductance݃ଶ்vs.gatevoltageܸ௚forQPCL(blue),QPCR(green) andPG(red).݃ଶ்wascalculatedbydividingthemeasuredcurrentܫbyaconstantexcitation voltageܸ௘௫of20ʅV.ThesepinchͲofftraceswereobtainedindependentlywhilekeepingthe other gates grounded. Both QPCL and QPCR were biased asymmetrically (only the upper sectionswereused),whereasPGwasbiasedsymmetrically.ThearrowshighlighttheQPC gatevoltageusedtoobtaintheblackCBtraceinFig.4.4atܸ௅=െ0.624Vandܸோ=െ0.204V.

ThedatasuggestthatQPCLismoreweaklycoupledtothedotthanQPCR.TheobservedQPC pinchoffcharacteristicsisexpectedtobequalitativelydifferentfromtheoneshowninFig. 1.5(b)becausethedotandtwoQPCsarepredefinedatthe2DEGusingthepatternedTG, causingelectroncoherentbackscatteringandinterferenceintheleads.

103 devices,wheretheseotherstructurescanbe“turnedoff”bygroundingtherespectivegates. As a result there will always be electron coherent backscattering that causes quantum interferencefluctuationsinourundopedetchedsamples.

Based on the dot geometry in Fig. 4.1, we expect the QPCs to pinch off conduction through the dot at a lower negative bias compared to the plunger gates. However, since the QPC gates were biased asymmetrically (only the upper part was used), weexpect theirpinchͲoffvoltage to beapproximately two times larger thanif theywerebiasedsymmetrically.ThedatainFig.4.2suggeststhatQPCLislesscoupledto thedotthanQPCR,asindicatedbythelargedifferencebetweentheirpinchͲoffvoltages ofെ1Vandെ0.5Vrespectively.ThiscouldbeunderstoodasaresultofnonͲuniform cappatterning.AsdiscussedinSection2.3.1,forunknownreasons,chemicalwetetchof thedopedcapbecomeshighlynonͲuniformifthesamplehadbeenannealedpreviously. DeviceAS08Nwasmadeundersuchconditions;asaresult,thecapwasetchedmultiple times in order for the gates to separate properly. We therefore hypothesize that the etchedregionbetweenQPCLandTGtobemuchdeeperthanthecorrespondingregion on the right. In this case, the larger pinchͲoff voltage can be explained due to the presenceofan“airgap”inthedeepertrenchregion,reducingthecapacitancebetween

QPCLandthedotchannel.ThisisillustratedschematicallyFig.4.3.Notethatwehave

Fig. 4.3:Illustration of the difference in etch depths between QPCL and QPCR to the dot channel. The deeper etched region on the left creates an “air gap”, which changes the dielectricconstantsignificantlyfrom̱12to̱1.Thismayexplaintheobserveddifference inthepinchͲoffvoltagesbetweenQPCLandQPCRinFig.4.2.

104 sinceoptimizedtheetchingprocessforsubsequentdevicespresentedinthisthesis.

To measure the Coulomb blockade oscillations,ܸ௅andܸோwere chosen to set similarconductancesatthepointcontactsaccordingtoFig.4.2,whilePGwassweptto tunetheelectronoccupancyuntilpinchͲoff(i.e.numberofelectronsonthedotܰ=0). This result is shown in Fig. 4.4, where a series of Coulomb blockade (CB) oscillations were observed superimposed on a decreasing conductance background. These oscillationshavecharacteristicsofCBoscillationsintheclassicalregime,wherethepeak maximadependmorestronglyonܸ௉ீthanthecouplingbetweentheleadsandthedot levels[17].Asܸ௉ீbecomesmorenegative,thechannelpinchedoffrapidly,andwedid notobserveanyCBoscillationswithminimaatzeroconductance.Thisislikelydueto thestrongcrosstalkbetweentheplungergateandthepointcontacts, shuttingdown one/bothchannel(s)completelyasthedotoccupancyistunedwithPG.Basedonthis observation, we made the separation between the dot and the plunger gate smaller (from100nmto50nm)inthesmallerdotdeviceAS43N,sothatonlyasmallbiasis requiredtochangethenumberofelectrons,reducingthecrosstalkbetweenPGandthe channels. In Section 4.3.2, a more thorough search was used to locate the Coulomb blockadeoscillationsforthesmalldotdevice.

Figure 4.4:Coulomb blockade oscillations observed in݃ଶ்vs.ܸ௉ீmeasured using seven different settings ofܸ௅andܸோ. The black sweep on the left corresponds to the trace obtainedatܸ௅ ൌെ0.624Vandܸோ=െ0.204V,ashighlightedbythearrowsinFig.4.2.The gatevoltageswerechosentosetasimilarconductanceineachoftheQPCsandso݃ଶ்(ܸ௅) ൎ݃ଶ்(ܸோ).Measurementswereobtainedwithaܸ௉ீresolutionof50ʅV.

105

4.2.2–Estimationofthechargingenergyforthelargedot

As discussed in Section 1.4.2, in order to observe Coulomb blockade, the charging energy of the dotܧ஼must be larger than the thermal energy݇஻ܶ. In the following,wewillprovideanestimateofܧ஼basedonasimpleparallelplatecapacitor model,andcompareitwiththeexperimentalresults.

ଶ ߑis dominated by the sum of theܥߑ, whereܥ஼ؠ݁ Ȁܧ,According to Eqn. 1.7 capacitances between the dot and all other gates. As a first approximation, we can assumethetopgatetohavethelargestcontributiontothetotalcapacitanceܥ்ீ ൎܥߑ, and apply the parallel plate capacitor modelܥ=݇߳଴ܣ݀Τ , where݇is the dielectric constant of AlGaAs̱12.15 [24] ,߳଴is the electric permeability of free space̱8.85 ൈ10Ͳ12F/m,ܣistheareaofthedoṯ0.64ൈ10Ͳ12m2and݀istheseparationbetween thebottomofthecapandthe2DEG=160nm.ThisassumptionisjustifiablebecauseTG has the largest overlapping area with the dot (largeܣ), and it is also located directly above the dot (small݀).Applying this model, we obtainedܥߑ̱430 aF, which corresponds toܧ௖= 0.37 meV (or 4.3 K in energy equivalent temperatureܶா=ܧȀ݇஻). Moreover,usingEqn.1.8theaveragesingleparticlelevelspacingοܧwasfoundtobe

11.2ʅeV(ܶா̱130mK),whichisonlyabout3%ofܧ௖.

Wecannowcomparetheaboveestimateofܧ஼withourexperimentaldata.Theperiod of CB oscillations in Fig. 4.4 is related to the capacitance between the dot and the plungergateܥ௉ீviaEqn.1.9.Thiscanbesimplifiedtoοܸ௉ீ ൌ݁Τ ܥ௉ீgiventhatοܧis onlyabout3%ofܧ௖fromtheaboveestimation.Inordertoobtainthetotalcapacitance

ܥఀandhenceܧ௖,weneedtoaddupthecapacitancesbetweenthedotandalltheother gates.Figure4.5showsthreesetsofCBoscillationsobtainedbysweeping(a)TG,(b)PG and(c)QPCR.TheextractedcapacitancescorrespondingtotheCBperiodsarepresented inTable4.1.Notehowever,wedidnothaveCBdataforallofthegates,thereforewe assumethecapacitancesbetweenthedotandallfourQPCgatestobethesame*.The totalcapacitancewasfoundtobe465.3aF,wherethetopgateconstitutesalmost80% ofit.Thecorrespondingܧ஼wasfoundtobe0.34meV(ܶா̱4K),whichagreeswithin10 %oftheestimatedvalueof0.37meVusingtheparallelplatecapacitormodelpreviously.

* Given the data in Fig. 4.2 suggesting that QPCL is less coupled to the dot than QPCR, the calculated

ܥఀof465.3aFisanoverestimation,whichwillmaketheagreementpresentedhereevenbetter.

106

Figure4.5:Coulombblockadeoscillationsobtainedfromsweeping (a)TG, (b)PGand(c)

QPCR.Thesesetsofoscillationsareusedtoextractthecorrespondinggatecapacitancesin ordertocalculatethechargingenergyܧ஼(seeTable4.1).Theinsetschematicshighlightthe gatessweptinyellow.

Plunger QPCleft Gates Topgate gate (top) ܥఀ 465.3aF οܸሾmVሿ 0.44 4.08 ̱10.34 ܧ 0.34meV ܥሾaFሿ 364 39.3 15.5 ஼

Table4.1:TheperiodοܸextractedfromtheCoulombblockadeoscillationsinFig.4.5for

AS08N.Thecorrespondingcapacitancesܥwerecalculatedusingtherelationοܸ௉ீ ൌ݁Τ ܥ௉ீ.

NotethattheCBoscillationdataforQPCL(bottomandtop)andQPCR(bottom)werenot available,andhenceweassumethattheyhavesimilarperiodstothatofQPCR(top)̱10.34 ଶ mV.Thechargingenergyܧ஼canbeestimatedusingܧ஼ ൌ݁ Ȁܥఀ,andwasfoundtobe̱ 0.34 meV, consistent with what we estimated using the parallel plate capacitor model earlier(0.37meV).

107

4.2.3–TemperatureanalysisoftheCBoscillationsfromthe largedot

As discussed in Section 2.4, for lattice temperatures well below 1 K, the electron temperaturecannotbesafelyassumedasequaltothelatticetemperature.Thereforeit isnecessarytoestimatetheelectrontemperaturebeforecomparingitwithܧ஼,toseeif theconditionܧ஼>>݇஻ܶwasmetinourmeasurements.Toachievethis,weperformed atemperaturedependenceanalysisoftheCBoscillationsobtainedfromsweepingthe plunger gate, as shown in Fig. 4.6. We observed that only whenܶ൐200 mK, the oscillationsbecomesuppressedasthetemperaturewasincreased.Thissuggeststhat the actualelectrontemperatureinthedotwasbetween200and300mK.Thiselectron temperatureisconsiderablyhigherthanthe70േ20mKreportedin[25].Thefactthat there was a significant amount of noise in the measurement (data shown in Fig. 4.6 weresmoothed)couldexplainthehigherthanexpectedelectrontemperatureobserved.

Given this effective electron temperature, we now have a situation whereܧ஼̱0.34 meV(4K)islargerthan݇஻̱ܶ22ʅeV(250mK),whichinturnislargerthanοܧ̱11 ʅeV(130mK).ThisclassifiesourCBpeakstobeintheclassicalregimeforthisdevice, consistentwiththeobservationmadeinFig.4.4.

Figure4.6:TemperaturedependenceoftheCBoscillations.Oscillationswereobtainedatܸ௅ =െ0.674Vandܸோ=െ0.254V,andaresimilartothoseshowninFig.4.4.Tracesareoffset andsmoothed,withadataresolutionof50ʅV.ThedatashowthatonlywhenT൒209mK, theCBpeaksappeartowiden,suggestinganelectrontemperaturetobebetween200and 300mK.

108

4.3–Measurementsofthesmallerquantumdot

InSection4.2,wepresentedtransportmeasurementsofthelargerquantumdot device,wheretheobservedCBoscillationswereshowntobeintheclassicalregime(ܧ஼

>݇஻ܶ>οܧ). As discussed in the chapter introduction, many applications of quantum dot rely on the discrete nature of the dot’s energy levels. It is therefore desirable to makequantumdotsthatoperateinthequantumregime(ܧ஼>οܧ>݇஻ܶ).Accordingto Eqn. 1.8, the average single level particle spacingοܧis inversely proportional to the areaofthedot.ThereforewefabricatedasmallerquantumdotdeviceAS43N,onthe sameundopedNBI30heterostructureasthelargerAS08Ndevice.

Figure4.7showsacolouredSEMofthedevice,itmeasures0.54ʅmby0.47ʅm lithographically,approximately3timessmallerthanthepreviousdevice.Asaresult,we expectedittohaveathreefoldhighersingleparticlelevelspacingof33ʅeV(ܶா̱0.4K). Electricalmeasurementswereperformedusingbothstandardacanddctechniques;the

Figure 4.7:Coloured SEM image of the smaller quantum dot device AS43N. A schematic representationoftheacsetupisalsoshown.Duringtheinitialgatecharacterization,both theleftQPCgates(inred)andtherightQPCgates(inblue)werebiasedsymmetrically(top and bottom), while only the lowert par of the PG gates (in black) was used. The central regioncolouredingreencorrespondstothetopgate,whereapositivevoltageisappliedto induceelectronpopulationanddefinethedot.Therectangleswithcrossesoneachsideof the device represent the Ohmic contacts. Components enclosed by a dashed rectangle represent a low pass RCR filter used in series with a 1:10 voltage divider to keep the electricalnoiseaslowaspossible.Thescalebarshownis500nmlong.

109 ac circuit setup used is similar to AS08N. However, in order to keep the electron temperaturelow,sothatοܧcanbeeasilyresolved(οܧ>݇஻ܶ),wehaveimplementeda fewimprovementsonourmeasurementsetuptoreducethenoise.Forexample,inthe acsetup,currentthroughthedotܫwasmeasuredusingaSR570currentpreamplifier, whichwasoperatedbyacarbatterytopreventgroundloopissues.Inthedcsetup,ܫ was measured with a Keithley K2000 multimeter through the Femto LCAͲ200Ͳ10G currentpreamplifier,andgatebiasingwasachievedusinga“quieter”Yokogawa7650dc source,insteadoftheDACoutputoftheSR830lockͲinamplifier(seeSection2.5.1for more info). A schematic of the ac setup is shown in Fig. 4.7. Measurements were performedontheKelvinoxK100dilutionrefrigeratoratUNSWwithabasetemperature of 40 mK, measured using a Nanoway cryoelectronics primary Coulomb blockade thermometermountedwiththedevice.

4.3.1–Gatecharacterizationforthesmalldot

In order to determine the operational range of the gates, we performed pinchͲoff measurements(݃ଶ்vs.ܸ௚forQPCL,QPCRandPG)similartothoseshownforthelarger dotdeviceinFig.4.2.TheresultsareplottedinFig.4.8,whereboththeleftandright QPCgateswereobservedtopinchoffatverysimilarvoltages.Thissuggeststhatboth gateshavesimilarcapacitancestothedot’schannel,demonstratingthesuccessofour improved etching technique for this device (“43N” in “AS43N” means it is the 43th devicemadeinthisseriesofdotontheNBI30wafer).Moreover,byusingonlythelower ofthetwoplungergatesandhavingitclosertotheTG(thetrenchwidthis50nminthis device compared to 100 nm for the larger dot), we were able to slowly depopulate electronswithoutpinchingoffthechanneltoorapidlyduetocrosstalk.Theconditionis evident from the large pinch off voltage of the PG, and the small pinch off (subͲ threshold)slope݀݃ଶ்Ȁܸ݀௉ீinFig.4.8.Notethattheplungergatestartedtoleakwhen

ܸ௉ீ<െ0.75V, as a result,ܸ௉ீwas restricted to larger thanെ0.6 V to avoid leakage problems.

110

Figure 4.8:PinchͲoff characteristics of PG (black), QPCR (blue) and QPCL (red) for device

AS43N,withthetwoͲterminalconductance݃ଶ்plottedvs.thegatebiasܸ௚.Thedatawere obtained using a standard twoͲterminal ac technique, where݃ଶ்was calculated from dividingthemeasuredcurrentܫacrossthewholedevice,bytheappliedexcitationvoltageof

100ʅV.Measurementswereobtainedat்ܸீ=0.95Vwithaܸ௚resolutionof1mV.

4.3.2–Coulombblockadeoscillationsforthesmallerdot

In order to find the optimum operating voltages to apply to the gates for observing CB oscillations, we monitored݃ଶ்as a function of different gate voltages.

Figure4.9showsacolourmapwhere݃ଶ்isplottedagainstܸோandܸ௉ீ,whileܸ௅=0V and்ܸீ=0.85V.ThebrightlinesrepresenttheCBpeaks,andcurrentisblockadedin the dark regions. According to Eqn. 1.9, the slope of the bright linesοܸோ /οܸ௉ீ representstheratioofthecapacitancebetweentheplungergateandthedotܥ௉ீ,to thecapacitancebetweenQPCRandthedotܥோ,(ܥ௉ீǡܥΤ ோ ̱ͳǤͶ).Thedatapresentedin Fig. 4.9 took several hours to acquire, and yet no signs of significant charge noise or random switching events were observed, demonstrating the stability of our undoped device. Moreover, the fact that CB oscillations were observed even whenܸ௅= 0 V indicatesstrongcrosstalkbetweentheplungergateandtheleftQPCatthegivenbias.

111

Figure4.9:AcolourmapshowingtwoͲterminalconductance݃ଶ்(colouraxis)asafunction ofܸோandܸ௉ீ, obtained at்ܸீ= 0.85 V andܸ௅= 0 V. The white dashed line shown at ܸோ=െ0.07VcorrespondstothesetofCBoscillationsinFig.4.10highlightedwithabracket.

TheslopeofthebrightlinesindicatestherelativecouplingbetweenQPCRandPGtothedot.

According to Eqn. 1.9, the slopeοܸோ /οܸ௉ீ =ܥ௉ீ /ܥோ , whereܥ௉ீ andܥோ represent the capacitancebetweentheplungergatetothedot,andthecapacitancebetweentheright

QPCtothedotrespectively.ܥ௉ீ/ܥோwasfoundtobe̱1.4.

Figure4.10:Aplotof݃ଶ்vs.ܸ௉ீshowingCBoscillationsofthesmalldotdeviceAS43N.The setofpeakshighlightedbythebracketinthetopaxiscorrespondtothedashedlineinFig. 4.9.Theasterisk*indicatesthesameCBpeakastheoneenclosedbyacircleinFig,4.11.For

ܸ௉ீ൏െ0.25V,theCBpeakshavezeroconductanceminima,anindicationoftransportin the quantum regime. The last few resolvable peaks with zero conductance minima are magnifiedandshownintheinset.Adcmeasurementsetupwasusedwithܸ௘௫=50ʅV.

112

Figure 4.10 shows a set of Coulomb blockade oscillations that correspond to thoseindicatedbythewhitedashedlineinFig.4.9,butoveralargerܸ௉ீrange.Similar totheCBoscillationsobservedinFig.4.4forthelargerdot,betweenܸ௉ீ=0andെ0.25 V, we observed CB oscillations superimposed on a varying background conductance.

However,forܸ௉ீ൏െ0.25V,theCBpeakshavezeroconductanceminimasittingat݃ଶ் =0,anindicationofoscillationsinthequantumregime.TheinsettoFig.4.10showsa magnifiedviewofthelastfewCBpeaks.

4.3.3–Biasspectroscopyforthesmalldot

Furtherinformationofthedottransportcanbeobtainedbyperformingabias spectroscopy study on the CB oscillations.Figure 4.11 shows a colour map where normalized differential conductance݃ᇱis plotted as a function of the dc sourceͲdrain biasܸௌ஽andܸ௉ீ. A sequence of Coulomb diamonds are formed in the dark regions whereelectrontransportisblockaded,theyarehighlightedbythewhitesolidlinesin

Figure 4.11:Bias spectroscopy of the CB peaks shown in Fig. 4.10, where normalized ᇱ differentialconductance݃ (colouraxis)isplottedagainsttheplungergatevoltageܸ௉ீ(xͲ axis) and the dc sourceͲdrain biasܸௌ஽ (yͲaxis). A sequence of Coulomb diamonds is highlightedbythewhitelines,whichindicateregionsoftransportblockade.Theblackand whitedashedlinesthatareparalleltoandoutsidethediamondedgesindicateexcitedstate transport.TheCBpeakhighlightedbythe*inFig.4.10andanalysedinFig.4.12andFig. 4.13islocatedinsidethewhitecircle.

113 thefigure.SimilartoFig.4.9,noevidenceofchargenoiseorrandomswitchingevents were found, demonstrating the stability of our device once again. The bright regions outside the Coulomb diamonds running parallel to the diamond edge (highlighted by white/black dashed lines) suggest transport via the excited states in the dot [16]. A comprehensive review of resonant tunneling features can be found in [26]. The level spacing߂ܧbetweenthegroundstateandtheexcitedstatescanbemeasuredbythe separation inܸௌ஽between the excited state line and the diamond edge (see Section 1.4.3fordetailedillustrations).Alevelspacingbetween180to240ʅeV(2.1–2.8K)was extracted. This is considerably larger than the 33ʅeV estimate obtained earlier, suggestingthattheeffectiveareaofthedotissignificantlysmallerthanitslithographic area.This is likely due to the lateral depletion caused by the plungerand QPC gates. Basedonthislevelspacing,thenumberofelectronsinthedotisestimatedtobe60.

