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Tunneling between two dimensional electron systems in a high magnetic field and crystalline phases of a two dimensional ...

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TABLEOFCONTENTSP age ACKNOWLEDGMENTS........................ii ABSTRACT..............................v CHAPTER 1INTRODUCTIONTOTHEQUANTUMHALLSYSTEM........1 1.1HistoryoftheQuantumHallEect....................1 1.2BilayerQuantumHallPhysics:ExperimentandTheory.........3 1.3CrystallinePhasesofthe2DElectronSystem:ExperimentandTheory9 2TUNNELINGCURRENTOFCOUPLEDBILAYERWIGNERCRYSTALS12 2.1SingleLayerEigenmodes..........................12 2.2BilayerEigenmodes.............................15 2.3CouplingtheBilayertoExternalElectrons:TunnelingCurrent.....18 2.3.1AnalyticSolution..........................23 2.3.2NumericalSolution.........................29 3QUANTUMHALLSYSTEMINTHEHARTREE-FOCKAPPROXIMATION32 3.1ElectronDynamicsinaPerpendicularMagneticField..........32 3.2Hartree-FockApproximation........................37 4ISOTROPICCRYSTALLINEPHASES................39 4.1StabilityAnalysisofIsotropic M -electronBubbleCrystals.......39 4.1.1ClassicalOrderParameterApproach...............44 4.1.2MicroscopicApproach.......................48 4.1.3NewState:BubbleCrystalwithBasis..............53 4.1.4NormalModesandZeroPointEnergy..............56 4.2EnergeticsofIsotropicCrystallinePhases.................59 4.2.1CohesiveEnergyofModiedCoulombInteraction:Classical Model................................61 4.2.2CohesiveEnergyofModiedCoulombInteraction:Microscopic Model................................62 iii

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5ANISOTROPICCRYSTALLINEPHASES..............66 5.1SolvingtheStaticHartree-FockEquation.................66 5.2IntroducingAnisotropyintotheCrystallineStates............69 5.3ElasticPropertiesofAnisotropicCrystals.................77 5.4AnalysisofExperimentalResults.....................80 6CONCLUSIONS..........................83 APPENDIX ABILAYERSYSTEMEIGENMODES.................86 A.1SingleLayerEigenmodes..........................86 A.2BilayerEigenmodes.............................93 A.3TunnelingCurrent.............................98 A.3.1CorrelationFunction........................100 A.3.2PropertiesoftheCorrelationFunction..............102 BISOTROPICCRYSTALS......................105 B.1FockTermCalculation...........................105 B.2MicroscopicPotential............................108 B.3BubblewithBasisDynamicalMatrix...................108 B.4NormalModes................................110 CBUILDINGTHESTATICHARTREE-FOCKEQUATION.......114 REFERENCES............................119 BIOGRAPHICALSKETCH......................124 iv

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy TUNNELINGBETWEENTWODIMENSIONALELECTRONSYSTEMSINA HIGHMAGNETICFIELDANDCRYSTALLINEPHASESOFATWO DIMENSIONALELECTRONSYSTEMINAMAGNETICFIELD By FilipposKlironomos May2005 Chairman:AlanT.Dorsey MajorDepartment:Physics WestudythebilayerquantumHallsystemintheincoherentregime.We modelthetwolayersascorrelatedWignercrystalsduetothepresenceofinterlayer interactionsandtakethecontinuumlimitthattreatsthesystemasanelasticmedium. Usingthisapproach,wendananalyticsolutionforthecollectivemodesofthesystem andcalculatethetunnelingcurrentassociatedwithexternalelectronscoupledtothese modes,reproducingexperimentalresults.Investigationoftheroleofinterlayerinteractionsintotheresponseofthesystemrevealsadualnature:theyintroducean excitationgapinthecollectivemodesandalsosoftentheeectofintralayerinteractions. Wefurtherstudythecollectivestatesformedbythe2DelectronsatlowLandau levelsbyworkingfromasemi-classicalandmicroscopicperspectiveandevaluating theelasticmoduli,normalmodes,andzero-pointandcohesiveenergiesofthedifferentcrystallinestructures.TheeectsofscreeningfromlledLandaulevelsand nitethicknessofthesamplearefoundnottoinuencetheoverallinterplayofthe phases.Whenprobingtheinternaldegreesofthecrystallinestructures,theenergy v