The charging energyܧ஼of the small dot can be directly extracted from the Coulombdiamondsbysubtracting߂ܧfromthehalfͲdiamondheight[16],anditranges between 0.44 and 0.45 meV (5.1 – 5.2 K), varying slightly with dot occupancy. This charging energy corresponds to a total dot capacitance of ׽ 360 aF. Note that the methodsforestimatingܧ஼presentedinSection4.2.2isonlyvalidforlargedotswhere

߂ܧissmallcomparedtoܧ஼,forthesmalldothere,weobtain߂ܧ/ܧ஼̱0.47,therefore thoseapproximationmethodsdonotapply.

4.3.4–LineshapeoftheCBpeaksinthequantumregime

Inordertofurtherverifythatoursmalldotoperatesinthequantumtransport regime,weperformedatemperaturedependenceanalysisontheCBoscillationsfrom

஼), theܧا஻ܶ݇ ا ܧFig. 4.10. As discussed Section 1.4.2.1, in the classical regime (߂ lineshapeofCBoscillationsisgivenbyEqn.1.10,wherethepeakconductance݃௣௘௔௞is temperatureindependentandthefullwidthathalfmaximumݓofthepeakincreases

஼) where the CB lineshape isܧاܧοاlinearly withܶ. In the quantum regime (݇஻ܶ given by Eqn. 1.11,݃௣௘௔௞ሺܶሻis inversely proportional toܶinstead, with the linear .[relationshipbetweenݓandܶmaintained[27

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Figure4.12(a)showsthetemperaturedependencefortheCBpeakcenteredat

ܸ௉ீ̱െ0.38 V, for temperatures between 40 and 590 mK. This peak was previously indicatedbyanasteriskinFig.4.10,andappearsinsidethewhitecircleinFig4.11.The solidlinesin4.12(a)arefitsofEqn.1.11tothedata,where݃௣௘௔௞wasusedasafree ିଵ parameter.InFig.4.12(b)weplot݃௣௘௔௞vs.ܶ,wherealinearrelationshipbetweenthe two quantities was observed at higherܶ. This confirms that the transport regime is quantumratherthanclassical.InFig.4.12(c),weplotݓagainstܶ,andfromthegradient

Figure4.12:(a)TemperaturedependenceoftheCBpeakcenteredatܸ௉ீ̱െ0.38V.Thisis thesamepeakindicatedbythe*inFig.4.10andinsidethewhitecircleinFig.4.11.The currentܫ(left axis) and twoͲterminal conductance݃ଶ்(right axis) are plotted against the voltage߂ܸrelative to the peak’s center. The solid lines are fits of Eqn. 1.11 to the ିଵ experimentaldata(points).(b)Theinversepeakconductance݃௠௔௫vs.ܶ.Thelineartrend demonstrates that the dot is in the quantum regime. (c) The peak full width at half maximum ݓ vs. ܶ . The saturation observed in (b/c) suggests a minimum electron temperatureof140mK.Datawereobtainedusingadcsetup.

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ݓȀ݀ܶofthelineartrendathigherܶ,wecanextracttheplungergateleverarmusing݀

=ఀ ൌͶ݇஻ሺξʹ ൅ ͳሻȀ݁ ൈ ݀ܶȀ݀ݓ[25].Weobtainedߙ௉ீܥ௉ீΤܥtherelationshipߙ௉ீ ؠ 0.059,whichisconsistentwith0.066,theaverageoftheleverarmsextractedfromthe leftandtherightdiamondsofthesameCBpeakinFig.4.11.Finally,inFigs.4.12(b/c)we ିଵ observed saturation of both ݃௣௘௔௞ and ݓ as ܶ ՜ 0, providing an estimate of the minimum electron temperature of ׽ 140 mK. This is a considerable improvementoverthepreviousestimateof200to300mK,formeasurementsofthe largedot,demonstratingthesuccessofourmodificationstotheelectricalsetup.Based on the above results, we have successfully demonstrated the working of an undoped quantumdotoperatinginthequantumregime.

4.3.5–Preliminarynoiseperformanceanalysis

InusingquantumdotsasultraͲsensitiveelectrometers[13,28],itisimportantto minimize the noise due to charge fluctuations in the device. The basic noise performanceofourquantumdotdevicecanbeassessedbysimplysittingtheplunger gateatafixedvoltage,monitoringthecurrentasafunctionoftime,andconvertingthe half peakͲtoͲpeak current fluctuation into an equivalent maximum charge noise.For

Figure4.13:(a)ThesameCoulombblockadepeakindicatedbythe*inFig.4.10isshown againhere.Thesolidcirclelocatedatܸ௉ீ=െ0.3427Vindicatesthepointwherevoltageis held fixed in order to obtain the measurement of current vs time presented in (b).This currentconsistsofaslowbackgroundcurrentdriftwithmultispectralnoisesuperimposed thereon, and corresponds to a maximum charge noise of 0.008݁, where݁eis th electron charge.

116 thisstudywehavechosenthesamepeakusedinthetemperaturedependencestudy above, it is shown again more closely in Fig. 4.13(a). To maximize the sensitivity in a quantum dot electrometer, we operated the device at a plunger voltage that correspondstothemiddleoftheriseintheCBpeak(seesolidcircleinFig.4.13a).Here the slope݀ܫȀܸ݀௉ீ is the greatest, and small fluctuations in the device’s charge environment appear strongly in the current.In Fig. 4.13(b) we show the time fluctuation of current measured at a sampling rate of 1 Hz over 15 minutes withܸ௉ீ fixedatെ0.3427V.Amultispectralnoisesuperimposedonaslowlyvaryingbackground driftinthecurrentwasobserved.Weusedtheslope݀ܫȀܸ݀௉ீattheoperatingpointto convert the half peakͲtoͲpeak current fluctuation into an effective charge noise. A changeof0.8%ofanelectronintheperiodof15minswasobtained,demonstrating thesuitabilityofourdevicesforultraͲsensitiveelectrometryapplications.

Tofurtherquantifythenoiseperformanceofourquantumdot,wetookthedata inFig.4.13(b)andperformedaFourieranalysistoextractthepowerspectraldensityܵ௤ (݂)asafunctionoffrequency݂,asplottedinFig.4.14inred.Notehowever,thisnoise spectrum contains contributions from both the device and the measuring circuit. In

Figure 4.14:Power spectral densityܵ௤as a function of frequency݂. The red trace was obtained from the data in Fig. 4.13(b) and represents the noise power spectrum of the deviceaswellastheelectricalsetup.Thebluetracescorrespondstothenoisepowerofthe measurementcircuitalone,obtainedat்ܸீ=0.85Vwithzeroexcitationvoltageandallof theothergatesgrounded.Thisdatasuggestthatthenoiselevelinthemeasurementsetup dominatesoverthenoiseinthesample.

117 ordertoexaminethenoiseperformanceofthedevicealone,andbeabletocompare directlywiththenoiseperformancefromotherquantumdotdevicesintheliterature, weassessthenoisepowerofthemeasuringcircuitbyobtainingacurrentfluctuation tracesimilartotheoneinFig.4.13(b),butmeasuredwithzerobiasesonallthegates

(except for்ܸீ ). The correspondingܵ௤ (݂) is plotted in Fig. 4.14 in blue, and is comparabletothatoftheredtrace.Thisresultsuggeststhatthenoiseinthemeasuring circuitdominatesoverthedevicenoise.Withoutfurtheroptimizingthecircuitnoiseand obtaining a large set of noise spectra for averaging; we cannot obtain meaningful information about the noise performance of the device. This work is currently being undertakenbyPhDstudentSarahMacLeodatUNSW.

For future work, we would like to compare the noise performance of an undopedSET,toitsmodulationͲdopedcounterpart.AnearlydesignofthemodulationͲ doped SET is shown in Fig. 4.15 with a nominally identical geometry as AS43N. Additionally, in order to investigate the noise power spectral density at a higher frequencyrange,measurementsshouldbeobtainedusingaRadioFrequency(RF)setup.

Figure 4.15:A modulationͲdoped prototype device, designed to have nominally identical geometrytothesmallquantumdotdevicepresentedinthischapter.AcolouredSEMimage isshownherewitha500nmscalebar.TheyellowregionscorrespondtodepositedTi/Au metalgate,whereasthegreyregionrepresentstheGaAssurface.

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4.4–SpinǦdependenttransportblockadeinundopedquantum dots

Up until now, the excitation ofܰͲelectron system can be described by the occupation of single electron state, where the correlations between electrons can be safelyignored.Inthissection,wepresentmeasurementsofoursmallundopedquantum dotwherespinͲdependenttransportblockadefeatureswereobserved.Inthiscase,the statesthatdescribetheܰcorrelatedelectronshavetobeused,withthetotalspinܵas a good quantum number [29]. In the following, we will firstdiscuss theexperimental signatures of these transport blockade effects, namely ground state suppression and negative differential conductance (NDC), in terms of the coupling rates between the leads and the levels in the dot; this is followed by discussions of the mechanisms involved.

4.4.1–Groundstatesuppressionandnegativedifferential conductance

In the sequential tunneling model of SET transport, the current through the device in ground state transitionܩܵሺܰሻ ՜ ܩܵሺܰ െ ͳሻis proportional to the coupling between the leads and the dot’s ground state, given byܫൌʹ݁߁௜௡߁௢௨௧Ȁሺʹ߁௜௡ ൅߁௢௨௧ሻ, where߁௜௡ሺ߁௢௨௧ሻdenotes the coupling rates between the dot state and the incoming (outgoing)lead[30].Groundstate(GS)suppressionoccurswhenthesecouplingrates, collectively referred to as߁ீௌ, become sufficiently low (see Fig. 4.16a) and hence the correspondingconductance݃peakwillbesuppressed.However,GSsuppressioncanbe

;߁ீௌbecomesenergeticallyaccessibleب liftedwhenanearbyexcitedstate(ES)with߁ாௌ thiscanbeachievedbymeansoftemperatureactivation,orbyincreasingthesourceͲ drain bias (see Fig. 4.16b & c). In some cases, this suppression can also be lifted by applyingamagneticfield[31].Conversely,inthecasewherethecouplingbetweenthe leadsandthegroundstateismuchlargerthanthecouplingbetweentheleadsandthe

߁ீௌ,݃is reduced when ES becomes available for transport. In aب excited state߁ாௌ stability diagram, this reduction of conductance correspond to negative differential conductance(NDC).

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Figure4.16:Thesediagramsillustratetheideasofgroundstate(GS)transportsuppression andnegativedifferentialconductance(NDC)inaquantumdot.(a)Groundstatesuppression occurswhenthecouplingbetweentheleadsandNͲelectrongroundstate߁ீௌisverysmall. Thisblockadecanbeliftedbyaccessinganearbyexcitedstate(ES)when߁ாௌ>>߁ீௌ.Thiscan byachievedbytemperatureܶactivationin(b),orbyincreasingthesourceͲdrainbiasin(c).

Conversely,when߁ாௌ<<߁ீௌ,regionsofNDCcanbeobservedinthecorrespondingstability diagram,thisisshownin(d).

Figure4.17(a)showsasetofCoulombblockadeoscillationsfromFig.4.10with eachpeaklabeled(peak1correspondstothepeakindicatedbyanasteriskinFig.4.10). To demonstrate that our dot device operates in the quantum regime, where wellͲ defined quantum states are resolved; we examine the CB peak spacingοܸ௉ீin Fig. 4.17(a) which is proportional to the addition energy, as a function of the applied magneticfieldܤperpendiculartothe2DEG.ThisanalysisisshowninFig.4.17(b)where thebottomtracecorrespondstotheseparationinܸ௉ீbetweenpeak1andpeak2(P1& P2).Qualitatively,ifweassumea2Dharmonicdotpotential,theeigenstatesofthedot are referred to as FockͲDarwin states [32Ͳ34], and the observed “crossing” of levels in (b) indicates that the orbital degeneracy is lifted. Consistent with the stability diagram and the temperature dependence analysis discussed previously, data in Fig. 4.17(b) suggests that orbital effects can clearly be resolved. Note however, the

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Figure4.17:(a)AsetoflabeledCoulombblockadeoscillationsasafunctionoftheplunger gatevoltageܸ௉ீmeasuredatܤ=0Tisshown.Thecurrentthroughthedeviceܫ,andthe twoͲterminal conductance݃ଶ்are shown on the left and right axis respectively. The data was obtained in a dc twoͲterminal configuration atܸ௅= 0 V,ܸோ=െ0.07 V as in Fig. 4.10. Peak 1 corresponds to the same peak indicated by an asterisk in Fig. 4.10. In (b), the correspondingpeakspacingοܸ௉ீwhichisproportionaltotheadditionenergyisshown.The bottomtracecorrespondstothespacingbetweenpeak1andpeak2,subsequenttracesare offsetby2.5mVforclarity.

conductancemaximaforP2toP5displayanomalousreduction,whichissuggestiveof ground state suppression. As discussed previously, ground state suppression can be liftedbyaccessingthe“bettercoupled”excitedstate.Inthefollowing,wewillverifythis by performing temperature and sourceͲdrain bias dependence analyses of these suppressedCBpeaks.

4.4.1.1–Temperatureactivatedlineartransport

Figure4.18showsthetemperatureܶdependenceanalysisofP1toP6fromFig.

4.17(a).As Tisincreased,themaximumpeakheights݃௣௘௔௞ofP1andP6decreaseas expectedforpeaksinthequantumregime[27](seeSection4.3.4).HoweverforP2to

P5,thereisananomalousincreasein݃௣௘௔௞asܶgoesup.Thisbehaviourisnotexpected

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Figure4.18:(a)TemperaturedependenceoftheCBpeaksP1toP6showninFig.4.17(a). Theminimumelectrontemperaturewasfoundpreviouslytobe̱140mK,thereforeonly data corresponding toܶ൐140 mK are shown here. (b) The maximum peak height݃௣௘௔௞ corresponding to (a) is plotted againstܶ. As expected for peaks in the quantum regime,

݃௣௘௔௞ofP1andP6dropasܶisincreased(inblue),whereas݃௣௘௔௞forP2toP5exhibitan anomalousincreaseasܶgoesup(inred).NotethatforP5,thisincreaseceaseswhenܶ൐ 350mK.

forquantumnorclassicalpeakswhentransportisviathegroundstate(intheclassical

CBregime,݃௣௘௔௞isnottemperaturedependent,seeSection1.4.2.1).Theseresultsare showninFig.4.18(b).Thedatain(b)suggeststhatgroundstatetransportsuppression can be lifted via temperature activation, consistent with the schematic shown in Fig. 4.16(b).

4.4.1.2–BiasspectroscopyofsuppressedCoulombblockadepeaks

AsillustratedinFig.4.16(c),anotherwaytoliftthegroundstatesuppressionis by applying a sourceͲdrain biasܸௌ஽such that the excited state lies within the bias window. In a stability diagram showing differential conductance݃ᇱas the colour axis, we expect to see “missing” diamond edges that corresponds to the ground state transport.Figure4.19(a)showssuchadiagramforP1toP6.Asexpected,forpeaksthat exhibitgroundstatesuppression(P2toP5),identifiedbythetemperatureanalysisinFig.

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Figure 4.19:(a) A stability diagram similar to Fig. 4.11 is shown for the CB peaks in Fig. 4.18(a).SignaturesofzeroͲbiasconductancesuppressionforP2toP5appearasgapsinthe diamonds.(b)Thesamestabilitydiagramin(a)isshownwithadiamondoverlaydrawnon top of it. Regions of suppressed ground state transport are highlighted by yellow dashed lines.Regionsofnegativedifferentialconductance(NDC)areenclosedbywhiteellipses.

4.18,missingdiamondedgesareobserved.ThisisshownmoreclearlyinFig.4.19(b), where a set of Coulomb diamond overlays are drawn on top of the same stability diagramin(a).Alsonotedin(b),stripsofNDCareobserved,theyareenclosedbythe whiteellipsesinthefigure.

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4.4.2–Spinblockadeandothersuppressionmechanisms

Signatures of ground state suppression and negative differential conductance have also been observed in others systems: in AlGaAs/GaAs split gate quantum dot devicesbyWeisetal.[35]andHütteletal.[31],inaquantumdotformedbyanimpurity potentialneara2DEGdefinedataAlGaAs/GaAsheterostructureby Nichollsetal.[36], andinanAlGaAs/GaAssuspendedquantumdotwithaphononcavitybyWeigetal.[37]. Bycomparingourexperimentalresultswiththosereportedintheliterature,webelieve that the mechanism for the observed GS suppression and NDC in our dot is spinͲ dependent.Inthefollowing,wewilldiscusstheconceptofspinblockade[29]andwhyit isthemostappropriateexplanationforourmeasurement.

4.4.2.1–Spinselectionrules

InthetheoreticaltreatmentofquantumdottransportcarriedoutbyWeinmann etal.[29],whichincludesanelectronspininteractingtermintheHamiltonianofthedot, NDC as well as GS transport suppression can be explained based on two sets of spin selection rules referred to as spin blockade typeͲ1 and typeͲ2.TypeͲ1 spin blockade relatestothepopulationofstateswithmaximalspinܵ=ܰ/2,whereܰcorrespondsto thenumberofelectronsinthedot.Thetransitions:

ே ቀܰǡ ܵ ൌ ቁ՜ሺܰെͳǡܵᇱሻሼͶǤͳሽ ଶ

decrease the electron number fromܰtoܰͲ1 and start with a spin polarized state ᇱ (ܵ ൌܵ௠௔௫ ൌܰȀʹ), will always reduce its total spin toܵ௠௔௫= (ܰͲ1)/2. In contrast, states withܵ൏ܰ/2 can either increase or decreaseܵ, therefore theܵ=ܰ/2 state is stableforrelativelylongtimecomparedtotheܵ൏ܰ/2state.Asaresult,whenaspinͲ polarizedexcitedstatebecomesaccessible,thecouplingratesbetweenESandtheleads isexpectedtobemuchsmallerthanthecouplingratesbetweenthenonͲspinpolarized groundstateandtheleadsሺ߁ாௌ<<߁ீௌ),andregionsofnegativedifferentialconductance

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(NDC)wouldappearinthecorrespondingstabilitydiagram;NDCisobservedinourdot inFig.4.19.

TypeͲ2spinblockadeontheotherhand,occursintransitionswherethetotal spinofthegroundstatesbetweenܰandܰͲ1differbymorethan1/2:

ଵ ܩܵሺܰǡ ܵሻ ՞ܩܵሺܰെͳǡܵᇱሻǡȁܵെܵᇱȁ ൐ ሼͶǤʹሽ ଶ

Forexample,ifܵ=3/2forܰ=5andܵǯ=0forܰ=4,thelinearconductancepeakthat corresponds to theܩܵሺܰൌͷሻ ՞ܩܵሺܰൌͶሻtransition will be suppressed (missing completely atܶ= 0). However, transport can be recovered by accessing the excited ଵ stateaslongasȁܵെܵᇱȁ ൑ ;thiscanbeachievedbymeansoftemperatureactivation, ଶ or increasing the sourceͲdrain bias. These qualitative behaviours are observed in our measurementsinFig.4.18and4.19respectively.

FurtherevidenceofspinͲdependenttransportblockadeinourmeasurementis provided in Fig. 4.20, where the magnetic fieldܤdependence of the ground state suppressionfeaturesisexamined.In(a),aplotofmaximumCBpeakheight݃௣௘௔௞vs.the appliedmagneticfieldܤperpendiculartothe2DEGisshown.ThesuppressedCBpeaks P2ͲP5 (in red) have recovered reasonably well atܤ= 0.5 T. At higherܤ, the overall conductancethroughthequantumdotdecreasebecauseofagradualcompressionof thedotsstatesbythemagneticfield[38].ForP1andP6,݃௣௘௔௞generallydecreasesasܤ israised.Thislowfielddependencemimicsthetemperaturedependence–thereare twokindsofbehaviours,withP1&P6showingdifferentܤͲdepfromP2–P5,justasthey showeddifferentܶͲdepinFig,4.18.

To further investigate theܤdependence of the spinͲblockade features, we comparethestabilitydiagramsthatcorrespondtoP4toP6,betweenB=0and1T,they areshowninFig.4.20(b)&(c)respectively.Asdiscussedearlier,thegapsnearܸௌ஽=0V representgroundstatesuppression;whenܤ=1Tisapplied,groundstatetransportis reͲenabled, and the gaps disappear. Qualitatively, since the dot levels cross at

125

Fig.4.20:(a)Themaximumpeakheight݃௣௘௔௞correspondingtoCBpeaksinFig.4.18(a)is plotted against the magnetic fieldܤ, applied perpendicular to the 2DEG. For peaks that exhibitgroundstatesuppression,namelyP2–P5(inred),݃௣௘௔௞recoversatܤ=0.5T.ForP1 andP6(inblue),݃௣௘௔௞graduallydecreasesasܤisincreased.(b)AstabilitydiagramforP4to P6,similartotheoneinFig.4.19(a)isshown.ThegroundstatesuppressionfeaturesatP4 andP5arerepresentedasmissingdiamondedges,andform“gaps”inthediagram.Regions of negative differential conductance (NDC) are enclosed by white ellipses. These spinͲ blockadefeaturesareliftedatܤ=1T,thecorrespondingstabilitydiagramisshownin(c), wherethegapsare“closed”,andregionsofNDCin(b)havedisappeared.

sufficientlyhighmagneticfield,assuggestedinFig.4.17(a),type1spinblockadewith ଵ ȁܵെܵᇱȁ ൐ nolongerapplies,thereforeGStransportisallowed.Additionally,regionsof ଶ NDC(enclosedbywhiteellipses)foundinFig.4.20(b)havealsodisappearedin(c).This canbesimilarlyunderstoodastheexcitedstatearenolongerspinpolarizedduetolevel crossingsatasufficientlyhighܤ[29].

,Finally,basedonthesinglelevelparticlespacingof׽210ʅVfromSection4.3.3 ourdotisestimatedtohaveabout60electrons.Althoughoneusuallydonotexpectto observestrongcorrelationeffectsinsuchlargesystem,spineffectsinquantumdotsof .theorderܰ׽10[39]and׽50[31]havebeenreported

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4.4.2.1–Othergroundstatesuppressionmechanism

Anotherpossiblesuppressionmechanismthataffectsthecouplingbetweenthe leadsandthelevelsinthedotrelatestotheenhancementorreductionofDOSinthe leads.InatwoͲdimensionalsystem,thedensityofstates(DOS)isessentiallyconstant. However,indiffusivenarrowchannelssuchastheleadsofananowirequantumdot, disorderͲinducedinterferenceeffectsmayleadtohighlynonͲuniformcharacterizationin theDOS[40].Indeed,conductancemodulationduetothequasiͲoneͲdimensional(Q1D) DOSwereobservedinnarrowMOSFETchannels[41Ͳ44],inthevicinityofanimpurity atominaGaAsquantumwell[45Ͳ47],andingatedsingledonortransistordevice48[47]. As recently reported in [48], the Q1D peaks (enhancement) in the reservoir of a quantum dot can lead to excited stateͲlike feature in its stability diagram. We can thereforeexpectgroundstatesuppressionbehaviourwhenaQ1Dminima(reduction) linesupwiththeelectrochemicalofthedot.However,inourcasewheretheelectron transportisballisticwithameanfreepathof2.1ʅmandtheleadsarerelativelywide (̱200nm),itishighlyunlikelythattheobservedGSsuppressionphenomenaisdueto thefluctuationsintheQ1Ddensityofstatesinthereservoirs.Finally,weexcludethe suppressionmechanismofexcitationinducedbylocalizedphononmodeconfinedina cavityproposedbyWeigetal.[37],aswedonothaveasuspendedcavity.

4.5–Chaptersummary

In summary, we have demonstrated the development and operation of two quantum dots fabricated on the NBI30 undoped AlGaAs/GaAs wafer. The smaller dot devices AS43N shows CB oscillations in the quantum regime with excited state resonancesappearinginthestabilitydiagrams.Ourpreliminarynoiseanalysissuggests thatoursmallerquantumdotAS43Nhasgoodnoiseperformance.Finally,wepresented experimentalevidenceofgroundstatesuppressionthatsuggeststhepresenceofspin blockade effects. In the next two chapters, we will study undoped quantum dots operatedintheopenregime.

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Bibliography

[1]“AGaAsgateheterojunctionFET”–P.M.Solomon,C.M.KnoedlerandS.L.Wright, IEEEElectronDev.Lett.,5,379(1984).

[2]“VariabledensityhighmobilitytwoͲdimensionalelectronandholegasesinagated

GaAs/AlxGa1ͲxAsheterostructure”–B.E.Kane,L.N.Pfeiffer,K.W.WestandC.K. Harnett,Appl.Phys.Lett.,63,2132(1993).

[3]“HighmobilityGaAsheterostructurefieldeffecttransistorfornanofabricationin whichdopantͲinduceddisorderiseliminated”–B.E.Kane,L.N.Pfeiffer,andK.W.West, Appl.Phys.Lett.,67,1262(1995).

[4]“FabricationofhighͲqualityoneͲandtwoͲdimensionalelectrongasesinundoped GaAs/AlGaAsheterostructures”–R.H.Harrell,K.S.Pyshkin,M.Y.Simmons,D.A.Ritchie, C.J.B.Ford,G.A.C.Jones,andM.Pepper,Appl.Phys.Lett.,74,2328(1999).

[5]“Veryhighquality2DEGSformedwithoutdopantinGaAs/AlGaAsheterostructures” –R.H.Harrell,J.H.Thompson,D.A.Ritchie,M.Y.Simmons,G.A.C.JonesandM. Pepper,J.ofCrystalGrowth,201,159(1999).

[6]“ResistivityofDilute2DElectronsinanUndopedGaAsHeterostructure”–M.P.Lilly, J.L.Reno,J.A.Simmons,I.B.Spielman,J.P.Eisenstein,L.N.Pfeiffer,K.W.West,E.H. Hwang,andS.DasSarma,Phys.Rev.Lett.,90,056806(2003).

[7]“TwoͲDimensionalMetalͲInsulatorTransitionas aPercolationTransitioninaHighͲ MobilityElectronSystem”–S.DasSarma,M.P.Lilly,E.H.Hwang,L.N.Pfeiffer,K.W. West,andJ.L.Reno,Phys.Rev.Lett.,94,136401(2005).

128

[8]“DisappearanceofMetalͲLikeBehaviorinGaAsTwoͲDimensionalHolesbelow30mK” –JianHuang,J.S.Xia,D.C.Tsui,L.N.Pfeiffer,andK.W.West,Phys.Rev.Lett.,98, 226801(2007).

[9]“QuantizedconductanceinquantumwireswithgateͲcontrolledwidthandelectron density”–B.E.Kane,G.R.Facer,A.S.Dzurak,N.E.Lumpkin,R.G.Clark,L.N.Pfeiffer andK.W.West,Appl.Phys.Lett.,72,3506(1998).

[10]“AlGaAs/GaAssingleelectrontransistorfabricatedwithoutmodulationdoping”–A. M.See,O.Klochan,A.R.Hamilton,A.P.Micolich,M.Aagesen,andP.E.Lindelof,Appl. Phys.Lett.,96,112104(2010).

[11]“SingleͲelectronchargingandperiodicconductanceresonancesinGaAs nanostructures”–U.Meirav,M.A.Kastner,andS.J.Wind,Phys.Rev.Lett.,65,771 (1990).

[12]“ArtificialAtoms”–M.A.Kastner,Phys.Today,46,24(1993).

[13]“ElectroncountingofsingleͲelectrontunnelingcurrent”–T.Fujisawa,T.Hayashi,Y. Hirayama,H.D.Cheong,andY.H.Jeong,Appl.Phys.Lett.,84,2343(2004).

[14]“ElectronicRefrigerationofaTwoͲDimensionalElectronGas”–J.രR.Prance,C.G. Smith,J.P.Griffiths,S.J.Chorley,D.Anderson,G.A.C.Jones,I.Farrer,andD.A.Ritchie, Phys.Rev.Lett.,102,146602(2009).

[15]“Quantumcomputationwithquantumdots”–D.LossandD.P.DiVincenzo,Phys. Rev.A,57,120(1998).

[16]“SpinsinfewͲelectronquantumdots”–R.Hanson,L.P.Kouwenhoven,J.R.Petta,S. Tarucha,andL.M.K.Vandersypen,Rev.Mod.Phys.,79,1217(2007).

[17]”Electrontransportinquantumdots”–L.P.Kouwenhoven,C.M.Marcus,P.L. McEuen,S.Tarucha,R.M.Westervelt,andN.S.Wingreen,inMesoscopicElectron Transport,NATOASISer.EVol.345,editedbyL.L.Sohn,L.P.KouwenhovenandG. Schön(Kluwer,Dordrecht,1997),pp.105–214.

129

[18]“ElectronmobilitiesinmodulationͲdopedsemiconductorheterojunction superlattices”ͲR.Dingle,H.L.Störmer,A.C.Gossard,andW.Wiegmann,Appl.Phys. Lett.,33,665(1978).

[19]“ResistancefluctuationsinGaAs/AlxGa1ͲxAsquantumpointcontactandHallbar structures”–C.Kurdak,C.J.Chen,D.C.Tsui,S.Parihar,andS.A.Lyon,Phys.Rev.B,56, 9813(1997).

[20]“OriginofswitchingnoiseinGaAs/AlxGa1оxAslateralgateddevices”–M.PioroͲ Ladrière,J.H.Davies,A.R.Long,A.S.Sachrajda,L.Gaudreau,P.Zawadzki,J.Lapointe,J. Gupta,Z.Wasilewski,andS.Studenikin,Phys.Rev.B,72,115331(2005).

[21]“InSituReductionofChargeNoiseinGaAs/AlxGa1ͲxAsSchottkyͲGatedDevices”–C. Buizert,F.H.L.Koppens,M.PioroͲLadrière,P.Tranitz,I.T.Vink,S.Tarucha,W. Wegscheider,andL.M.K.Vandersypen,Phys.Rev.Lett.,101,226603(2008).

[22]“SingleelectronchargingandperiodicconductanceoscillationsinGaAs nanostructures”–U.Meirav,Ph.D.thesis,MIT,1990.

[23]“Newphenomenaincoupledtransportbetween2Dand3DelectronͲgaslayers”–P. M.Solomon,P.J.Price,D.J.Frank,andD.C.LaTulipe,Phys.Rev.Lett.,63,2508(1989).

[24]“GaAs,AlAs,andAlxGa1оxAs:Materialparametersforuseinresearchanddevice applications”–S.Adachi,J.Appl.Phys.,58,R1(1985).

[25]“StatisticsandParametricCorrelationsofCoulombBlockadePeakFluctuationsin QuantumDots”–J.A.Folk,S.R.Patel,S.F.Godijn,A.G.Huibers,S.M.Cronenwett,C. M.Marcus,K.Campman,andA.C.Gossard,Phys.Rev.Lett.,76,1699(1996).

[26]“Resonanttunnelingfeaturesinquantumdots”–C.CEscott,F.A.Zwanenburgand A.Morello,Nanotechnology,21274018(2010).

[27]“TheoryofCoulombͲblockadeoscillationsintheconductanceofaquantumdot”–C. W.J.Beenakker,Phys.Rev.B,44,1646(1991).

[28]“BidirectionalCountingofSingleElectrons”ͲT.Fujisawa,T.Hayashi,R.Tomitaand Y.Hirayama,Science,312,1634(2006).

130

[29]“SpinBlockadesinLinearandNonlinearTransportthroughQuantumDots”–D. Weinmann,W.Häusler,B.Kramer,Phys.Rev.Lett.,74,984(1995).

[30]“Solvingrateequationsforelectrontunnelingviadiscretequantumstates”–E. Bonet,M.M.DeshmukhandD.C.Ralph,Phys.Rev.B, 65,045317(2002).

[31]“SpinblockadeingroundͲstateresonanceofaquantumdot”–A.K.Hüttel,H.Qin, A.W.Holleitner,R.H.Blick,K.Neumaier,D.Weinmann,K.Eberl,andJ.P.Kotthaus Europhys.Lett.,62,712(2003).

[32]“BemerkungzurQuantelungdesharmonischenOszillatorsimMagnetfeld”–V. Fock,Z.Phys.,47,446(1928).

[33]“EffectsofCoulombinteractionsonspinstatesinverticalsemiconductorquantum dots”–S.Tarucha,D.G.Austing,S.Sasaki,Y.Tokura,W.vanderWiel,W,L.P. Kouwenhoven,Appl.Phys.A,71,367(2000).

[34]“CollapseoftheSpinͲSingletPhaseinQuantumDots”–M.Ciorga,A.Wensauer,M. PioroͲLadriere,M.Korkusinski,J.Kyriakidis,A.S.Sachrajda,andP.Hawrylak,Phys.Rev. Lett.,88,256804(2002)

[35]“CompetingchannelsinsingleͲelectrontunnelingthroughaquantumdot”–J.Weis, R.J.Haug,K.v.Klitzing,andK.Ploog,Phys.Rev.Lett.,71,4019(1993).

[36]“Chargingeffectsandtheexcitationspectrumofaquantumdotformedbyan impuritypotential”–J.T.Nicholls,J.E.F.Frost,M.Pepper,D.A.Ritchie,M.P.Grimshaw, andG.A.C.Jones,Phys.Rev.B,48,8866(1993).

[37]“SingleͲElectronͲPhononInteractioninaSuspendedQuantumDotPhononCavity”– E.M.Weig,R.H.Blick,T.Brandes,J.Kirschbaum,W.Wegscheider,M.Bichler,andJ.P. Kotthaus,Phys.Rev.Lett.,92,046804(2004).

[38]“EffectsofquantumlevelsontransportthroughaCoulombisland”–E.B.Foxman, P.L.McEuen,U.Meirav,NedS.Wingreen,YigalMeir,PaulA.Belk,N.R.Belk,andM.A. Kastner,Phys.Rev.B,47,10020–10023(1993).

131

[39]“SpincoherenceinMultiͲelectronGaAsdoubledots”–A.P.Higginbotham,F. Kuemmeth,C.Barthel,C.M.Marcus,M.P.HansonandA.C.Gossard,APSmarch meeting,57,1(2012).

[40]“FewͲElectronQuantumDotsinNanowires”–M.T.Björk,C.Thelander,A.E. Hansen,L.E.Jensen,M.W.Larsson,L.R.WallenbergandL.Samuelson,NanoLetters,4, 1621(2004).

[41]“QuasiOneͲDimensionalConductioninMultiple,ParallelInversionLines”–A.C. Warren,D.A.Antoniadis,andH.I.Smith,Phys.Rev.Lett.,56,1858(1986).

[42]“FabricationandTransportPropertiesofSiliconQuantumWireGateͲAllͲAround Transistor”–K.Morimoto,Y.Hirai,K.Yuki,andK.Morita,Jpn.J.Appl.Phys.,35,853 (1996).

[43]“MesoscopictransportinSimetalͲoxideͲsemiconductorfieldͲeffecttransistorswith adualͲgatestructure”–H.Matsuoka,T.Ichiguchi,T.Yoshimura,andE.Takeda,J.Appl. Phys.,76,5561(1994).

[44]“PeriodicconductancefluctuationsinquasiͲoneͲdimensionalmetalͲoxideͲ semiconductorfieldͲeffecttransistorswithshallowͲtrenchisolations”–K.Takeuchiand R.Newbury,Phys.Rev.B,43,7324(1991).

[45]“EnhancedFluctuationsofthe TunnelingDensityofStatesneartheBottomofa LandauBandMeasuredbyaLocalSpectrometer”–J.P.Holder,A.K.Savchenko,V.I. Fal’ko,B.Jouault,G.Faini,F.Laruelle,andE.Bedel,Phys.Rev.Lett.,84,1563(2000).

[46]“ObservationoftheLocalStructureofLandauBandsinaDisorderedConductor”–T. Schmidt,R.J.Haug,V.I.Fal’ko,K.v.Klitzing,A.Förster,andH.Lüth,Phys.Rev.Lett.,78, 1540(1997).

[47]“Spectroscopyoflocaldensityofstatesfluctuationsinadisorderedconductor”–T. Schmidt,R.J.Haug,V.I.Fal'ko,K.v.Klitzing,A.Förster,andH.Lüth,Europhys.Lett.,36, 61(1996).

132

[48]“ProbeandcontrolofthereservoirdensityofstatesinsingleͲelectrondevices”–M. Möttönen, K. Y. Tan, K. W. Chan, F. A. Zwanenburg, W. H. Lim, C. C. Escott, J. M. Pirkkalainen,A.Morello,C.Yang,J.A.vanDonkelaar,A.D.C.Alves,D.N.Jamieson,L.C. L.Hollenberg,andA.S.Dzurak,Phys.Rev.B,81,161304(R)(2010).

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Chapter5

ComparisonofthermalstabilityofMCFin undopedandmodulationǦdopedbilliards

5.1–Introduction

In Chapter 4, we demonstrated two quantum dots fabricated on an undoped heterostructure and operating in the Coulomb blockade regime. In operating the quantum dot as a single electron transistor, the point contacts were biased to form tunneljunctions,wherethenumberofelectronsinthedotܰisalwaysa wellͲdefined number[1].Inthischapter,wepresentmeasurementsofthesmallquantumdotAS43N operatedintheopenregimeasabilliarddevice.

Inanopenquantumdotwherethepointcontactsarebiasedtoallowelectron transmissionviaanumberofquantizedtransversemodes,theCoulombblockadeeffect issuppressedduetoadecreaseintheelectron’sdwelltime.Thedecreaseindwelltime andhigherconductivityoftheQPC’smeanthatthenumberofelectronsonthedot,ܰ, is no longer a precisely defined number. As discussed in detail in Section 1.5.3, it is widelyacceptedthatwhentheelasticmeanfreepathislargerthanthewidthofthedot, the electronic motion within it is considered predominately ballistic in nature, with

134 largeͲanglescatteringeventsoccurringprimarilyattheconfiningwalls[2].Inthisregime, openquantumdotsrepresentaphysicalimplementationofwhatisknownasabilliard. Atmillikelvintemperatureswherethephasecoherencelengthislargerthanthewidth of the device, quantum interference pattern produces complex fluctuations in a billiard’s magnetoconductance [3], arising from the AharonovͲBohm effect [4]. It is widelyacceptedthatthesefluctuationsareadirectfingerprintoftheelectrondynamics establishedbythedot’swalls[3,5,6].Thisideaiscentraltoquantumchaosexperiments [3,5,7Ͳ13]proposingthatthepowerspectrumoftheconductancefluctuations[3]and thelineshape[5]ofthezeroͲfieldconductanceminimumcanbeusedasnovelprobesof thebilliarddynamics.DetaileddiscussionsoftheseexperimentscanbefoundinSection 1.5.5.

However, it has been acknowledged that in a traditional modulationͲdoped billiardthecharacteristicsoftheMCFchange afterthermalcyclingtoroomtemperature [14]. As a result, MCF experiments were often performed in a single cooldown. This contradictsthecommonviewofMCFbeingrepresentativeofthedevicegeometryonly, since the shape of the billiard does not change between cooldowns. More recently, Scannelletal.reportedathermalcyclingexperimentwheretheMCFcharacteristicsof anetchedGaInAs/InPmodulationͲdopedbilliardwerefoundtodecorrelaterapidlyafter warmingthedeviceupto120K[15].Wewillreturntodiscussionsofthisexperiment with our collaborative input in Chapter 6. This result strongly suggests that remote ionized impurityscatteringcannotbeconsideredtoonlycausesmallperturbationsin theoverallbilliarddynamics.

In this chapter we will demonstrate that remote ionized impurities in a modulationͲdopedbilliardplayadominantroleindeterminingthecharacteristicsofthe MCF.WeachievethisbyperformingcomparativeMCFstudies*betweenanundoped and a modulationͲdoped billiard with nominally identical geometry, electron density

*Notethatthedepthsofthetwo2DEGsbeneaththeheterostructuresurfacedifferbetweentheundoped and modulationͲdoped heterostructures (185 nm vs. 70 nm, respectively), see Section 2.2. This depth difference is not the cause of the effects we observe, but is important to bear in mind in making comparisons.

135 and mobility. In Chapter 6, we will look at the MCF decorrelation process of our modulationͲdopedbilliardmoreclosely,aswellaspresentingstudiesofusingMCFasa probetodeterminetheeffectofunscreenedsurfacestatesinanundopedbilliard.

5.2–MCFmeasurementsofanundopedAlGaAs/GaAsbilliard

MagnetoͲconductance fluctuation measurements have been historically performedonmodulationͲdopedbilliards[3,5,6].Inthefollowing,wepresentthefirst studyofMCFfromanundopedsemiconductorbilliardwithoutmodulationdoping.

5.2.1–Deviceandmeasurementsetup

Theundopeddeviceusedinthischapter,AS43N,isthesamedeviceusedasa singleelectrontransistor(SET)inChapter4.IntheSETconfiguration,thepointcontacts wereusedastunneljunctions.Thiswasachievedeitherdirectly,byapplyinganegative biasontheQPCgates,orindirectly,by relyingonthecapacitivecouplingbetweenthe plungergateandtheconductionchannels.HereAS43Nisusedasabilliardwhereeach QPCpassesmorethanonequantizedtransversemode.Weachievedthisbykeepingthe QPCgatesandtheplungergategrounded.MagnetoͲconductancemeasurementsofthe billiardwereperformedusingafourͲterminalacsetupinaconstantexcitationvoltage configuration.Figure5.1showsaschematicrepresentationoftheelectricalsetup.With performing multiple thermal cycling runs in mind, the device was measured in our HelioxVLpumped 3HesystematUNSWwithathermalcyclingturnaroundtimeof̱2 hrsandabasetemperatureof̱250mK.OperationoftheHelioxsystemisdiscussedin detailinSection2.4.3.