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Figure1.1:DissipativeandHallresistivitiesofaGaAssample.ReprintedfromStormer,PhysicaB177,401(1992).Copyright(1992),withpermissionfromElsevier.theLandaulevelsplitting,givingrisetogaplesssingleparticleexcitations.IntheFQHE,thephysicsisofamanybodynature.Theelectronsformanincompressiblegroundstatewithagap(whichissmallerthantheIQHE)duetotheirinteractionsthatbecomedominantwhentheirkineticenergyisquenchedbytheappliedmagneticeld.Thegroundstatecanbeaccuratelydescribed(inthesymmetricgauge)byLaughlin'swavefunction[4](z1;;zN)=NYi>j=1(zizj)1 4NPi=1jzij2;(1.1)where=N

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Figure1.2:Symbolicgraphofabilayerstructureofinterlayerdistanced.TheelectronscomingfromtheSidonorsaretrappedintheGaAs-AlGaAsinterfaceandformthe2Delectrongas.havebeendevelopedbasedonLaughlin-likewavefunctions[5,6],microscopiceld-theoreticaltreatments[7],orcompositefermiontheory[8],whichhavebeenquitesuccessful.1.2BilayerQuantumHallPhysics:ExperimentandTheoryHighlyinterestingandintriguingphysicsarisesiftwo2DESarebroughtwithinanonzeroseparationdistanced.Experimentallythesestructurescanbegrownbymolecular-beamepitaxywheretwosemiconductors(usuallyGaAsanddopedAlGaAs)formaquantumwellattheirinterface(100-1000Awide)whereelectrons,comingfromSi-dopedlayersoccupy,formthe2Delectrongas.WhenanundopedAlGaAsinterfaceseparatesthetwoquantumwellsbyadistanced,thentheratiod=`Bbecomesameasureoftheinterlayerinteractionstrength.WeshowinFig.(1.2)aschematicgraphofthebilayerstructure.Whatmakestheseheterostructuressointerestingisthattheyexhibit\forbidden"QHplateaus.InreportedexperimentsbySuenetal.[9]andEisensteinetal.[10],thebilayersystemexhibitsplateausattotalllingfactorsT=1andT=1=2.Thisisa\violation"oftheodddenominatorconstraintforthe

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Figure1.3:PhasediagramatT=1ofinterlayerCoulombinteractionstrengthvssingleparticletunnelingstrength.EnergymeasuresinunitsoftheintralayerCoulombinteraction.SolidsymbolsindicatesamplesshowingQHEbehavior,opensymbolsdenotethosethatdonot.Reprintedinsetofgure1withpermissionfromS.Q.Murphy,J.P.Eisenstein,G.S.Boebinger,L.N.Pfeier,andK.W.West,Phys.Rev.Lett.72,728(1994).Copyright(1994)bytheAmericanPhysicalSociety.FQHE.ThesenewFQHEstatesareattributedtotheextradegreeoffreedom(thelayerindex)eachelectronpossesses;andtothefactthatthespin-polarizationofthe2DESisrelaxed,leadingtospin-textures[11].Yoshioka,MacDonaldandGirvin[12]haveproposedthesocalled3;3;1state[13]asacandidategroundstateforadoublelayerFQHEatT=1 2.ThisisaLaughlin-likestatewhichintroducescorrelationsamongtheelectronsinthetwolayersandkeepsthemfromoccupyingthesamepositioninthe2Dplanes,asiftheywerelyingonthesameplane.Asexperimentaldataindicate,thebilayerstructureshaveaninterestingphysicalbehaviorthatorig-inatesfromtheinterplayoftheintralayerCoulombelectroninteraction(withinthesamelayer)andtheinterlayerCoulombelectroninteraction(betweenthetwolayers).AphasediagramfortheQHE,experimentallyproducedbyMurphyetal.[14],isshowninFig.(1.3).ThestrengthofinterlayerCoulombinteraction,relativetotheintralayerone,isplottedasafunctionofthetunnelingstrengthSAS(measuredinthesameunits)thatdeterminestheenergydierenceofasingleelectronassociatedwith