136

Figure5.1:SchematicfourͲterminalmeasurementsetupfortheAS43Nbilliard.AS43Nisthe samedeviceusedasasingleelectrontransistorinChapter4.Thetopgate(TG)isshadedin green,whiletheQPCgates(QPCLandQPCR)andtheplunger(PG)areshadedinredandblue respectively. The unͲshaded (light grey) gate at the top was not used and remained groundedforalloftheexperiments.Ohmiccontactsareshownasrectangleswithcrosses. DetailedmeasurementsetupinformationcanbefoundinSection2.5.Thescalebarshown is500nmlong.

5.2.2–Thethermalcyclingexperiment

Asdiscussedintheintroductiontothischapter,foramodulationͲdopedbilliard, the characteristics of the MCF were observed to be different after thermal cycling to roomtemperature[14,15].ThefirsttestweperformedontheundopedbilliardAS43N was to investigate this thermal reproducibility. To proceed with the thermal cycling that we compared to MCFכܯexperiment, we first measured a reference MCF trace כ tracesfromthesubsequentcooldowns.Theܯ tracewasobtainedat்ܸீ=1.05V,with allothergatesgrounded.Thistopgatebiascorrespondstoanelectrondensityof̱2.5 ×1011cmͲ2,amobilityof386,000cm2VͲ1sͲ1andameanfreepathof3.19ʅm.

measured backͲtoͲback for opposite , כܯFigure 5.2 shows two traces of sweeping directions of the applied magnetic fieldܤperpendicular to the 2D plane to demonstrate the repeatability of the MCF in a fixed gate configuration and a single cooldown.Thedotconductance݃wasobtainedfromdividingthepotentialdifferenceܸ betweenthevoltageprobeslocatedoneachsideofthedot,bythecurrentܫacrossthe wholesamplemeasuredwithaconstantvoltagesetup.Notethattheregionenclosedby

137

are shown backͲtoͲback with opposite sweepכܯFigure: 5.2:Two reference MCF traces directionsoftheappliedperpendicularmagneticfieldܤ.Thedotconductancewasobtained bydividingthepotentialdifferenceܸbetweenvoltageprobeslocatedoneachsideonthe dot,bythecurrentܫflowingthroughthewholedevice,measuredwithaconstant excitation voltageof100ʅV.Thedataresolutionis0.8mTandthebluetraceisoffsetupwardsby0.35 ൈ2݁ଶ/݄for clarity. The observed fluctuations are repeatable and symmetric inܤ. This referenceMCFtracewasusedtocomparewiththeMCFtracesfromsubsequentcooldowns.

thevoltageprobesincludedsmallsectionsoftheHallbarmesa.However,sincetheir resistances (3 squares at 65ё/ප at்ܸீ= 1.05 V) only makes up 1.5 % of the total resistance (13 kё), their contribution is negligible and does not affect our qualitative analysis. The data in Fig. 5.2 shows aperiodic oscillations that are repeatable and symmetricinܤ.

As discussed in Section 1.5.2, only conductance fluctuations in the low field regimecorrespondtotheelectrontrajectoriesthatcanaccessallregionsofthebilliard. Asܤis increased further, the electrons paths “skip” along the walls and the entire internalareaofthebilliardcannolongerbeaccessedbytrajectoriesthatinterceptthe QPCs. The upper bound of this low field regime can be calculated by setting the cyclotrondiameter(2ൈݎ௖௬௖)equaltothewidthofthebilliardܹ௕.Thisupperboundis referred to as the cyclotron fieldܤ௖௬௖=ʹ԰݇ிȀܹ݁௕. Substitutingܹ௕= 500 nm for the

11 Ͳ2 billiardwidthandusing݇ி=ξʹߨ݊where݊=2.5×10 cm ,ܤ௖௬௖wasfoundtobe̱ .כܯ0.33Tfor

138

Another consideration that needs to be taken into account when determining theappropriaterangeofMCFthatcanmeaningfullybeanalysedistheoccurrenceofthe Shubnikov de Haas oscillations arising from the adjacent sections of Hall bar mesa enclosedbythevoltageprobes.Fig.5.3presentsaplotsthedatashowninFig.5.2asa function ofͳȀܤ. The set of oscillations, betweenͳȀܤ= 3 and 1 TͲ1 (highlighted by a bracketinthefigure)appearstobeperiodic,whichisanindicationofSdHoscillations discussed in Section 1.3.1. To verify this, we extracted the electron density from the 11 Ͳ2 periodoftheseoscillations,and݊wasfoundtobe2.47ൈ10 cm ,consistentwith݊(்ܸீ =1.05V)=2.5ൈ1011cmͲ2.Wethereforeexcludetheconductancefluctuationsthatappear atܤ>1/3̱0.33T,thesamerangeweobtainedbyusingܤ௖௬௖.

Toproceedwiththethermalcyclingexperiment,wewarmeduptheundoped billiardtoܶ=300KbyliftingtheHelioxinsertoutoftheheliumdewarcompletely.The devicewasleftinsidetheInnerVacuumCan(IVC)inalowpressureheliumenvironment forabout2hours,withawarm,condensationͲfreeouterIVCsurfaceindicatingthatthe sample temperature had reached 300 K. We then recooled the system back to base

shown in Fig. 5.2. Periodic oscillations כܯ for ܤFigure 5.3:Conductance ݃ vs. ͳȀ corresponding to SdH oscillations appear betweenͳȀܤ= 3 and 1 TͲ1, as highlighted by a bracket near the top axis. The electron density calculated from the period of these oscillations in 1/ܤis 2.47ൈ1011 cmͲ2, consistent with݊calculated from the parallel capacitormodel(discussedinSection3.2.1),at2.5ൈ1011cmͲ2.

139

.fromthe1stcooldownshownpreviouslyinFigכܯFigure:5.4:(a)ThereferenceMCFtrace nd 5.2,obtainedat்ܸீ=1.050V.(b)AMCFtracefromthe2 cooldowntakenatthesametop gatevoltageas(a).Althoughthegeneralbackgroundconductancebetweentracesin(a)and (b)aresimilar,thefinescaleoscillationsdonotmatchverywell.(c)ThreeMCFtracesfrom nd the 2 cooldown are shown. They were obtained between்ܸீ= 1.050 V (bottom) and 1.060V(top)instepsof5mV.Forclarity,thetopandbottomMCFtracesareoffsetin݃by± כ ଶ 0.35ൈ2݁ /݄respectively.ThemiddleMCFtracetakenat்ܸீ=1.055Vmatchesܯ almost perfectly. This thermal reproducibility has not been previously reported in the literature. Datain(b/c)hasaresolutionof2.5mT.

temperature(235mK).Inthis2ndcooldown,anotherMCFtracewasobtainedusingthe כ same top gate bias at்ܸீ= 1.05 V, and compared withܯ ǤFigure 5.4 shows this is displayed in (a), and thecorresponding MCF traces taken atכܯcomparison, where thesame்ܸீisshownin(b).Althoughthegeneralbackgroundconductancetrendsof thetwotracesaresimilar,thefinescaleoscillationsdonotmatchverywell.However,if כܯweappliedaslightlyhighertopgatevoltageof1.055V,wewereabletoreproduce withhighfidelity.ThisresultispresentedinFig.5.4(c),whichshowsthreeMCFtraces between்ܸீ= 1.050 V (bottom) and 1.060 V (top) in steps of 5 mV. This thermal reproducibilityofMCFhasnotbeenpreviouslyreportedintheliterature.

To further verify our discovery, we performed yet another round of thermal cycling.Thistimethedevicewasleftatܶ=300KinsidetheIVCoftheHelioxinsertfor9 daysbeforeitwascooleddownagain.Fig.5.5(c)showstwoMCFtracesobtainedinthe

140

Figure5.5:MCFcomparisonsbetweenallthreecooldowns.In(a/b),datafromthe1stand 2ndcooldowninFig.5.4areshownhereagainforcomparison.The3rdand2ndcooldowns wereseparatedby9days,withthesamplestayedinsidetheIVCunderalowpressureHe inthe3rdcooldown,atכܯenvironmentduringtheperiod.In(c)wewereabletoreproduce st ்ܸீ= 1.045 mV, 5 mV less than what was used in the 1 cooldown. We attribute this differencetothechargefluctuationofthescreenedsurfacestatesdiscussedinthetext.Also, כܯboth upper traces in (b/c) obtained at 5 mV above the bias needed to producible resembleeachotherwell.

tracesfromthe1stand2ndcooldownsכܯ3rdcooldown.Forcomparison,thereproduced was found to be כܯ,are shown again in (a) and (b) respectively. Once again reproducible, even after we thermally cycled the device to room temperature twice. These results demonstrate the remarkable thermal stability of our undoped device. nd Notethat,similartotheresultinthe2 cooldown,anoffsetofο்ܸீ=െ5mV(at்ܸீ= rd כ 1.045V)wasneededtoreproduceܯ inthe3 cooldown.Weinterpretthisο்ܸீasan offsetinthegatebiasnecessarytoobtainthesameelectrondensity.Inthisregard,the characteristics of the MCF can be used as a tool to determine whether the densities werethesame.

As discussed in Section 3.2.1, for the operation of our undoped device, a minimumtopgatebiasof0.32Vwasneededtocounterthesurfacestates,andstart populatingelectronstothe2DEG.Thisminimumturnonvoltagewillthereforedepend onthechargestatusofthesurfacestates,andmaydifferfromdevicetodevice.The data in Fig. 5.5 suggest that the surface states on the top gate do not have a stable

141 chargeconfigurationatroomtemperature.Betweencooldowns,aslightlydifferent(5 mV) bias was required to obtain the same electron density. However, these surface states were screened by the݊ାcap of the wafer, therefore characteristics of MCF remainedreproducible.Wewillreturntoinvestigationsoftheeffectofthesesurface statesinSection6.4.

Also noted in Fig. 5.5, we observed a strong resemblance between the two upperMCFtracesin(b)and(c).Theywerebothobtainedatο்ܸீ=5mVabovethe்ܸீ traces were taken. This result further verifies the MCFכܯat which the reproduced reproducibilityofourundopedbilliard.InFig.5.5(c),theamplitudesoftheoscillations appeartobeslightlylargerthanthatof(a)and(b).Thissuggeststhataslightlylower electrontemperaturewasachievedduringthe3rdcooldown.

5.2.3–SensitivityofMCFtolocallyinduceddisorder

The results in Section 5.2.2 demonstrate the reproducibility of MCF in our undopedbilliard.Thefindingshowsthattheprecisedistributionofdisorderinsuchan undopeddevicecanremainunchangedbetweencooldowns.WhathappenstotheMCF ifwecanchangethedisorderpotentialofourundopeddevice?Inthefollowing,wewill answer this question and show how sensitive the characteristics of the MCF are to disorder. To achieve this, we caused a leakage current of electrons between the rightQPCgateandthe2DEG#,whereconductionwasexpectedtooccurviahoppingat largegatebias[16].IthasbeenreportedthatlatticedefectsinGaAssuchasthearsenic antisite [17,18] are responsible for this hopping conduction [19,20]. Therefore it is reasonabletohypothesizethatthechargeconfigurationofthetrapsalongthehopping chainwasalteredasaresultofthisleakage.Althoughwedidnotobserveanyswitching events after leakage within our measurement bandwidth, the change in the trap configurationhadcausedanobservablechangeintheMCFcharacteristicsofourdevice. Thefollowingprovidestheresultsofthisinvestigation.

#Basedonthemeasurementsetupused,itisnotpossibletodifferentiatewhethertheleakagecurrent flowed between QPCR and the 2DEG, or between QPCR and the top gate. However, our analysis in this sectionappliestobothcases,aslongasahoppingpathwascreated.

142

Inearlierdevicecharacterization,weobservedasmallleakageofcurrentwhen therightQPCgatesweresymmetricallybiasedtoܸோ<െ0.2V.AfterthereferenceMCF st כ traceܯ was obtained in the 1 cooldown, we biased QPCR toെ0.4 V until a small leakage current of 20 pA was observed. We then sweptܸோback to 0 V and obtained anotherMCFtraceusingthesametopgatebiasforcomparison.Figure5.6showsthese twoMCFtracesobtained்ܸீ=1.05V,before(a)andafter(b)gateleakage.Achangein theMCFcharacteristicswasobserved.Thischangepersistedfortheremainderofthe inthe2ndcooldownshowninFig.5.4(c)makesכܯcooldown,butthereproducibilityof itclearthatthecharacteristicsrecoveredreasonablywellafterthermalcyclingtoroom temperature.

asshownpreviouslyinFig.5.4(a).Thetracein(b)wasobtainedusingtheכܯ(Figure5.6:(a samesetupin(a),afteraslightleakageofcurrentfromtheQPCRhadoccurred.Achangein theMCFcharacteristicswasobserved.Thetracesin(a/b)areplottedonthesamescaleas tracesinFig.5.7forcomparison.

Tofurtherinvestigatethiseffect,afterwehadconfirmedthereproducibilityof in the 3rd cooldown, we caused another leakage using the same procedure asכܯ ,(in(aכܯpreviouslydescribed.Figure5.7showsacomparisonbetweenthereproduced andin(b),asetoftracestakenbetween்ܸீ=1.03V(bottom)and1.06V(top)in5mV

143

Figure5.7:ThisfigureshowsacomparisonofthechangesinMCFbetweenthe1st(a)and3rd

(b)cooldownafteraslightleakageofcurrentfromQPCRtothe2DEG.In(a),theMCFshown wasobtainedat்ܸீ=1.050V.In(b),sevenMCFtracesareshownobtainedbetween்ܸீ= 1.060V (top) and 1.030 V (bottom) in steps of 5 mV are shown. The middle trace was ଶ obtainedat்ܸீ=1.050Vandtherestofthetracesareoffsetsequentiallyby0.35ൈ2݁ /݄ in݃forclarity.

stepsafterleakage.ThedatainFig.5.7showthatnoneoftheMCFtracesin(b)matches theonein(a).ThisresultisanextensionofthecomparisoninFig.5.6becausenotonly haveweshownthatthecharacteristicsofMCFhadchangedafterleakage,wehavealso shownthatthischangeincharacteristicscannotberecoveredbyanoffsetin்ܸீ.This makessensebecauseunlikethescreenedsurfacestatesinvolvedinFig.5.5,wherean offsetin்ܸீissufficienttocounterthechargeimbalance,here,thetrapstateschange theoveralldisorderpotentialofthebilliard.Moreover,noneoftheMCFtracesinFig. 5.7(b)matchestheoneinFig.5.6(b),suggestingthattheleakagefromthe1stand3rd cooldownshadtakendifferenthoppingpathsbetweenthegateandthe2DEG.

Toseeifwecanfurtherchangethisdisorderpotential,aleakagecurrentof20 pAwasallowedtopassaftermeasurementsinFig.5.7(b),forapproximately10seconds 9 before the bias on QPCR was swept back to 0 V. This corresponds to̱1.3 x 10 ݁of charge passing from gate to 2DEG. A comparison of the MCF before and after this secondleakagecurrentisshowninFig.5.8.Asshown,nofurtherchangesintheMCF

144

Figure 5.8:In (a), three MCF traces from Fig. 5.7(b) obtained between்ܸீ= 1.040 V (bottom)and1.050V(top)areshownwithanincrementof5mV.In(b),wepresentdata fromareͲmeasurementofthesamesetsofMCFaftersettingupanotherleakagecurrent from the QPCR following the measurements in (a). For clarity, the top and bottom MCF tracesareoffsetupwardsby൅0.35ൈ2e2/handെ0.35ൈ2݁2/݄respectively.

wereobservedwhichsuggeststhatthepopulationoftrapsalongthisleakagepathhas saturated.Aninterestingfutureexperimentwouldbetoexplorethissaturationprocess at lower current, however this would be a challenging experiment due to the tiny currentinvolved.

5.3–MCFmeasurementsofamodulationǦdopedAlGaAs/GaAs billiard

InSection5.2.2wepresentedmeasurementsofanundopedbilliard,wherethe correspondingcharacteristicsoftheMCFdonotchangeevenafterthermalcyclingthe devicetoroomtemperature.Inordertoemphasizetheimplicationofthisfinding,we presentadirectcomparativestudyofamodulationͲdopedbilliarddeviceC2275ͲCwith nominally identical geometry. We have chosen the C2275 modulationͲdoped heterostructure because the C2275 wafer has a comparable electron density and mobilitytoNBI30.Ashortsummaryoftheextracted݊andߤareshowninTable5.1for

145 quickreference.TheC2275waferlayoutanditscharacterizationdetailsarepresentedin Section2.2.2&3.2respectively.

Wafer/ ݊[1011cmͲ2] ߤ[1000cm2/Vs] Temperature(K)

C2275/0.24 2.34 336

C2275/1.1 2.34 333

*NBI30/1.4 2.33 367

Table 5.1: Comparisons of݊andߤbetween the C2275 modulationͲdoped and NBI30 undopedAlGaAs/GaAsheterostructures.DatawereextractedfromtheHallandShubnikov de Haas measurements discussed in Section 3.2. *Density of 2.33ൈ1011cmͲ2 in NBI30 correspondsto்ܸீ̱1V.

5.3.1–Deviceandmeasurementsetup

Not only does the C2275ͲC device have a comparable electron density and mobilitytothatoftheundopedcounterpartAS43N,wealsomadesurethattheyhave verysimilardevicegeometries.Precisematchingofthebilliard’sdimensionsisdifficult to achieve because the exact billiard area depends on factors such as the vertical distancebetweenthegatesand2DEG,thegatebiasandelectrondensity.Therefore,we simply made C2275ͲC to have a similar lithographic area to AS43N, and ignored the effect of any surface depletion. Figure 5.9 shows a coloured SEM image of a device similar to C2275ͲC, with a schematic of the measurements setup. Note that we intentionally did not image any of the splitͲgate devices used in the thesis, due to concernsaboutmobilitydegradation.TheC2275ͲCdevicemeasures0.53ʅm×0.44ʅm lithographically,andhasameanfreepathof2.68ʅm.ThefourͲterminalelectricalsetup usedisessentiallythesameasAS43N.MeasurementswerealsoperformedinourHeliox VLpumped3HesystematUNSWwithabasetemperatureof̱250mK.

146

Figure5.9:AcolouredSEMimagesimilartoC2275ͲCisshownwithascalebarof500nm. C2275ͲCwasdesignedtohaveasimilargeometryanddimensionstoAS43Nforcomparative studies.Thebilliardisdefinedbynegativelybiasingthesidegatesܸௌீshowninyellow.The grey region in the SEM image corresponds to the GaAs surface of the C2275 wafer. The measurementsetupusedisverysimilartothatforAS43N.

5.3.2–Billiarddefinitionandsidegatevoltagecharacteristics

Inordertodefinethebilliard,weappliedanegativebiastothesidegatesܸௌீto deplete the 2DEG at 70 nm below the surface. The top and bottom side gate were biased equally. Figure 5.10 shows conductance of the dot݃as a function ofܸௌீ. The conductance drops forെ0.22 <ܸௌீ< 0 V and atܸௌீ=െ0.22 V all of the electrons underneaththemetalgateshavebeendepleted.Alargerܸௌீisrequiredtodepletethe electrons that remain in between the splitͲgates, as evident from the different slope

݀݃Ȁܸ݀ௌீobservedforܸௌீ<െ0.22V.Thebilliardisthereforesaidtobedefinedatܸௌீ̱ െ0.22V,thisvoltageisoftenreferredtoasthe“kneevoltage”.TheinsetinFig.5.10 showsamagnifiedversionofthesamepinchͲofftracesobtainedat235mKinredandat 1.2Kinblue.Here,insteadofquantizedconductancesteps,asexpectedinthecaseofa singleQPCpinchͲoff[21,22],weobservedfineͲscaleaperiodicoscillations(redtrace)on topofa“staircaseͲlike”background(bluetrace).Thisisexpectedbecausethesidegates definethreeseparateentitiescollectively:twopointcontactsandonebilliard.

147

Figure5.10:Conductance݃asafunctionofܸௌீ.Thebilliardisfullydefinedforܸௌீ<Ͳ0.22V, asindicatedbya“knee”inthepinchͲofftrace.Theinsetfiguremagnifiestherangeforܸௌீ< Ͳ0.2V.ThebluetracerepresentsthesamepinchͲofftraceobtainedatahighertemperature of1.2K.Itisshownwithanoffsetof൅0.5ൈ2݁ଶ/݄in݃forclarity.

Assuggestedbythecomparisonbetweenthetwopinchofftracesintheinsetof Fig.5.10,thefineͲscaleoscillationsalsooriginatefromquantuminterference.Theyarise asthesizevariation,causedbyincreasingthenegativeܸௌீ,forcessuccessivedotstates to sweep past the Fermi surface, thus modulating the coupling between these states andtheentranceandexitQPCs[23].

5.3.3–ThermalcyclingofmodulationǦdopedbilliarddevice

To proceed with the thermal cycling experiment, a reference MCF trace was ଶ obtainedatܸௌீ=െ0.4965V,wherethezeroͲfieldconductance݃ሺܤ ൌ Ͳሻ̱1ൈ2݁ /݄, fortheundopeddevicediscussedinSection5.2.2.Thisreferenceכܯissimilartothatof MCFtraceisshowninFig.5.11,andacyclotronfieldof̱0.32Twascalculatedforthis devicewithadensityof2.34ൈ1011cmͲ2andthebilliardwidthof500nm.

A thermal cycling was performed on C2275ͲC device, with the device was left insidetheIVCoftheHelioxinsertfoṟ2hrsbeforecoolingbackdownagain.Figure5.12 shows a comparison ofܸௌீcharacteristics between the cooldowns. While the general

148

Figure5.11:AreferenceMCFtraceforthemodulationͲdopedbilliardC2275ͲCobtainedat st ܸௌீ=െ0.4965Vinthe1 cooldownisshown.Thecyclotronfieldcalculatedforthe500nm widebilliardis0.32T.Thedatawereobtainedwithafieldresolutionof0.5mT.

backgroundtrendsofthetwotracesareverysimilar,thefineͲscaleoscillationsareare clearlydifferent.ThepinchͲoffvoltageinthe2ndcooldownhasalsoshiftedby̱23mV. AswiththeMCF,weattributethisdecorrelationtotherandomizationofimpuritiesin

Figure5.12:SidegatevoltagecharacteristicsbeforeandafterthermalcyclingforC2275ͲC.

The݃vs.ܸௌீtracebeforethermalcyclingisshowninred(alsoshownpreviouslyinFig.5.10 asaninset).Theequivalenttraceafterthermalcyclingisshowninblack,andisshownwith an offset of൅0.5ൈ2e2/h in݃for clarity. The fineͲscale oscillations have evolved due to reconfigurationoftheremoteionizedimpuritiesbetweencooldowns.

149 the dopant layer after thermal cycling [15].As we will discuss later in Section 6.3.1, these side gate characteristics can also be used to determine the thermal stability of disorderinthedevice.

In order to obtain a MCF trace that corresponds to the trace from the 1st cooldownforcomparison,wecouldsetthesidegatesatthesamebias.However,as seeninFig.5.12,thesideͲgatecharacteristicschangedsignificantlyafterthermalcycling. Inordertokeepthebilliardpotentialsassimilaraspossible,MCFcomparisonsofthe modulationͲdoped device were performed with matched zerod Ͳfiel conductance݃ሺͲሻ instead.Figure5.13showstheMCFtracesobtainedfromtwodifferentcooldowns.The MCF traces in (a/b) are݃ሺͲሻmatched; where their difference in݃is less than 2.9 %, comparedto24%ifwematchedܸௌீatെ0.4965Vinstead.Thedatain(a/b)clearly shows that the characteristics of the MCF had become considerably different after thermalcycling.

Figure 5.13:MCF comparisons before (a) and (b/c) after thermal cycling. In (a), the referenceMCFwasobtainedatܸௌீ=െ0.4965Vwithazerofieldconductance݃ሺͲሻof̱1 oftheundopeddevice.In(b)a݃ሺͲሻmatchedMCFtraceobtainedatכܯൈ2݁ଶ/݄,similarto

െ0.4695V.In(c),asetofMCFweretakenbetweenܸௌீ=െ0.4775(bottom)and–0.4615V (top)instepsof2mV.Tracesaresequentiallyoffsetby0.2ൈ2݁ଶ/݄in݃forclarity.Noneof theMCFtracesin(c)matchestheonein(a).

AsseenfromFig.5.12,thebilliardconductanceisverysensitivetosmallchanges ingatevoltage.ToensurethatmatchingMCFtraceswerenotaccidentallymissed,we

150 haveextendedthesingletracecomparisoninFig.5.13(a/b)tocomparingthetracein(a), withalargersetofMCFtracesin(c).Startingfromthetracein(b),westeppedܸௌீin bothpositive(tracesathigher݃)andnegative(tracesatlower݃)incrementsof2mV.In doing so, we span a݃ሺͲሻrange from 0.81 to 1.54ൈʹ݁ଶ/݄.The corresponding MCF tracesareplottedinFig.5.13(c).Tobeevenmorerigorous,inFig.5.14(b),weextend thissearchbysteppingtheuppersidegatewhilekeepingthelowergateheldatܸௌீ= െ0.4695 V, and vice versa in Fig. 5.14(c). We have observed that under all cases, no matchingMCFtraceswerefound.

Figure5.14:ComparisonofMCFtraces(a)beforeand(b/c)afterthermalcycling.In(b),a setofMCFtraceswereobtainedbysteppingtheuppergateinincrementsof2mV,while keepingthelowergatefixedatܸௌீatെ0.4965V,andviceversain(c).Tracesin(b/c)are offsetsequentiallyby0.15ൈ2݁ଶ/݄in݃forclarity.Thesteppinggateisindicatedinyellow intheschematicsin(b/c).Noneofthetracesin(b/c)matchestheonein(a).

BasedontheresultsfromHallmeasurements,therewasasmallbutmeasurable change in the electron density (̱0.09 %) in C2275ͲC after thermal cycling. Since݊ affectstheFermiwavelengthwhichinturnsaffectstheMCF,inordertoaccountforthis difference,weusedthebottomplate oftheLCC20packageasa“backgate”tovarythe densitybyapplyingabiastoit.Figure5.15showsthemeasureddensityasafunctionof 8 Ͳ2 thebackgatebiasܸ஻ீ.Thedatasuggestthatwewereabletochange݊by1.6ൈ10 cm (̱0.07 %) for every volt that we applied to the back gate, which is sufficient to compensate the observed change in݊between cooldowns. In Fig. 5.16, we extended

151 theMCFcomparisonbetweencooldownsinFig.5.13(a/b)tocomparingthetracein(a), withalargesetofMCFtracesobtainedbetweenܸ஻ீ=൅10V(top)andെ10V(bottom) inincrementsof1V.ThecorrespondingtracesareshowninFig.5.16(a/b).Notethatthe densityofthebluetracein(b)matchesthatofthetracein(a)towithin0.02%,itisclear from the data that despite both݃ሺͲሻand݊for the two traces were closely matched, their MCF characteristics are significantly different. Also noted in Fig. 5.16, characteristics of the MCF for this device are very sensitive to small changes in the density.

Figure 5.15:Measured electron density݊vs. back gate biasܸ஻ீ. This back gate (BG) corresponds the bottom of a LCC20 package holding the C2275ͲC device. The red square indicates the density before thermal cycling, extracted by a Hall measurement. The data showsthatthereisasmallchangeiṉ݊0.09%betweencooldowns,andwewereableto change݊by1.6×108cmͲ2(̱0.07%)foreveryvoltthatweappliedonthebackgate.

152

Figure5.16:ComparisonofMCFtraces(a)beforeand(b)afterthermalcycling.Thesetof

MCFtracesin(b)wereobtainedbetweenܸ஻ீ=െ10V(bottom)and൅10V(top)instepsof 1V.TheblueMCFtracein(b)obtainedatܸ஻ீ=1Vandhasalmostthesamedensity(within 0.02%)astheMCFin(a).Tracesin(b)aresequentiallyoffsetby0.05ൈ2݁ଶ/݄in݃forclarity. Despitehavingcompensatedforthechangesindensitybetweencooldowns,itisclearthat noneofthetracesin(b)matchestheonein(a).

5.4–Chaptersummary

Tosummarise,wehaveshowninthischapterthatthecharacteristicsoftheMCF obtained from an undoped billiard are reproducible between cooldowns with high fidelity. In contrast, the corresponding MCF characteristics from a modulationͲdoped devicewithnominallyidenticalgeometrychangesignificantlyafterthermalcycling.The sensitivity of MCF to disorder is also demonstrated by the observed changes in the characteristics after a leakage from the doped semiconductor gate forming the rightͲ handQPCtothe2DEGhadoccurred.Finally,duetothesensitivityoftheMCFtoܸௌீin modulationͲdoped devices, we performed a thorough search where a reference MCF traceobtainedinthe1stcooldown wascomparedtoaseriesofMCFtracesobtained withasmalloffsetinܸௌீandܸ஻ீ,inordertocompensateforanypotentialgateshifts afterthermalcycling.Despiteallthis,nomatchingMCFtraceswerefound.

153

Bibliography

[1]“Electrontransportinquantumdots”–L.P.Kouwenhoven,C.M.Marcus,P.L. McEuen,S.Tarucha,R.M.Westervelt,andN.S.Wingreen,inMesoscopicElectron Transport,NATOASISer.EVol.345,editedbyL.L.Sohn,L.P.KouwenhovenandG. Schön(Kluwer,Dordrecht,1997),pp.105–214.

[2]“SmoothͲdisordereffectsinballisticmicrostructures”–K.Richter,D.UllmoandR.A. Jalabert,Phys.Rev.B,54,R5219(1996).

[3]“ConductanceFluctuationsandChaoticScatteringinBallisticMicrostructures”–C.M. Marcus,A.J.Rimberg,R.M.Westervelt,P.F.Hopkins,andA.C.Gossard,Phys.Rev.Lett., 69,506(1992).

[4]“SignificanceofElectromagneticPotentialsintheQuantumTheory”–Y.Aharonov andD.Bohm,Phys.Rev.,115,485(1959).

[5]“WeakLocalizationinChaoticversusNonchaoticCavities:AStrikingDifferenceinthe LineShape”–A.M.Chang,U.H.Baranger,L.N.PfeifferandK.W.West,Phys.Rev.Lett., 73,2111(1994).

[6]“UniversalconductancefluctuationsinthemagnetoresistanceofsubmicronͲsizen+Ͳ GaAswiresandlaterallyconfinednоͲGaAs/(AlGa)Asheterostructures”–R.P.Taylor,M. L.Leadbeater,G.P.Whittington,P.C.Main,L.Eaves,S.P.Beaumont,I.McIntyre,S. ThomsandC.D.W.Wilkinson,Surf.Sci.,196,52(1988).

[7]“WeakLocalizationandInstabilityinBallisticCavities”–H.U.Baranger,R.A. Jalabert,andA.D.Stone,Phys.Rev.Lett.,70,3876(1993).

154

[8]“ConductanceFluctuationsintheBallisticRegime:aProbeofQuantumChaos?”–R. A.Jalabert,H.U.Baranger,andA.D.Stone,Phys.Rev.Lett.,65,2442(1990).

[9]“ClassicalandquantumballisticͲtransportanomaliesinmicrojunctions”–H.U. BarangerD.P.DiVincenzo,R.A.JalabertandA.D.Stone,Phys.Rev.B,44,10637(1991).

[10]“QuantumͲchaoticscatteringeffectsinsemiconductormicrostructures”–H.U. Baranger,R.A.Jalabert,andA.D.Stone,Chaos,3,665(1993).

[11]“Phasebreakinginballisticquantumdots:Experimentandanalysisbasedon chaoticscattering”–C.M.MarcusandR.M.Westervelt,P.F.HopkinsandA.C.Gossard, Phys.Rev.B,48,2460(1993)

[12]“Conductancefluctuationsinaquantumdotinthetunnelingregime:Crossover fromaperiodictoregularbehaviour”–C.M.Marcus,R.M.Westervelt,P.F.Hopkins,A.C. Gossard,Surf.0Sci.,305,48 (1994).

[13]“Conductancefluctuationsandquantumchaoticscatteringinsemiconductor microstructures”–C.M.Marcus,R.M.Westervelt,P.F.Hopkins,andA.C.Gossard, Chaos,3,643(1993).

[14]“Coherentelectronicbackscatteringinballisticmicrostructures”–M.J.Berry,J.H. Baskey,R. M.WesterveltandA.C.Gossard,Phys.Rev.B,50,8857(1994).

[15]“Chaoticelectrontransportinsemiconductordevices”–B.Scannell,PhDthesis, (2010).

[16]“TimeͲirreversiblerandomtelegraphsignalduetocurrentalongasinglehopping chain”–D.H.Cobden,A.Savchenko,M.Pepper,N.K.Patel,D.A.Ritchie,J.E.F.Frost, andG.A.C.Jones,Phys.Rev.Lett.,69,502(1992).

[17]“StructuralpropertiesofAsͲrichGaAsgrownbymolecularbeamepitaxyatlow temperatures”–M.Kaminska,Z.LilientalͲWeber,E.R.Weber,T.George,J.B.Kortright, F.W.Smith,BͲY.Tsaur,andA.R.Calawa,Appl.Phys.Lett.,54,1881(1989).

[18]“DependenceofthecarrierlifetimeonacceptorconcentrationinGaAsgrownat lowͲtemperatureunderdifferentgrowthandannealingconditions”–M.Stellmacher,J.

155

Nagle,J.F.Lampin,P.Santoro,J.Vaneecloo,andA.Alexandrou,J.Appl.Phys.,88,6026 (2000).

[19]“NonalloyedohmiccontactsonlowͲtemperaturemolecularbeamepitaxialGaAs: Influenceofdeepdonorband”–H.Yamamoto,Z.Q.Fang,andD.C.Look,Appl.Phys. Lett.,57,1537.(1990)

[20]“ElectricalpropertiesinarsenicͲionͲimplantedGaAs”–W.C.ChenandC.S.Chang,J. Appl.Phys.,80,1600(1996).

[21]“ImpactofSmallͲAngleScatteringonBallisticTransportinQuantumDots”–A.M. See,I.Pilgrim,B.C.Scannell,R.Montgomery,O.Klochan,A.M.Burke,M.Aagesen,P.E. Lindelof,I.Farrer,D.A.Ritchie,R.P.Taylor,A.R.HamiltonandA.P.Micolich,Phys.Rev. Lett.,108,196807(2012).

[22]“TheQuantumHallEffectinOpenConductors”–M.Büttiker,Semiconductorand Semimetals,35,191(1991).

[23]“ OneͲdimensionaltransportandthequantisationoftheballisticresistance”–D.A. Wharam,T.J.Thornton,R.Newbury,M.Pepper,H.Ahmed,J.E.F.Frost,D.G.Hasko,D. C.Peacock,D.A.RitchieandG.A.C.Jones,J.Phys.C,21,L209(1988).

156

Chapter6

StudiesofdisorderusingmagnetoǦconductance fluctuations

6.1–Introduction

ThekeyfindinginChapter5wasthatthemagnetoͲconductancefluctuationsof the undoped billiard remained unchanged even after thermal cycling to room temperature. In contrast, the MCF characteristics of a modulationͲdoped device with nominally identical geometry evolved significantly in comparison. These results point strongly to the smallͲangle scattering from the remote ionized impurities playing a dominant role in determining the MCF. This contradicts the assumption of ballistic transportinasemiͲclassicalbilliard,questioningthevalidityofquantumchaosstudies [1Ͳ9].Inthischapterweaimtounderstandtheimplicationoftheresultsfromthelast chapter.Specifically,wewanttoanswerthefollowingthreequestions:

First,atwhattemperaturedoesMCFfromamodulationͲdopedbilliardstartto decorrelate?Byextractingthisactivationenergy,wewillhaveabetterunderstandingof themechanisminvolved.

157

Thesecondquestionwewanttoaddressisrelatedtothevalidityoftheresults fromthelastchapter.IntheMCFofbothbilliardsfromthelastchapter,therewasalack of spectral content in the conductance fluctuations betweenേܤ௖௬௖. This is probably related to our dot design which results a large number of electron trajectories to go straight through between the QPCs without traversing the whole billiard area. In this chapter, we will briefly verify the results from the last chapter by repeating similar comparisonsbetweensampleswithamoreconventionalbilliarddesign[10Ͳ17].Wewill alsointroduceacorrelationfunction[18]toperformquantitativecomparisonbetween pairsofMCFtraces.

Finally,wepresentstudiesofotherformsofdisorderthatarestillpresentinthe undopedheterostructureusingthecharacteristicsoftheMCFasaprobe.Inparticular, weareinterestedintheeffectofsurfacestatesthatarenotscreenedbythecap.To study the effect of surface oxidation, we compared the MCF of an undoped device between cooldowns that were separated by 30 days, leaving the sample out of the Helioxwithexposuretoroomairenvironment.Furthermore,inanattempttoincrease theamountofunscreenedsurfacestates,wefabricatedalargerundopedbilliardabout 4 times the size as the one discussed in the Chapter 5, and investigate the thermal reproducibilityoftheMCFproducedbyit.

6.1.1–Billiarddevicesusedinthethesis

Inthischapter,atotaloffournewbilliarddevicesareintroduced.Inorderto avoid confusion for the readers, we will address those using more descriptive names ratherthanthosewereoriginallyassigned.Table6.1providesalistoftheallthebilliards used in Chapter 5 and 6, showing their names, sizes, subjects of study and chapter appearances. For each type of the heterostructure – undoped (UD) or modulationͲ doped(MD),threebilliardsofdifferentsizeswerefabricated:theonespresentedinthe lastchapterhavethesmallestareas(MDͲS&UDͲS),whereastheonesintroducedinthis chapter,areofmedium(M)andlarge(L)insize.Allbilliardswerefabricatedusingthe Semiconductor Nanofabrication Facility (SNF) at the University of New South Wales between2009and2010bymyself.

158

Figure6.1:ColouredSEMimagesofallsixbilliarddevicesusedinthisthesis.Thenamesin red correspond to the names that we will be referring to in this chapter. They have the formatof“(heterostructuretype)–(size)”,wheretheformeriseitherUD(undoped)orMD (modulationͲdoped),andsize=S(small),M(medium)orL(large).Sizesareshownwiththe conventionofwidth(horizontal)ൈlength(vertical).Thegreentintedregionsintheundoped billiards correspond to the top gates (TG), where positive biases were applied to induce electrons and form billiards at the 2DEG 160 nm below. The yellow tinted regions in the modulationͲdoped billiards correspond to the side gates (SG). In each cases, they were biasedsymmetrically(upperandlowerpart)todepleteelectronsandfromabilliardatthe 2DEG 70 nm below. Scale bars shown are 500 nm in length. All billiard devices were fabricatedatUNSWbetween2009and2010.

159

6.2–ActivationenergyofdopantsintheC2275billiard

Sofar,resultsfromthemodulationͲdopedbilliardssuggestthatthecorresponding MCFcharacteristicschangesignificantlyafterwarmingthedevicetoܶ=300K.Butat whattemperaturedoesthisdecorrelationprocessbegin?

6.2.1–PreviousresultsfromaGaInAs/InPbilliarddevice

Scannelletal.[19]studiedtheeffectofremoteionizedimpuritiesinamodulationͲ dopedbilliardbycomparingMCFtracesmeasuredat240mKafterwarmingthedevice to an intermediate temperatureܶ௜between 0.24 K and 300 K for a fixed time of 30 minutes. A schematic representation of the GaInAs/InP heterostructure used in their “thermal cycling experiment” is shown in Fig. 6.2(a). The heterostructure used was grownusingmetalorganicvaporphaseepitaxy(MOVPE)[20],andthe2DEGisformedin anInGaAsquantumwell20nmbelowtheSidonorlayer.Figure6.2(b)showsanAFM image of the micronͲsized square billiard device used in that study. The walls were formedusingelectronbeamlithographyincombinationwithanonͲselective,HBrͲbased wet etch [21]. SelfͲconsistent numerical simulation suggests that the etched billiards haveamuchstronger[22](about20times)confinementpotentialthanourmodulationͲ

Figure6.2:DeviceusedinthethermalcyclingexperimentbyScannelletal.[19].In(a),layer detailsoftheGaInAs/InPheterostructureisshown.Detaileddiscussionofthedevicecanbe foundin[19,21].(b)AnAFMimageoftheetchedmicronͲsizedbilliarddeviceisshown.The billiardwallswereformusingEBLfollowedbyaHBrwetetch.

160 doped billiards, which were defined using the splitͲgate technique [23]. In their experiment, it was found that the MCF characteristics started to decorrelate rapidly whenܶ௜ш120K.

In order to determine this activation temperatureܶ௔for our C2275 modulationͲ dopedbilliard,andthereforegainabetterunderstandingofthedisorderinvolvedinthe decorrelationprocess,IfabricatedalargermicronͲsizedbilliarddeviceMDͲL(seeFig.6.1) and performed a thermal cycling analysis, similar to the one discussed above, in collaborationwithScannelletal.attheUniversityofOregon.WhileImadethebilliard device and performed the initial characterization measurements at UO, the following dataandanalysiswereobtainedbyI.Pilgrim,andareavailablein[24].

6.2.2–ThermalcyclinganalysisforMDǦL

ToproceedwiththethermalcyclinganalysisforMDͲL,firstofall,areference MCF trace was obtained atܶ= 240 mK. The device was then warmed to various intermediatetemperaturesܶ௜forafixedperiodof30minutesbeforeitwascooledback tobase,whereanotherMCFtracecorrespondingtoܶ௜wasobtainedforcomparison.In ordertoinvestigatetheevolutionofMCFcharacteristicsatincreasingܶ௜,itisessential tovaryܶ௜sequentially.ThisexperimentalprocedureisfurtherillustratedinFig.6.3(a).