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Figure1.4:Zerobiaspeakanomalyinthetunnelingconductanceofabilayersystem.Leftpanel:TunnelingconductancedI dVvsinterlayervoltageVatT=1.Rightpanel:TemperaturedependenceofthezerobiastunnelingconductanceatT=1athighandlowdensities.Reprintedgures1and3withpermissionfromI.B.Spielman,J.P.Eisenstein,L.N.Pfeier,andK.W.WestPhys.Rev.Lett.84,5808(2000).Copyright(2000)bytheAmericanPhysicalSociety.symmetricorantisymmetricoccupationofthebilayerquantumwellsystem.WhatisastonishingisthattheQHEpersistsevenwhenSASapproacheszero,inotherwordswhentunnelingisturnedo.Thephaseboundaryintersectstheverticalaxisatanonzerovalued=lB'2.Thissigniestheonsetofcorrelationeectsbetweenthetwo2DES.Ontheotherhand,whenstrongtunnelingispresent,electronstunnelbackandforthrapidly,assumingsymmetricstateswithrespecttothetwolayers.Correlationsarenotimportantinthislimitandthesystembehavesasasingle-layer2DES,wheretheelectronsareconnedinawiderquantumwell.Ifd=lBbecomeslargethough,theantisymmetric(andmorelocalizedinindividualwells)statebecomesfavorableagaindestroyingSASandtheQHEaltogether.Thissuggestsaquantumphasetransitionfromanincompressibletoacompressiblestate.Furtherinvestigationintothebilayerstructureshasrevealeddirectevidenceofthisinterlayercoherence.InanexperimentconductedbySpielmanetal.[15]measuringthetunnelingconductanceinGaAs/AlxGa1xAsdoublequantumwellheterostructures,awellpronouncedfeatureappearsattotalcarrierdensitiesofNT

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Figure1.5:EvidenceofcoherenceinabilayerquantumHalldragexperiment.Leftpanel:ConventionalandCoulombdragresistancesofalowdensitydoublelayer2DES.TraceAistheconventionallongitudinalresistanceRxxmeasuredwithacurrentinbothlayers.TraceBistheHalldragresistanceRxy;D.TraceCisthelongitudinaldragresistanceRxx;D,andtraceDistheHallresistanceRxyofthesinglecurrent-carryinglayer.Rightpanel:CollapseofT=1Halldragquantizationandsecondh=e2plateauinRxy.Reprintedgures1and2withpermissionfromM.Kellogg,I.B.Spielman,J.PEisenstein,L.N.Pfeier,andK.W.WestPhys.Rev.Lett.88,126804(2002).Copyright(2002)bytheAmericanPhysicalSociety.introducingthepossibilitythatwayforelectronstoformacoherentstatewherebothquantumwellswillbesymmetricallyoccupied.MostrecentlyaquantumHalldragexperimentconductedbyKelloggetal.[22]gavedirectevidenceofcoherenceinthetransportpropertiesofthebilayersystem.Inthatexperiment,anexcitationcurrentwasdriventhroughonelayer,andaHallvoltagedropwasreportedontheotherlayer.ThesamequantizationofHalldragwasobserved,evenwhentherolesofthetwolayerswereinterchanged.ThesignoftheHalldragwasthesameasintheconventionalHalleect.However,thelongitudinaldragvoltagewasoppositeinsigntothelongitudinalresistivevoltagedropinthecurrent-carryinglayer.Thissigndierencesigniestheforcebalancebetweenthelayers,resultingfromtheconstraintthatnocurrentowisallowedinthedraglayer.RemarkablythesephenomenaareobservedattotalllingfactorT=1.ThereisaquantumHallplateauandacorrespondingvoltagedropevenwhenacurrentisdriven