InFig.6.3(b),thefourMCFtracescorrespondingtoܶ௜=0.24K,115K,170Kand

295 K are shown, while the corresponding݃vs.ܸௌீtraces with the same colour are shownin(c).DatainFig.6.3(b/c)clearlysuggestthatahighdegreeofcorrelationwas maintainedintheMCFand݃vs.ܸௌீbetweenܶ௜=0.24Kandܶ௜=115K.Aboveܶ௜=115 K however, the traces become decorrelated. Based on this result, the activation temperaturewasestimatedtobebetween115Kand170K.

161

Figure 6.3:(a) Schematic representation of the measurement procedure used for the thermalcyclinganalysisdiscussedinthetext.Toobtainthetracesin(b)and(c),theMDͲL devicewasfirstcooledtobaseatܶ=0.24K,whereareferencetracewasobtained(reddot).

The device was then warmed to intermediate temperaturesܶ௜for a fixed period of 30 minutesbeforeitwascooleditbackdowntobaseܶwherethecorrespondingMCF(b)and sidegatecharacteristics(c)wereobtained.Thesamecolourcodesareusedin(a)and(b/c).

In(b/c),comparisonsof(b)݃vs.ܤand(c)݃vs.ܸௌீbetweenܶ௜=0.24K,115K,170Kand 295Kareshown.TheMCFtracesin(a)wereobtainedwithamatched݃(0)=3ൈ2݁ଶ/݄. The traces are offset upwards sequentially by 0.25 and 0.5ൈ2݁ଶ /݄in (b) and (c) respectively.Theresolutionofthedatais0.5mTin(b)and0.1mVin(c).Thedataabove suggeststhatahighdegreeofcorrelationbetweentraceswasmaintainedforܶ௜൑115K.An activationtemperaturebetween115Kand170Kwasestimated.In(c),thearrowsindicate the fifth and sixth plateau (counting from bottom up), where the correspondingܸௌீwas usedtoobtainMCFtracesfortheanalysisinFig.6.4(b).

162

In order to obtain a more accurate estimate ofܶ௔, two things are needed: a metrictoquantifythecorrelationbetweentwoMCFtraces,andmoreMCFtraceswith finer step size inܶ௜. A correlation functionܨ[18] defined in Eqn. 6.1, is used for the

ଶ ஻ۄ ሻሿܤሻ െ݃ଶሺܤሾ݃ଵሺۃ ܨ ൌ ඨͳെ ሼ͸Ǥͳሽ ܰ

ଶ ஻ wasۄ ሻ൧ܤሻ െ݃௬ሺܤൣ݃௫ሺۃ = ܰ former purpose, where the normalization constant calculatedbyaveragingthesquareofthedifferencebetween15pairsofMCFtraces, where(݃௫ሺܤ)and݃௬ሺܤሻ)wereobtainedfromcooldownsthatareseparatedbythermal cycling to room temperature, i.e. MCF atܶ௜=0.24 K and MCF atܶ௜= 300 K. By this definition,apairofidenticalMCFtracesyieldsܨ=1,andapairofMCFtracesseparated bythermalcyclingtoroomtemperaturegivesܨ=Ͳ.Wecalculatedܨforthe640data pointsofeachMCFtracespanningbetweenേ0.16T,anestimateofܤ௖௬௖,calculatedfor thedevice.InSection6.3.3,wewilluseadifferentformofcorrelationequation(defined inEqn.6.2)fortheMCFcomparison,Eqn.6.1wasusedtokeepthecomparisonofthe GaInAs/InPetchedbilliardandMDͲLin[24]consistentwithanearlierstudyofaGaAs wirein[25].

Wewillnowdescribethemethodthatweusedtoobtainܨvs.ܶ௜.Firstofall,we defineonethermalcycletobetheperiodbetweentheinitialcooldowntobaseandܶ௜൒

300K.Withinthisthermalcycle,ܨcanbecalculatedbycomparingMCFatܶ௜<300K andthereferenceMCFtraceatܶ௜=0.24K.Onceܶ௜൒300K,wehaveenteredanother thermal cycle, where another reference MCF trace is needed for subsequent comparisons in this cycle. This method of generating MCF traces is illustrated in Fig. 6.4(a)forclarity.

In Fig. 6.4(b), we presentܨvs.ܶ௜for whereܨwas calculated between MCF tracesfromatotalof36thermalcyclesandacrosstwonominallyidenticalMDͲLdevices (asecondMDͲLdevicewasmadebecausethefirstonewasaccidentallydamagedatUO). The MCF pairs used to calculateܨwere obtained at three different conditions. They

163

Figure6.4:(a)Agraphicalillustrationshowing3differentthermalcycles.Correlationswere only calculated between MCF traces obtained from the same thermal cycle. The yellow circlesindicatedMCFseparatedbyafullthermalcycletoroomtemperature,andwereused to calculate the normalization constantܰin Eqn. 6.1. (b) Decorrelation plot for MDͲL showingܨ(definedinEqn.6.1)asafunctionofܶ௜.IdenticalMCFtracesgiveܨcloseto1and MCFtracesseparatedbythermalcyclingto300Karedefinedascompletelydecorrelatedfor ܨ=0.Thedatawereobtainedfromatotalof36thermalcyclesandspantwonominally identicalMDͲLdevices.TheMCFtraceswereobtainedatthreedifferentconditions:witha matched݃ሺͲሻ=3ൈ2݁ଶ/݄ǡandwithamatchedplateauͲlikefeature(plateau#5and#6,see Fig.6.3c).Datain(a)werethereforecalculatedusing4differentnormalizationconstants: oneforMDͲLat݃=3ൈ2݁ଶ/݄,threeforMDͲL(2nd)at݃=3ൈ2݁ଶ/݄ǡplateau#5andplateau

#6respectively.Datain(b)showthatMCFbecomesquicklydecorrelatedforܶ௜ ൒150K.A largenumberofdatapointsbetweenܶ௜=100and200Kareusedtoaveragethestatistical variationsofܨinthisrange,whichisthemostsensitivepartofthedecorrelationcurve.The fittingerrorinthistemperaturerangehowever,isdominatedbytheerrorinܶ,whichcanbe approximated by the maximum temperature fluctuation during the MCF measurements. Thisisatmost2%

164 either have matched zeroͲfield conductance݃= 3ൈ2݁ଶ/݄or matched plateauͲlike featuresin݃vs.ܸௌீ(plateau#5and#6highlightedinFig.6.3c)wereused.Asaresult, fourdifferentnormalizationconstantswereused,oneforMDͲLat݃=3ൈ2݁ଶ/݄,three forMDͲL(2nd)at݃=3ǘ2݁ଶ/݄,plateau#5andplateau#6respectively.DatainFig.6.4(b) suggeststhatthecharacteristicsoftheMCFstarttodecorrelaterapidlywhenܶ௜ ൒150K, consistentwiththeearlierestimationofܶ௔tobebetween115and170K.Thefittedline has the form ܨ ൌ ሾെߟሺെߚȀ݇஻ܶ௜ሻሿ where ߟ and ߚ were used as fitting parameters,whichequalto5.2ൈ106and230meVrespectively.Bothparametersaffect theshapeofthedecorrelationcurve,withߟbeingweightedtowardstheslopeofcurve, andߚtowards the onset ofܶ௔. This equation was previously used in analysing the influenceofthesidewallsonthequantumtransmissionthroughaGaAsnanowire[25]. Forthepurposeofourdiscussions,weonlyfocusontheactivationtemperaturewhere characteristics of the MCF start to decorrelate. Detailed information of this fitting equationcanbedfoun in[19,25].

Letusnowconsiderthenatureofthisdecorrelationprocess.Weknowthata changeintheoverallpotentialofthedotοܸௗ௢௧willaltertheelectrontrajectories.Inthe process of generating MCF, the sensitivity of electron paths toοܸௗ௢௧is amplified by reflectionsoffthebilliardwalls.Therefore,achangeintheMCFcharacteristicsreflects changes inοܸௗ௢௧. But what causes this change in the potential landscape in the first place?

6.2.3–Chargerelocationmechanismsinthedopantlayerof C2275

WehaveestablishedpreviouslythattheMCFdecorrelationprocessisrelatedto a change inthe dot’s disorder potentialǤSince externalparameters such as݊,ߤandܶ remainthesameintheMCFcomparison,weattributeοܸௗ௢௧solelytotheoutcomeof chargerelocation[26,27]inthedopantlayerofC2275heterostructure.Inthefollowing, wewillbelookingatafewpossiblemechanismsresponsibleforthischargerelocation. First of all, as a first approximation, for every electron in the 2DEG, there is a correspondingionizedSiatomintheAlGaAsdopantlayer.However,aswehaveshown

165 inSection3.2,notalloftheSidopantsareionized.GiventheionizationenergyofSito be about 5.6 meV (̱65 K), which is considerably less that the observed 150 K (̱13 meV).SincewedidnotobserveanyMCFdecorrelationnearܶ௜=65K,weruleoutSi ionizationasthedecorreationmechanism.Ontheothersideofthescale,diffusionofSi ,[dopantshasbeenreportedtooccurattemperaturesof׽500°CinɷͲdopedGaAs[28 andmuchhighertemperaturesinmodulationͲdopedAlGaAs/GaAs[29].Thereforewe also rule out the physical relocation of dopants to be the mechanism responsible for decorrelationofMCF.

SiliconinAlGaAscanalsoexistinasecondstate,knownastheDXcenter[30]. Initiallyconjecturedasthecombinationofadonor(D)andanunknownlatticedefect(X), possiblyanAsvacancy[31,32],aDXcenterisacomplexstateinwhichadonoratom bindsanextraelectron.Thelatticerelaxesaroundthedonorwhenanelectronarrives, releasing extra energy so the electron is more deeply bounded. DX centers are also known to give rise to persistent photoconductivity (PPC) [30]. Although the exact microscopicstructureoftheDXcenterhasbeenasubjectofdebate[30],nevertheless, itwasfoundexperimentallythattheoccupationoftheDXcenters“freezes”outatlow temperaturesbelow150K[30].Thisenergyscaleforelectrontrappingandreleasingis inexcellentagreementwithwhatwedfoun inthedecorrelationanalysisabove.

6.3–ExtendedcomparativeMCFstudiesbetweenUDǦMand MDǦM

Thequantumdotdevicespresentedinthelastchapterhavea“straightͲthrough” hexagonal geometry design, where the entrance and exit QPCs are located symmetrically in the middle. Furthermore, the vertical dimension of the dot is not substantiallylargerthanthewidthofitsQPCchannels.Asaresult,weattributethelack ofspectralcontentobservedinUDͲSandMDͲS,tothelackofelectrontrajectoriesthat reflectoffthebilliardwalls.

InordertoobtainMCFwithahigherspectralcontent,weadoptedthesquare billiard design used extensively by the Nanoelectronics Materials Laboratory at RIKEN [13,33Ͳ37], and later at Arizona State University, where the QPCs are located in the

166 upperpartofthebilliard.Figure6.5(a)showsanexampleofsuchsquarebilliardusedin [38].Inthisdesign,thefringingfieldfromtheupperpartofthebilliardgate“pushes” theelectronstowardsthelowerbilliardwalls,astheyareinjected.Figure6.5(b)showsa simulationoftheelectroninjectionprobabilitydensitydistributionforasquarebilliard calculatedbyBirdetal.[38].Allbilliardspresentedinthechapterhavethisdesign.

Figure6.5:(a)ASEMimageofthesquarebilliardusedin[38]withascalebarof1ʅm.The billiards discussed in this chapter (see Fig. 6.1) were fabricated based on this design. (b) Probability density distribution for electron injection, calculated for the dot shown in (a). Threemodesareassumedtobeoccupiedbythepointcontacts,andthedotsizeusedfor thiscalculationis0.8ʅmby0.8ʅmtoaccountforedgedepletion.Figuresreproducedfrom [38].

6.3.1–Thermalstabilityofgatecharacteristics

As discussed in Section 5.3.2, the gate characteristics ݃ vs. ܸௌீ of our modulationͲdoped device originated from quantum interference effects. They arise from the modulation of the Fermi wavelength as a result of changes in the electron densityandeffectivedotarea[39].Asaresult,wecanusethesefluctuationsinasimilar waytotheMCFasaprobeforchargerelocationeventsinthebilliards.Similarlyforour undoped devices, the correspondinggate characteristics݃vs.்ܸீcan also be used as boththesidegateandthetopgateaffectthedotareaandlocalcarrierdensity.We have shown in Fig. 5.12 that the gate characteristics of MDͲS were not reproducible afterthermalcycling.Inthefollowing,wewillcomparethegatefluctuationsbetween theUDͲMandMDͲMbilliarddevices.

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Figure 6.6:These figures compare the thermal stability between our undoped and modulationͲdoped billiards.(a)Topgatecharacteristics (݃vs.்ܸீ)forUDͲMisshownfor two different cooldowns. The red trace represents݃vs.்ܸீbefore thermal cycling toܶ= 300 K, while the black trace was obtained after thermal cycling. Data were taken with a resolutionof0.5mV.Theblacktraceisoffsetupwardsby0.2ൈ2݁ଶ/݄forclarity.Datain(a) demonstratethestabilityofourundopeddevice, consistentwith the MCFreproducibility result discussed in Section 5.2.2. (b) Comparison of݃vs.ܸௌீfor MDͲM from 3 different cooldowns.Theblackandorangetracesareoffsetupwardsby0.5and1ൈ2݁ଶ/݄forclarity. Thesidegatecharacteristicshaveclearlyevolved,andweattributethistochargerelocation inMDͲMafterthermalcycling.Thesmallscaleoscillationsinboth(a)and(b)originatefrom the modulation of Fermi wavelength as a result of changes in the electron density and effectivedotarea[39].

Figure6.6(a)showscomparisonsof݃vs.்ܸீforUDͲMbetweentwocooldowns, while the corresponding comparisons of݃vs.ܸௌீfor MDͲM is shown in Fig. 6.6(b). ThermalcyclingwasperformedthesamewayasforUDͲSandMDͲS:bywarmingupthe sampletoroomtemperatureinalowpressureHeliumenvironment(insidetheIVC)for

168 abouttwohoursbeforecoolingdownagain.Themeasurementsetupusedissimilarto theonedescribedinSection5.2.1and5.3.1.Thedatafrom(a)indicatesremarkable thermalreproducibility,consistentwithwhatweobservedintheMCFofUDͲSinSection

5.2.2. In contrast, fluctuations in݃vs.ܸௌீfor the MDͲM device in (b) have evolved significantlyafterthermalcyclingtoroomtemperature.Weattributethistothecharge relocationinC2275,randomizingthedotpotentialbetweencooldowns.

6.3.2–ThermalstabilityofmagnetoǦconductancefluctuations

WenowcomparecharacteristicsoftheMCFbetweencooldownsforUDͲMand

MDͲM.Figure6.7(a)showsasetofMCFtracesfortheUDͲMdevice,takenbetween்ܸீ

=0.930V(bottom)and0.955V(top)withastepsizeο்ܸீ=5mV.Thecorresponding MCFtracesobtainedafterthermalcyclingareshownsideͲbyͲsidein(b).Onceagain,the data demonstrate remarkably reproducibility of the MCF in the undoped sample.

Furthermore,withinthecalculatedaverageܤ௖௬௖of0.22Tat்ܸீ=0.943V(recall݊=݊

ሺ்ܸீሻfromSection3.2.1),thenumberofaperiodicfluctuationsissignificantlygreater thanfortheMCFfromUDͲS.GiventhatUDͲSandUDͲMweremeasuredatthesame temperatures with similar electron density, we expect the two to have similar phase coherence [40]. An increase in the number of oscillations implies an increase in the spectralcontent,demonstratingthesuccessofourchoiceofbilliarddesign.

Figure6.7(c/d)comparescharacteristicsofMCFbetweencooldownsforMDͲM.

AswecanseeInFig.6.6(b),݃isverysensitivetoܸௌீ,thereforeifwecomparetheMCF traces obtained at the sameܸௌீas shown in Fig. 6.7(c), the difference between their zeroͲfieldconductanceis0.31ൈ2݁ଶ/݄.InFig.6.7(d),wecomparetwoMCFtraceswith a matched݃(0), where the difference in݃is less than 2.9 %. In both (c) and (d), the characteristicsofMCFhavebecomesignificantlydifferentafterthermalcycling.These resultsareconsistentwiththoseshowninFig.5.13fortheMDͲSdevice.

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Figure6.7:(a/b)ThisfigurecomparesMCFtracestakenbefore(a)andafter(b)thermal cyclingUDͲMtoroomtemperature.Theloweranduppertracesinbothpanelsweretaken at்ܸீ=0.930Vand0.955V,respectively,witharesolutionof0.42mT.Tracesabovethe ்ܸீ=0.930Vtraceweretakenwithastepsizeof5mVin்ܸீ,andaresequentiallyoffset upwardsby0.4ൈ2݁ଶ/݄forclarity.Comparisonsbetweentracesin(a)&(b)suggestthatthe disorderintheundopedsystemdoesnotchangeafterwarmingupthesampletoܶ=300K.

In(c),wecomparetheMCFtracesofMDͲMfromdifferentcooldownstakenatthesameܸௌீ =െ0.4435 V. The zeroͲfield conductance݃ሺͲሻbetween the two traces differs by 0.31ൈ 2݁ଶ/݄.Theblacktraceisoffsetdownwardsby0.6ൈ2݁ଶ/݄forclarity.In(d),wecompare thetwoMCFtraceswiththeir݃(0)matchedtowithin2.9%atܸௌீ=Ͳ0.430V(red)andͲ 0.451V(black).Theblacktraceisoffsetupwardsby0.5ൈ2݁ଶ/݄forclarity.Inboth(c)and (d),theMCFcharacteristicsbecamequitedifferentafterthermalcycling.

170

Duetothissensitivityof݃inܸௌீ,weperformedamorethoroughsearchforthe matchingMCFtraceincaseitwasaccidentallymissed.Thisisthesametreatmentwe performed on MDͲS in Fig. 5.14. Figure 6.8 shows the result of this search. Despite performingthisextensivesearch,noneofthe51MCFtracesfromthesecondcooldown showninFig.6.8(bͲd),resemblestheMCFtraceobtainedinthefirstcooldownin(a).

Figure6.8:ThetracesinthisfigurerepresentMCFfromtwodifferentcooldowns(aandbͲd). Theblacktracein(a)andthesamepurpletracesin(bͲd)were݃(0)matchedandusedinFig.

6.7(d).Panel(b)showsMCFtracestakenwitharangeofܸௌீfromͲ0.4710V(bottom)toͲ 0.4310V(top)instepsof2.5mV.Panel(c)and(d)containsMCFtracestakenatthesame rangeofܸௌீasin(b),exceptthegateswerebiasedasymmetrically.Theyareindicatedby theschematicsatthetopofthepanels.Thegatecolouredinyellowrepresentstheonethat wasstepped,whilethegateshowningreywasheldatͲ0.4510V.TheMCFtracesthatbest resembles the trace) in (a is highlighted in green in (d), as suggested by the correlation functiondescribedinthetextinSection6.3.3.Alltracespresentedherehavearesolutionof 0.83mT.TheMCFtracesaboveandbelowthepurpleonesaresequentiallyoffsetin݃by േ0.3ൈ2݁ଶ/݄respectively.AsimilartreatmentforMDͲScanbefoundinFig.5.14.

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6.3.3–MCFcorrelationanalysisbetweenUDǦMandMDǦM

InordertoputtheMCFcomparisonsinFig.6.7(a/b)forUDͲM,andinFig.6.8for MDͲM into quantitative context, we introduce a crossͲcorrelation function that was previously used in studies of selfͲsimilarity [18], to numerically compare the כܨcharacteristicsoftheMCFbeforeandafterthermalcycling.Thiscorrelationfunction is defined in Eqn. 6.2, where݃ͳሺܤሻand݃ʹሺܤሻrepresent the two MCF traces under

ଶۄሻܤሻ െ݃ ሺܤሺ ݃ۃඥ ൌͳെ ଵ ଶ ሼ͸Ǥʹሽ כܨ כܰ

was obtained by calculating 1000 pairs ofכcomparison. The normalization constantܰ randomlygeneratedMCFtraces.Theserandomtraceshavethesamenumberofdata points and resolution as the real MCF data and were bounded by the maximum and כ minimumconductanceof݃ͳ(ܤ)and݃ʹ(ܤ).Basedonthisdefinitionofܰ ǡtwoidentical .andF̱0fortworandomtraces,1̱כܨtraceswillgive

Thiscorrelationanalysiswasperformedovertheܤ rangebetweenേ0.5Tfor data in Fig. 6.7(a/b) and Fig. 6.8. The results for UDͲM are shown in Table 6.1. We

כܨ (ீ்ܸ)MCFpairs

0.955 0.92302

0.950 0.91926

0.945 0.91860

0.940 0.91229

0.930 0.93917

Table6.1:ResultsofthecorrelationanalysisperformedondatainFig.6.7(a/b),usingEqn. 6.2. The MCF traces obtained at the same top gate voltage were compared between .between0.91229and0.93917wasobtainedכܨcooldowns.Arangeof

172

forכܨbetween 0.91229 and 0.93917. In contrast, the highestכܨobtained a range of MDͲMwith51comparisonsmadebetweentheMCFtraceinFig6.8(a),totherestofthe tracesin(bͲd)is0.75658.TheMCFtracethatcorrespondstothiscorrelationvalueis it is ,כܨhighlighted in green in Fig. 6.8(d). Despite this relatively large value of visuallyevidentthatthetwoMCFtraceshaveverydifferencecharacteristics.Thislarge value wasmainly due to the fact that the MCF traces in Fig. 6.8 have very similarכܨ background conductances, and the randomly generated traces used to calculate the normalization constant do not have any common background. This lack of sensitivity indicatesthelimitationofourapproach.