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Figure1.6:MicrowaveresonanceresponseofaquantumHallsystem.Leftpanel:Realpartofxxvsfrequencyffordierentllingfactorvalues(osetforclarity).Theinsetisreproductionofselectivellingfactorvaluesandatanexpandedscale.Rightpanel:Peakfrequencyvsllingfactorforthetworesonancesshowntocoexistontheleftpanel.Reprintedgures1and3withpermissionfromR.M.Lewis,YongChen,L.W.Engel,D.C.Tsui,P.D.Ye,L.N.Pfeier,andK.W.WestPhys.Rev.Lett.93,176808(2004).Copyright(2004)bytheAmericanPhysicalSociety.interlayerinteractions.Ourapproachhasbeenrewardingbecauseithasprovidedinsightintothedualroleoftheinterlayerinteractions.Wehavestudiedonlytheincoherentregimeofthesystem,butthiskindofsystematicmodelinghaspavedthewayforlaterattemptstoincludecoherenceandreproducethefascinatingI-Vresponseshownearlier.1.3CrystallinePhasesofthe2DElectronSystem:ExperimentandTheoryThepreceedingintroductionintothephysicsofquantumHallsystems(whetherbilayerorsinglelayer)shouldhaveconvincedthereaderthatsystemslikethesehavearichvarietyofstatesorphasesthatcanpotentiallymanifestthemselves,dependingonthedierentparametervaluesassociatedwithsuchsystems(suchasdisorder,carrier

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2m_u2e_uA(u) 2n0Zd2r0[ru(r)][r0u(r0)]e2

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2m_u2T+1 2m_u2L+1 2m!c[_uTuL_uLuT]1 2m!2Lu2L1 2m!2Tu2T;(2.4)whereweusetherealeldpropertyu(q)=u(q)andtheconventionjuj2u(q)u(q).Weseethatthemagneticeldentersinthedynamicsonlythroughthethirdtermwhichmixesthetransverseandlongitudinalmodesasexpected.Forthecyclotronfrequencywehave!c=eB=mwhilethelongitudinalandtransversezeromagneticeldeigenfrequenciesarerespectivelygivenby!L=r 2!2c+!2T+!2Lq

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2K n0(uAuB)2n0Zd2r0e2[ruA(r)][r0uB(r0)] 4p

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n0=e2=4d l2;(2.12)whereisdimensionlessandl=p 2u;(2.13)uB=v+1 2u;(2.14)whichturnouttobetheeigenmodesofit.Asaresult,thetwocoupledsinglelayerdynamicsofEq.(2.11)decomposetouncoupled\eectivesinglelayer"dynamicsofin-phaseandout-of-phasenature.Westartedwithasystemofatotaloffourmodes,soweexpecttwoin-phaseandtwoout-of-phasemodesasaresult.DetailsofthecalculationaregivenintheAppendix.Theresultforthetwoout-of-phase

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2!2c+2T+2Lq mn0;(2.16)2L=c2Lq2+2K mn0+e2n0 2!2c+O2T+O2Lq

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qiquA(q)eqduB(q):(2.22)Inotherwords,thecouplingterminthebilayersystemassociatedwithanindependentelectroninjectedintothebulkofeitherofthetwoquantumHallsystemshastheformHcoupling=cyAcAZd2q 21edquLcyBcBZd2q 21edquL:(2.24)Usingtheanalyticexpressions,derivedintheprevioussection,fortheoperatorformofthein-phaseandout-of-phasemodesweshowintheAppendixthattheabovecouplingtermcanbewrittenintermsofcreationandannihilationoperatorsofthebilayerquantumHalleigenmodesasHcoupling=cyAcAi4Xs=1MsAaysas+cyBcBi4Xs=1MsBaysas;(2.25)