6.3.4–StatisticsofMCFfluctuations

In this section, we will extend the comparison between UDͲM and MDͲM by considering the statistical properties of their magnetoͲconductance fluctuations. In particular, we will examine the corresponding Fourier power spectra and

஻ . Forۄሿۄሻܤሺ݃ۃ ሻ െܤ൅οܤ஻ሿሾ݃ሺۄሻܤሺ݃ۃ ሻ െܤሾ݃ሺۃ ሻ ൌܤሺοܥ autocorrelation functions UDͲM,wehavechosenthesetofMCFtracesshowninFig.6.7(a);whereasforUDͲM,a selectionoftracesfromFig.6.8(b)wereused.Fortheseanalyses,theMCFareextracted bysymmetrizingthedata,andremovingathirdorderpolynomialfitasper[2]between ܤ=0and0.5T.Theaveragerootmeansquare(rms)fluctuationamplitudesaresimilar betweenthetwodevices(0.0877and0.0951ൈʹ݁ଶȀ݄forUDͲMandMDͲM),however theundopedsampleappearstocontainhigherfrequencyfluctuationsasindicatedby the Fourier analysis shown in Fig. 6.9. This observation is consistent with the correspondingautocorrelationanalysesplottedasinsets,whereܥሺοܤሻforUDͲMclearly dropsmorerapidlythanthatofMDͲM(theaveragecorrelationfieldis7.2mTforUDͲM and11.1mTforMDͲM).

173

Figure6.9:Fourierpowerspectraof݃(ܤ)fordevice(a)UDͲMand(b)MDͲM.Calculations wereperformedonsymmetrized݃(ܤ)witha3rdorderpolynomialbackgroundsubtracted. Datain(a)werebasedontheMCFtracesinFig.6.7(a),whereasin(b),MCFtraceswere obtainedfromFig.6.8(b),bytakingeverythirdtracefromthetop(inclusive).TheFourier transformationswerecalculatedbyinterpolating݃(ܤ)tohave8192pointsbetweenܤ=0 and 0.5 T. The corresponding autocorrelation functionsܥሺοܤሻare shown as inset. The FourieranalysessuggestthatUDͲMcontainshigherfrequenciesfluctuationscomparedto

MDͲM,whichisconsistentwithܥሺοܤሻ,showinganaveragecorrelationfieldܤ஼(definedas ܥሺܤ஼ሻ=ܥ(0)/2)of7.2mTforUDͲM,and11.1mTforMDͲM.

According to the AharonovͲBohm effect discussed in 1.5.3, when a perpendicular magnetic fieldܤis applied, the areaܣenclosed by the electron partial wavesisinverselyproportionaltotheperiodofoscillationοܤin݃(ܤ).Asaresult,richer highfrequenciescontentintheFourierpowerspectrumindicatesalargerproportionof bigloopareasformedinthebilliard.Qualitatively,wecanpicturethemodulationͲdoped devicetosupportasmallerproportionoflargeloopareasduetothepresenceofionized dopant disorder potential. Note however, in order to properly examine the effect of remote ionized impurities in MCF statistics, a more vigorously approachthat involves matchingtheareasofbilliardsprecisely,measurementsofalargersampleset,aswellas lookingatthecorrespondingphasebreakinganalysesisrequired.Thisbodyofworkis beyondthescopeofthethesisandiscurrentlybeingundertakenbyPh.D.studentIan PilgrimattheUniversityofOregon.

174

6.4–Effectsofdisorderinanundopedbilliarddevice

While we have eliminated remote ionized impurities in our undoped heterostructure,otherdisordersuchasbackgroundimpurities,surfacestates,interface roughness,andlatticedefectsarestillpresentinthesystem.Inthefollowingstudy,we investigate the effect of surface states on the MCF reproducibility of our medium undoped billiard UDͲM. Before we begin discussing of the experimental method, we pointoutthatsurfacestateslocatedabovetheheavilydoped݊ାcapwillbescreened. Asaresult,theydonotaffecttheelectrontransportatthe2DEGandhavenoinfluence onthereproduciblyoftheMCF.AswehaveseenforUDͲSinSection5.2.2,thesesurface statesabovethecapwillonlydeterminehowmuchvoltageweneedtoapplytothetop gateinordertogetthesameelectrondensityatthe2DEG.Asaresult,anoffsetof்ܸீ = 5 mV was observed in obtaining reproducible MCF trace between difference cooldowns(seeFig.5.5).Inthissectionhowever,weareonlyinterestedinthesurface states that are not screened by the cap. They will act as smallͲangle scatterers for electrons in the 2DEG just as the remote ionize impurities do. So where should we expectthesesurfacestatesinourundopedbilliards?Fig.6.10showsacrossͲsectional viewofourNBI30undopeddevice.AsdiscussedinSection2.3.1,thefabricationofthe billiardsinvolvedpatterningofthecapusingEBLandwetchemicaletching.Thisetching processusuallystopswithinthespacerlayer,approximately5Ͳ10nmabovetheAlGaAs

Figure6.10:(a)AcrossͲsectionalviewofourundopedNBI30billiarddevicetakenbyslicing along the white solid line shown in (b). The blue dashed lines at the heterojunction representthe2DEG.Onlyregionsunderthepositivelybiasedtopgatearepopulatedwith electrons.Thesurfacestatesthatare ontopof thecaparescreened, they areshownas greendashedlines.Thesurfacestatesthatareontopoftheetchtrenches(asreddashed lines) however, are not screened and can affect the billiard transport at the 2DEG. Only thesestatesareofinterestinthissection.

175 layer.Asaconsequence,surfacestateslocatedattheetchedtrenchesarenotscreened by the cap. Furthermore, if the trenches have nonͲuniform etch depths, regions of AlGaAscanbeexposedtotheairandtherewillbesurfaceoxidation[41]andevensome subsurfaceoxidation.

WehavealreadyseenfromMCFreproducibilityresultsofUDͲSandUDͲMthat thesurfacestatesdidnotaffectthereproducibilityofMCF,asweobtainedexcellent MCF correlation between cooldowns. Here we devised two experiments, trying to amplifytheeffectsoftheunscreenedsurfacestates.

6.4.1–EffectofsurfaceoxidationonMCFreproducibility

The first experiment was to compare the MCF traces of UDͲM between cooldownsthatare30daysapart.Duringthisperiod,thedevicewaskeptinaroomair environment,unlikewhatwehavedoneintherestofMCFreproducibilityexperiment, wherethedevicewaskeptinanoxygenͲ freeenvironment,insidetheIVCoftheHeliox. ThisMCFcomparisonisshowninFigure6.11(a/b)wheretheblackMCFtracein(a)was takenat்ܸீ=0.940V,reproducedfromFig.6.7(b)beforeoxidation.Thecorresponding

MCFtracetakenatthesame்ܸீ30dayslaterisshowndirectlytoitsrightinFig.6.11(b). ComparisonbetweenthesetwotracesrevealsthattheMCFcharacteristicshaveevolved significantly.Byvisualinspection,thisdecorrelationisjustasstrongasthoseobservedin the modulationͲdoped billiards. The rest of the traces in Fig. 6.10(b) were obtained between்ܸீ=0.930V(bottom)and0.950V(top)instepsof5mV.Thisistoensure thattheobservedcannotbecompensatedbyanoffsetin்ܸீ,asobservedinthecaseof

UDͲS. Figure 6.11(c) shows݃vs.்ܸீfor the corresponding MCF traces in (a/b). As expected,wealsoobserveddecorrelationofthetopgatecharacteristics,consistentwith theMCFresults.

176

Figure6.11:In(a/b),wecomparetheMCFofUDͲMbetweentwocooldownsseparatedby 30daysinanopenairenvironment.Unliketherestofthermalcyclingexperimentswhere thedevicewasleftintheIVCwithasmallamountofHeexchangegaspresent.UDͲMwas kept in the open air environment in this period of 30 days. Therefore we expect surface oxidationtohavetakenplace.Theblacktracein(a)representsthesameMCFtracefromFig. 6.7(b)obtainedat்ܸீ=0.940V,whilethebluetracesinFig.6.11(b)correspondtoasetof MCF taken in a subsequent cooldown after 30 days with a range of்ܸீ, from 0.930 V (bottom) to 0.950 V (top) in steps of 5 mV. This comparison is to ensure the observed difference in MCF was not due to a shift in்ܸீbetween the two cooldowns. In (c), a comparisonof݃vs.்ܸீwiththesameconditionasin(a/b)isshown.Thebluetracesabove andbelowthemiddletracein(b),areoffsetbyേ0.5ൈ2݁ଶ/݄in݃forclarity.Whilein(c), thebluetraceisoffsetupwardsby0.4ൈ2݁ଶ/݄.Both(a/c)and(c)showahighdegreeof decorrelationafter30days.

177

WehavediscussedinSection6.2.3 thatachangeinthecharacteristicsofthe MCFcorrespondtoachangeintheoverallpotentialofthedot.Inthisstudy,wefound evidencethattheexistenceofthesurfaceoxidehadalteredtheoveralldotpotential. This is achieved by the using the MCF as a sensitive probe for the precise charge configuration in the system. However, in order to investigate the exact surface chemistry responsible for the observed changes in the MCF between cooldowns, we suggestusingconventionalsurfacesciencemethods[41]suchasAugerdepthprofiling andxͲrayphotoelectronspectroscopy(XPS).

In order to determine whether these surface oxides have a stable charge configurationatroomtemperature,werepeatedthesamethermalreproducibilitytest witha2hrthermalcyclingturnaroundtime,keepingthesampleunderHeaswedid previously,andcomparedthecorrespondingMCFandthetopgatecharacteristics.Fig. 6.12 shows the results of this experiment. Both comparison of݃vs.ܤin (a) and݃vs.

்ܸீ in (b) indicate that the correlations between cooldowns are not as great as previouslymeasuredinFig.6.7(a/b)andFig.6.6(a/b).Inparticularwelocatedseveral placesinFig.6.12(a/b)wherefeatureshaveclearlyevolvedafterthermalcycling,they arehighlightedbythearrowsinthefigure.Notehowever,theMCFcorrelationisstill much better than the modulationͲdoped counterpart MDͲM. We interpret this as a resultofhavingasmallnumberofchargerelocationeventsamongthesurfacestates that are detectable by the method of comparing MCF of different cooldowns after surface oxidation. Although the exact chemistry responsible for it is unknown, there existsstudyofchargespreadingofaluminiumoxidereportedintheliterature[42].

As a suggestion for future studies to verify the effect of surface oxidation observedabove,wecouldpreparetwonominallyidenticalundopedbilliarddevicesand measure the thermal reproducibility of their MCF immediately after fabrication. We thenstorethemseparatelywithoneinanoxygenfreeenvironment.Afterafewmonths, wecouldmeasuretheirthermalreproduciblyagainandcomparetheirMCFcorrelations to the results previously measured. We would expect to see in drop in the MCF correlationforthebilliardthatwasexposedtoairduringthisstorageperiod.

178

Figure6.12:Thesetwopanelsshowcomparisonsofthe(a)MCFand(b)݃vs.்ܸீforUDͲM between two cooldowns after data in Fig. 6.11(b) were obtained. Thermal cycling was performedwiththeusual2hrturnaroundtimeinsidetheIVC.Thecorrelationsbetweenthe twocooldownsinboth(a)and(b)arenotasgoodasthecorrespondingonesinFig.6.6(a/b) and Fig.6.7(a/b).In(a),theuppertracesareoffsetby0.45ൈ2݁ଶ/݄in݃forclarity.Whilein (b),thecorrespondingtracesareoffsetupwardsby0.35ൈ2݁ଶ/݄.Dataresolutionis0.83mT in(a),and0.5mVin(b)respectively.Weattributethereductionofcorrelationtopossible changerelocationeventsamongtheunscreenedoxidizedsurfacestates.

179

6.4.2–MCFreproducibilityofalargeundopedbilliard

Thesecondexperimentaimingtoamplifytheeffectofsurfacestatesinvolves thelargeundopedbilliarddeviceUDͲL.ThisdevicehasfourtimestheareaoftheUDͲS device discussed in Chapter 5 (see Fig. 6.1 for more information). By having a larger billiard,therearemoreunscreenedsurfacestates,duetolongeretchedtrenches.We performedthesameMCFreproducibilityexperimentonUDͲLasthoseshownabovein

Fig. 6.12. The comparison of݃vs.ܸௌீbetween two different cooldowns for UDͲL is shown in Fig. 6.13(a).The corresponding MCF traces obtained at்ܸீ= 0.95 V are

Figure 6.13:These plots measure the stability of the larger UDͲL billiard device by comparing(a)݃vs.ܸௌீand(b)݃vs.ܤobtainedfromdifferentcooldowns.Tracesobtained inthesamecooldownareshowninthesamecolour.TheblackMCFtracein(b)isoffset ଶ upwardsby0.25ൈ2݁ /݄forclarity.Inbothcases,்ܸீ=0.95Vwasused.Theresolutionin (a)&(b)are5mVand0.5mTrespectively.IncomparisonwiththesmallerUDͲSandUDͲM devices,thethermalreproducibilityofUDͲLisnotasgood(asindicatedbythearrows).In

(b), the two traces were taken at different ܸௌீ in order to match their zeroͲfield conductance.TheoscillationsenclosedbythebluedashedcirclearenoiseͲrelatedbecause theyarenotsymmetricinܤ.

180 showninFig.6.13(b).Datain(a)showthatthesidegatecharacteristicshaveevolved afterthermalcyclingtoܶ=300K.IntheMCFcomparison,wethereforehadtochoose twoslightlydifferentܸௌீinordertomatchtheirzeroͲfieldconductance.Consistentwith whatweobservedin(a),theMCFtracesin(b)donotmatchexactly.Thereexistfeatures thathaveevolvedafterthermalcycling,theyareindicatedbyarrowsinthefigure.We have also observed some measurement noise in the MCF traces. For example, the oscillationshighlightedbythebluecirclein(b)arenotsymmetricinܤandthereforeare noiserelated.

ThedatainFig.6.13suggestthattheMCFreproducibilityofthislargerbilliard device(UDͲL)isclearlynotasgoodasthesmallerones(UDͲSandUDͲM).Apartfromthe factthattherearemoreunscreenedsurfacestatespresentinthislargerdevice,there arealsotwoothersmechanismsworthconsidering.Firstofall,theUDͲLsamplewasleft inaroomairenvironmentforafewweeksbetweenfabricationandmeasurement.Asa result,therecouldbeasignificantamountofoxidizedsurfacestatesthatdonothavea stablechargeconfigurationatroomtemperatureasdiscussedpreviously.Secondly,this reduction in MCF correlation can also be attributed to an increased number of thermallyͲreconfigurablebackgroundimpuritieslevelinUDͲL:givenafixedbackground impuritydensityinasystem,thelargerthebilliard,themoretheimpurityatomsfound within close proximity of the device. The background impurities that are sufficiently closetothebilliardtocauseasignificantchangeintheelectronmomentumwillaffect themobility,whereastheonesthatarefurtherawaywillactassmallͲanglescatterers. However, we emphasize that only impurities that do not have a stable charge configuration at room temperature will affect the reproducibility of MCF. Here, we hypothesizethatsiliconatomcouldbeapossiblecandidatebecauseitwasusedinthe MBEgrowthprocessofourNBI30heterostructure(dopedcap),andstudies[43,44]have shownthatAlGaAscontainsupto3timeshigherbackgroundimpuritylevelthanthatof GaAs,backgroundimpuritiescontainingSiatomthatendedupintheAlGaAscanform DXstatesandaffectstheMCFreproducibilitybetweencooldowns.

181

6.5–Summeryanddiscussions

In Chapter 5 and 6, we have demonstrated that MCF in modulationͲdoped billiardsarenotcompletelydeterminedbythedevicegeometryalone,butsensitively affectedbytherandomionizationofremoteimpuritiesinthemodulationdoping.We demonstrated this result by performing comparative MCF studies between billiards formedonamodulationͲdopedandundopedheterostructureswithnominallyidentical geometries. Both the comparisons of MCF and gate voltage characteristics between cooldownssuggestthatmodulationͲdopedsamplescontaindisorderthataresubjectto change upon thermal cycling, while the undoped samples do not. Our decorrelation analysis shows that disorder in the modulationͲdoped sample has an activation temperatureof150K.ThisisinexcellentagreementwiththeenergyscaleofDXcenters [30],whicharestatesthatSidopantsexistinAlGaAs.Wethereforeconcludethatthe dominantcontributiontoMCFinamodulation–dopeddeviceissmallͲanglescattering ratherthanthebilliardgeometry.

In addition, during our comparative MCF studies of billiards, we have also demonstrated the ability to use MCF as a sensitive probe to determine the thermal stabilityofdisorderinvolvedinasystem.Inparticular,weobservedachangeinMCF characteristicsupontheformationofsurfaceoxideintheetchtrenches,whicharenot screened by the݊ାcap. With these surface oxides formed, the MCF characteristics betweensubsequentcooldownsbecamelesscorrelated,althoughtheircorrelationsare stillmuchbetterthanthemodulationͲdopedcounterparts.

6.5.1–Implicationsofourfinding

To what extent does our finding affect the understanding of previous MCF experiments? Two aspects are discussed here: the length scale, and the effect of experimental signatures of the electron dynamics. In terms of length scale, in the semiclassical view, there is a wide distribution of lengths in the set of all possible electrontrajectorieswithinthedotthatintercepttheQPCsandthuscontributeto݃(ܤ). OnewouldexpecttheeffectofsmallͲanglescatteringtoincreasewithtrajectorylength, andbecomeminimalfortheshortestpaths.Indeed,ScanningGateMicroscopy(SGM)

182 studies[45,46]provideclearevidencethattheshorterballistictrajectoriessurvivethe diffusiveeffectofsmallͲanglescattering.Thereforestudiesthatinvoolvedtheinfluence ofshortperiodicorbitson݃(ܤ)spectralcontent[39]arelikelytoberobust,partlydue tothereducedimpactofsmallͲanglescatteringforshorttrajectories,,butalsobecause quantuminterferencemayenabletheseorbitstosurvivedespitethediffusiveeffectof smallanglescattering[45,47,48].

RegardingtheeffectofsmallͲanglescatteringonexperimentalsignatureofthe electrondynamics,let’sconsidertheexperimentperformedbyBerryyetal.[49],where ݃(ܤ) was used as a magnetoͲfingerprint to detect changes in the electron dynamics inducedbyaddinganarrowbarriertotheinteriorofamodulationͲdopedAlGaAs/GaAs circularquantumdot.Suchexperimentsrequiretwoseparatedevices,andourfinding showsthattheequivalentchangein݃(ܤ)couldhaveresultedfromroomtemperature thermalcyclingofeitherdevice.Thesameproblemexistsformorereecentworkby[50].

Fiigure6.14:ThesefiguresshowthepowerspectraofthemagnetoͲconduuctancefluctuations ݃(ܤ)forthemodulationͲdopedcircleͲandstadiumͲshapedbilliards.Thestudyinvolvedtwo chips (1 & 2), each withone stadium billiard and one circle billiard on it. Measurements wereperformedat20mK.Thedata suggestthatalthoughthereisasignificantdifference between thepowerspectraofthetwogeometries,this differenceisassignificantasthe onesbetweenthetwosamplesofthesamegeometry,especiallyfortheecircle.Thisimplies thatsmallͲanglescatteringsignificantlymasktheelectrondynamicsofthegeometryin݃(ܤ) inthesedevices.FigurereproducedfromFig.7of[2].

183

Experiments that rely on simple statistical measure are also affected. For example, Marcus et al. [2] presents power spectra for two separate devices, each containsonecircularandonestadiumͲshapedmodulationͲdopedquantumdot(seeFig. 6.14).IfsmallͲanglescatteringwasnegligible,onewouldexpectthepowerspectrafor thetwodotsofthesamegeometrytomatch,buttheydonot.Indeed,thedifferencein power spectra between nominally identical dots is as significant as between dots of differentgeometries.Itisimportanttoemphasizethatinbothcasesabove,wedonot claimthatthedotgeometryhasnoeffectonelectrondynamicsatall;onlythatsmallͲ anglescatteringmasksitseffecton݃(ܤ)asanexperimentalsignatureofdynamics.One approachthatmightpossiblyovercomeisthatemployedbyChangetal.[3],wherea6 ൈ8arrayofnominallyidenticaldotswasmeasuredtoaverageoutthefluctuationsin ݃(B).

6.5.2–Correlationbetweenthermalrobustnessandcharge stability

Inthisthesis,wereportcomparativestudiesbetweenundopedandmodulationͲ doped quantum dots and claim that undoped devices are ultraͲstable with thermally robust quantum properties. A natural question arises: can the (lack of) robustness to thermalcyclingberelatedtochargestability(instability)atlowtemperature?Letus consider these effects qualitatively. The lack of thermal robustness of MCF in modulationͲdopeddevicesisshowntoberelatedtoaparticularstateofsilicondopant in the AlGaAs layer, known as the DX center [30]. During thermal cycling, the distribution of DX centers is randomized, and this is reflected experimentally in the correspondingmagnetoͲconductance.Ontheotherhand,chargestabilityisbelievedto berelatedtotheswitchingofchargestates[51]atvariousfrequencies.Despitethefact thatDXcentershaveanactivationtemperatureof150K,theprobabilityofDXswitching atcryogenictemperatureisnonͲzero, andtheeffectbecomessignificantwhenthereisa largequantity.Asaresult,wesuspectthatthereisacorrelationbetweenrobustnessto thermalcyclingandchargestability.However,untilwecandemonstrateexperimentally thatundopeddeviceshavebetternoiseperformancethanthemodulationcounterparts, theabovestatementremainsa speculation.

184

6.5.3–Futurework

Finally, we would like to point out that even in the case of undoped billiards where dopantͲinduced disorder is eliminated; there exists disorder such as antisite defect,whichhasstablechargeconfigurationsatroomtemperature.Suchdisorderwill inevitably contribute to the MCF, but does not affect the thermal robustness of the system. In order to investigate the geometryͲinduced electron dynamics in a billiard system, we propose to measure݃(ܤ) from an array of nominally identical undoped billiards,similartotheapproachusedbyChangetal.[3].Thiswouldaverageoutany ݃ ( ܤ ) contributions from disorder (whether or not they have stable charge configurationsatroomtemperature).Furthermore,inordertominimisetheeffectof backgroundscatteringwiththebilliards,suchadeviceshouldideallybefabricatedonan undopedheterostructurewithhighelectronmobility.

Additionally, it would also be interesting to study modulationͲdoped billiards withthickerspacerlayers(i.e.>200nm),wheretheeffectofsmallͲanglescatteringis believedtobereduced,andexaminethethermalrobustnessofMCFinsuchsystems.

185

Bibliography

[1]“WeakLocalizationandInstabilityinBallisticCavities”–H.U.Baranger,R.A. Jalabert,andA.D.Stone,Phys.Rev.Lett.,70,3876(1993).

[2]“ConductanceFluctuationsandChaoticScatteringinBallisticMicrostructures”–C.M. Marcus,A.J.Rimberg,R.M.Westervelt,P.F.Hopkins,andA.C.Gossard,Phys.Rev.Lett., 69,506Ͳ509(1992).

[3]“WeakLocalizationinChaoticversusNonchaoticCavities:AStrikingDifferenceinthe LineShape”–A.M.Chang,U.H.Baranger,L.N.PfeifferandK.W.West,Phys.Rev.Lett., 73,2111Ͳ2114(1994).

[4]“ConductanceFluctuationsintheBallisticRegime:aProbeofQuantumChaos?”–R. A.Jalabert,H.U.Baranger,andA.D.Stone,Phys.Rev.Lett.,65,2442Ͳ2445(1990).

[5]“ClassicalandquantumballisticͲtransportanomaliesinmicrojunctions”–H.U. BarangerD.P.DiVincenzo,R.A.JalabertandA.D.Stone,Phys.Rev.B,44,10637(1991).

[6]“QuantumͲchaoticscatteringeffectsinsemiconductormicrostructures”–HaroldU. Baranger,RodolfoA.Jalabert,andA.DouglasStone,Chaos,3,665(1993).

[7]“Phasebreakinginballisticquantumdots:Experimentandanalysisbasedonchaotic scattering”–C.M.MarcusandR.M.Westervelt,P.F.HopkinsandA.C.Gossard,Phys. Rev.B,48,2460(1993)

[8]“Conductancefluctuationsinaquantumdotinthetunnelingregime:Crossoverfrom aperiodictoregularbehavior”–C.M.Marcus,R.M.Westervelt,P.F.Hopkins,A.C. Gossard,Surf.Sci.,305,480(1994).

186

[9]“Conductancefluctuationsandquantumchaoticscatteringinsemiconductor microstructures”–C.M.Marcus,R.M.Westervelt,P.F.Hopkins,andA.C.Gossard, Chaos,3,643(1993).

[10]“CoulombblockadeoftheAharonovͲBohmeffectinGaAs/AlxGa1ͲxAsquantumdots” –J.P.Bird,K.Ishibashi,M.Stopa,Y.Aoyagi,andT.Sugano,Phys.Rev.B,50,14983 (1994).

[11]“MagnetoͲCoulomboscillations”–J.P.Bird,K.Ishibashi,M.Stopa,R.P.Taylor,Y. Aoyagi,andT.Sugano,Phys.Rev.B,49,11488(1994).

[12]“Spectralcharacteristicsofconductancefluctuationsinballisticquantumdots:The influenceoffinitemagneticfieldandtemperature”–J.P.BirdandK.Ishibashi,D.K. Ferry,Y.Ochiai,Y.AoyagiandT.Sugano,Phys.Rev.B,52,8295(1995).

[13]“LeadͲinducedtransitiontochaos inballisticmesoscopicbilliards”–J.P.Bird,D.M. Olatona,R.Newbury,R.P.Taylor,K.Ishibashi,M.Stopa,Y.Aoyagi,T.SuganoandY. Ochiai,Phys.Rev.B,52,R14336(1995).

[14]“PreciseperioddoublingoftheAharonovͲBohmeffectinaquantumdotathigh magnetic fields”–J.P.Bird,K.Ishibashi,Y.Aoyagi,andT.Sugano,Phys.Rev.B,53,3642 (1996).

[15]“Magnetotransportfluctuationsinregularsemiconductorballisticquantumdots”– R.Akis,D.K.FerryandJ.P.Bird,Phys.Rev.B,54,17705(1996).

[16]“Weaklocalization inballisticquantumdots”–R.Akis,D.K.Ferry,J.P.Bird,andD. Vasileska,Phys.Rev.B,60,2680(1999).

[17]“Magnetoprobingofthediscretelevelspectrumofopenquantumdots”–J.P.Bird, R.Akis,andD.K.Ferry,Phys.Rev.B,60, 13676(1999).

[18]“CorrelationanalysisofselfͲsimilarityinsemiconductorbilliards”–R.P.Taylor,A.P. Micolich,andR.NewburyandT.M.Fromhold,Phys.Rev.B,56,R12733(1997).

187

[19]“Chaoticelectrontransportinsemiconductordevices”–B.Scannell,PhDthesis, UniversityofOregon,2010.

[20]“Ga0.25In0.75As/InPquantumwellswithextremelyhighandanisotropictwoͲ dimensionalelectrongasmobilities”–P.Ramvall,N.Carlsson,P.Omling,L.Samuelson, W.Seifert,M.Stolze,andQ.Wang,Appl.Phys.Lett,.68,1111(1996).

[21]“Unifiedmodeloffractalconductancefluctuationsfordiffusiveandballistic semiconductordevices”–C.A.Marlow,R.P.Taylor,T.P.Martin,B.C.Scannell,H.Linke, M.S.Fairbanks,G.D.R.Hall,I.Shorubalko,L.Samuelson,T.M.Fromhold,C.V.Brown,B. Hackens,S.Faniel,C.Gustin,V.Bayot,X.Wallart,S.Bollaert,andA.Cappy,Phys.Rev.B, 73,195318(2006).

[22]“Theinfluenceofconfiningwallprofileonquantuminterferenceeffectsinetched

Ga0.25In0.75As/InPbilliards”–T.P.Martin,R.P.Taylora,H.Linkea,C.A.Marlowa,G.D.R. Halla,I.Shorubalkob,I.Maximovb,W.Seifertb,L.SamuelsonbandT.M.Fromholdc, SuperlatticesMicrostruct.,34,179(2003).

[23]“TheroleofsurfaceͲgatetechnologyforAlGaAs/GaAsnanostructures”–R.P.Taylor, Nanotechnology,5,183(1994).

[24]“ProbingtheSensitivityofElectronWaveInterferencetoDisorderͲInduced ScatteringinSolidͲStateDevices”–B.C.Scannell,I.Pilgrim,A.M.See,R.D.Montgomery, P.K.Morse,M.S.Fairbanks,C.A.Marlow,H.Linke,I.Farrer,D.A.Ritchie,A.R. Hamilton,A.P.Micolich,L.Eaves,andR.P.Taylor,–inPressforPhys.Rev.B, arXiv:1106.5823.

[25]“QuantuminterferenceeffectsasatooltoprobethesidewallsofsubͲmicrometreͲ sizen+GaAschannels”–R.P.Taylor,P.C.Main,L.Eaves,S.Thoms,S.P.BeaumontandC. D.W.Wilkinson,Can.J.Phys.,70,979(1992).

[26]“Potentialfluctuationsinheterostructuredevices”–J.A.NixonandJ.H.Davies, Phys.Rev.B,41,7929(1990).

[27] “Negativedifferentialconductanceinquantumwaveguides”–C.Berven,M.N. Wybourne,A.EckerandS.M.Goodnick,Phys.Rev.B,50,14639(1994).

188

[28]“PostͲgrowthdiffusionofSiindeltaͲdopedGaAsgrownbyMBE”–R.B.Beall,J.B. Clegg,J.Castagn`e,J.J.Harris,R.Murray,andR.C.Newman,Semicond.Sci.Technol.,4, 1171(1989).

[29]“Influenceofrapidthermalannealingonmodulationdopedstructures”–D.Inoue, S.Matsushita,K.Matsumura,M.Sawada,K.Yodoshi,andY.Harada,SolidͲState Electronics,ProceedingsoftheTopicalWorkshoponHeterostructureofMicrolectronics, 41,1475(1997).

[30]“Deepdonorlevels(DXcenters)inIIIͲVsemiconductors”–P.M.Mooney,J.Appl. Phys.,67,R1(1990).

[31]“LargeͲLatticeͲRelaxationModelforPersistentPhotoconductivityinCompound Semiconductor”–D.V.LangandR.A.Logan,Phys.Rev.Lett.,39,635(1977).

[32]“TrappingcharacteristicsandadonorͲcomplex(DX)modelforthepersistentͲ photoconductivitytrappingcenterinTeͲdopedAlxGa1ͲxAs”–D.V.Lang,R.A.Loganand M.Jaros,Phys.Rev.B,19,1015(1979).

[33]“GiantbackͲscatteringresonancesinedgeͲstatetransportthroughquantumdots” –J.P.Bird,M.Stopa,K.Connolly,D.P.Pivin,Jr.,D.K.Ferry,Y.AoyagiandT.Sugano, Phys.Rev.B,56,7477(1997).

[34]“Magnetotransportfluctuationsinregularsemiconductorballisticquantumdots”– R.AkisandD.K.FerryandJ.P.Bird,Phys.Rev.B,54,17705(1996).

[35]“Phasebreakingofnonequilibriumelectronsinaballisticquantumdot”–H.Linke,J. P.Bird,J.Cooper,P.Omling,Y.AoyagiandT.Sugano,Phys.Rev.B,56,14937(1997).

[36]“PreciseperioddoublingoftheAharonovͲBohmeffectinaquantumdotathigh magneticfields”–J.P.Bird,K.Ishibashi,Y.Aoyagi,andT.Sugano,Phys.Rev.B,53,3642 (1996).

[37]“SignaturesofthediscretelevelspectrumintemperatureͲdependenttransport throughopenquantumͲdotarrays”–A.Shailos,C.Prasad,M.Elhassan,R.Akis,D.K. Ferry,J.P.Bird,N.Aoki,L.ͲH.Lin,Y.Ochiai,K.IshibashiandY.Aoyagi,Phys.Rev.B,64, 193302Ͳ1(2001).

189

[38]“QuantumTransportinOpenMesoscopicCavities”–J.P.Bird,K.Ishibashi,Y. Aoyagi,T.Sugano,R.Akis,D.K.Ferry,D.P.PivinJr,K.M.Connolly,R.P.Taylorc,R. Newbury,D.M.Olatona,A.Micolich,R.Wirtz,Y.OchiaiandY.Okubo,Chaos.Solitons& Fractals,8,1299(1997).

[39]“LeadͲOrientationͲDependentWaveFunctionScarringinOpenQuantumDots”–J. P.Bird,R.Akis,D.K.Ferry,D.Vasileska,J.Cooper,Y.AoyagiandT.Sugano,Phys.Rev. Lett.,82,4691(1999).

[40]“Recentexperimentalstudiesofelectrondephasinginmetaland semiconductor mesoscopicstructures”–J.J.LinandJ.P.Bird,J.Phys.:Condens.Matter,14,R501 (2002).

[41]“StorageconditionsforhighͲaccuracycompositionstandardsofAlGaAs”–K.A. Bertnessa,A.Roshko,S.E.AsherandC.L.Perkins,J.Vac.Sci.Technol.B,23,1267(2005).

[42]“MethodforCharacterizingChargeSpreadinginSinteredAluminabySecondary ElectronEmission:EffectofMicrostructureandTemperature”–K.Zarbout,A.SiAhmed, G.Moya,G.Damamme,J.BernardiniandA.Kallel,AnnualReportConferenceon ElectricalInsulationandDielectricPhenomena,457(2007).

[43]“DistinguishingimpurityconcentrationsdinGaAsan AlGaAsusingveryshallow undopedheterostructures”–W.Y.Mak,K.DasGupta,H.E.Beere,I.Farrer,F.Sfigakis, andD.A.Ritchie,Appl.Phys.Lett.,97,242107(2010).

[44]“RoleofbackgroundimpuritiesinthesingleͲparticlerelaxationlifetimeofatwoͲ dimensionalelectrongas”–S.J.MacLeod,K.Chan,T.P.Martin,A.R.Hamilton,A.See,A. P.Micolich,M.Aagesen,andP.E.Lindelof,Phys.Rev.B,80,035310(2009).

45[45]“PeriodicScarredStatesinOpenQuantumDotsasEvidenceofQuantum Darwinism”–A.M.Burke,R.Akis,T.E.Day,GilSpeyer,D.K.Ferry,andB.R.Bennett, Phys.Rev.Lett.,104,176801(2010).

46[46]“ImagingFractalConductanceFluctuationsandScarredWaveFunctionsina QuantumBilliard”–R.Crook,C.G.Smith,A.C.Graham,I.Farrer,H.E.BeereandD.A. Ritchie,Phys.Rev.Lett.,91,246803(2003).

190

[47]“BoundͲStateEigenfunctionsofClassicallyChaoticHamiltonianSystems:Scarsof PeriodicOrbits”–E.J.Heller,Phys.Rev.Lett.,53,1515(1984).

[48]“FluxͲcancellationeffectonnarrowͲchannelmagnetoresistancefluctuations”–C. W.J.BeenakkerandH.vanHouten,Phys.Rev.B,37,6544(1988).

[49]“Influenceofshapeonelectrontransportinballisticquantumdots”–M.J.Berry,J. A.Katine,R.M.WesterveltandA.C.Gossard,Phys.Rev.B,50,17721(1994).

[50]“Threekeyquestionsonfractalconductancefluctuations:Dynamics,quantization, andcoherence”–A.P.Micolich,R.P.Taylor,T.P.Martin,R.Newbury,T.M.Fromhold,A. G.Davies,H.Linke,W.R.Tribe,L.D.Macks,C.G.Smith,E.H.Linfield,andD.A.Ritchie, Phys.Rev.B,70,085302(2004).

[51]“OriginofswitchingnoiseinGaAs/AlxGa1оxAslateralgateddevices”–M.PioroͲ Ladrière,J.H.Davies,A.R.Long,A.S.Sachrajda,L.Gaudreau,P.Zawadzki,J.Lapointe,J. Gupta,Z.Wasilewski,andS.Studenikin,Phys.Rev.B,72,115331(2005).

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AppendixA–Deviceprocessingtechniques

A1–AssessmentofNBI30undopeddevices

TheOhmiccontactsinaNBI30undopeddeviceareformedusingaselfͲaligned process.AsdiscussedasSection2.3.1,themainproblemwiththeselfͲalignedtechnique is that Ohmic contacts sometimes short to the݊ାdoped cap and create leakage problems.Inordertoidentifyworkingsamplespriortoexperiments,devicesareoften assessedbyimmersioninliquidheliumattemperature̱4.2K.FigureA1(a)showsa schematicofatypicalundopeddevicewherethe“TͲgate”providesanelectricalcontact tothedopedcap (topgate).ToassesstheOhmics,firstwegroundedoneoftheOhmic contactswhilekeepingtheothersfloated.Atopgatebias(்ܸீ)wasthenappliedviaa

Keithley K2400 source meter while the leakage currentܫ௟௘௔௞was monitored. Figure

A.1(b)showsܫ௟௘௔௞vs.்ܸீforthecorrespondingOhmicslabelled“2”,“20”,“18”and“14”

FigureA1:(a)SchematicofaNBI30devicebeingassessed.Theblueregioncorrespondsto theHallbarmesawherethesurfaceisadegenerately݊ାdopedcap(topgate).Theyellow regionsrepresentTi/AudepositedtomakeelectricalcontactstotheOhmicsandtopgate.(b)

Leakagecurrentܫ௟௘௔௞vs.topgatevoltage்ܸீcorrespondstotheOhmiccontactslabelledin (a). The top gate bias was supplied by a K2400 Keithley source meter, whileܫ௟௘௔௞was monitoredbysettinganOhmiccontacttogroundandkeepingtheothersfloating.Thedata in(b)suggestthatOhmic#2leaksat்ܸீ̱0.1V,whiletheothersturnonat்ܸீ=0.3– 0.35VandthecorrespondingcurrentsfollowthesamecurrentpathsasOhmic#2athigher

்ܸீ.

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in(a).Fromthedata,weobservethatOhmic#2startedtoleakat்ܸீ̱0.1V.Inthe case where all Ohmic contacts are connected via the 2DEG, when we performed the sameleakagetestonOhmic#20,#18and#14,assoonassufficientelectrondensityat the2DEGwasobtained,theOhmics“turnedͲon”andallowedcurrentstoflowtothetop gateviaOhmic#2.ThiscanbeobservedinFig.A1(b)wheretheܫvs.்ܸீtracesoverlap for்ܸீ>0.35V.Ontheotherhand,iftheOhmiccontactsdidnotsufficientlydiffuse intotheAlGaAs/GaAsheterojunction,theywillnotturnonandnoleakagecurrentwas observed.Ideallyforanundopeddevice,wewanttheOhmicstoturnonatlow்ܸீ,and onlyleakathigh்ܸீ.Thetypicallyoperatingrangeforourundopeddevicesisbetween 11 Ͳ2 ்ܸீ=0.4to1.1V,whichcorrespondstoanelectrondensityof0.3to2.67ൈ10 cm .

A2–MethodsforremovingleakingOhmiccontacts

IfanOhmiccontactfromaNBI30devicewasfoundleakingtothetopgateprior toturningon,wecanremoveitandseeiftherestofthedevicestillwork.Toachieve this,ascribercanbeusedtocuttheleakingOhmicsoffsothattheresteofth devicecan still be used. However, it is not possible to locate the initial scribing point precisely enoughusingthemicroscopeattachedtothescribersystembecausethediamondtipis initsway.Idevelopedasolutiontothisproblem,whichwastouseasecondmicroscope

Figure A2:(a) Scriber setup used to scribe leaking Ohmics off the NBI30 device. The photographshowsasecondmicroscopewasusedtohelplocatingtheinitialscribingpoint. (b)ImageofanunodpeddeviceafterscribingshowingoneleakingOhmiccontactsonthe left,andthreeoftheright.

193 pointingatthesamplefromtheside.FigureA2(a)showsaphotographofthescriber plustheextramicroscope,whileanimageofanNBi30chipwithitsleakingOhmiccut offisshownin(b).AnothermuchmoreaccuratewaytocuttheleakingOhmicoffisto performachemicaletch.Toproceed,wefirstspinalayerofpositivephotoresistsuchas S1813,andthenwemakeuseofthevariablesizeapertureintheopticalmicroscopeto target the arm of a leaking Ohmics and expose it for 20 sec at full intensity. This is followed by standard development and etching processing steps. Figure A3 shows a photographofthismethodafterdevelopment.

FigureA3:AphotographofanexposedholeinaphotoresistalignedtotheHallbararmtoa leaking Ohmic contact. The exposure is performed using an optical microscope with a variableaperture.TheOhmicisseveredusingaHCletch.

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