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[+x(1ex)]3=21

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mn0c2Tq20= (lq0)2e2

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+x(1ex)1

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2r+2 2:(2.49)Also,itisconvenienttoswitchfrequency(!)tobiasvoltage(V)(measuringinmV)byintroducingthechange!=eV=1000~.ThatwaytheargumentofC(!)willmeasureinmVwhilethecorrespondingAnsatzparameterintheexponentofEq.(2.45)willacquiretheform=e 2Ne 2r+1 2=2;(2.50)1 2Ne 2r+2 2=2c1;(2.51)1 2Ne 2r+3 2=2c2+c21:(2.52)WecandivideEq.(2.51)andEq.(2.52)byEq.(2.50)toget=e 2 22;(2.53)=e 2 2;(2.54)orequatingthetwo2r+2 2 2=r+1 2

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Figure2.1:TunnelingcurrentcurvesfordierentmagneticeldvaluesusingthemomentexpansionsolutionofEq.(2.41).ThelegendshowsthepeakbiasvaluescalculatedbyEq.(2.57).tunnelingcurrentcorrelationfunctionassumestheformC(V)=Ne

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Figure2.2:NormalizedtunnelingcurrentcurvesfordierentinterlayerseparationdistancesdmeasuredinA.tothestablehexagonallatticeconguration(n0=2=p

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Figure2.3:TunnelingcurrentcurvesfordierentmagneticeldvaluesproducedbynumericallyintegratingEq.(2.41).Thelegendshowsthepeakbiasvaluesobtainedwiththisapproach.Theyareinstrongagreementwiththeanalyticresults.inthefollowingdimensionlessformzC(z)=Z10dxf(x)!(x)

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2m(^pxeBy)2+1 2m^p2y(x;y)=E(x;y):(3.1)Sinceinthisgaugechoice,themagneticelddoesnotfullycouplethetwodirections,aplanewavesolutionisexpectedinoneofthem(x-direction)resultinginawavefunctiondecouplingoftheform:(x;y)=eikxx(y).Thiskindofdecouplingproducesadisplacedharmonicoscillatorequationforthey-directiongivenby^p2y 2m!2c(yY)2(y)=~!cn+1 2(y):(3.2)32

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2`2HnyY `;(3.3)wherenisthesocalledLandaulevelindex,associatedwithkineticenergyexcitationsofthenon-interactingelectrons,andHn(x)istheusualHermitepolynomialsofordern.Theeectoftheappliedmagneticeldistoquenchthekineticenergyoftheelectronsinthe2Dsystem,whichresults(intherealsystem)inanenhancementoftheroleofinteractionsamongelectrons.Thiscanbecomeprominentathighmagneticeldvalues,wherekineticenergyexcitations(oftheorderof~!c)mightexceedthethermalenergyrange(oftheorderofkBT),andasaresultbecomeinaccessible.Thisisthemagneticeldrangethatthekineticenergybecomesirrelevant,andonlyinter-electroninteractionsaecttheenergeticsofthesystemandintroducealargeclassofhierarchicalstates,asthemagneticeldisvaried.Thedegeneracyassociatedwiththeplanewaveeigenstatesinthex-directionallowsamacroscopicnumberofelectrons(fermionicparticles)tooccupythesamekineticenergyeigenstate(eveninthenon-interactinglimit).Thespindegreeoffreedomisassumedtobefrozenatthesehighmagneticeldvalues.ForasystemwithnitelengthLxinthex-directionthedegeneracygcanbefoundtobeg=Lx 2`2= 0;(3.4)whereisthetotalareaofthesystem,and0=h=eistheuxquantum(associatedwiththequantumHallsystemwhichistwicethevalueofthesuperconductinguxquantum).ThequantummechanicaloperatorexpressionsfortheCMcoordinatesX,Y(thatenterintothedynamicsoftheelectrons)canbederivedfromtheirclassical

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i~[x;H]=^pxeBy;(3.7)^y=m^_y=m i~[y;H]=^py:(3.8)CombiningtheabovedenitionstogetherwecanderivetheCMcoordinateformsintermsoftheusualquantummechanicaloperators:^X=^x^py

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4Lnq2`2 2cyn;Ycn;Y+qx`2=Fn(q)(q);(3.14)whereLn(x)isthen-thorderLaguerrepolynomial.ThesestructurefactorhastheformFn(q)=eq2`2 4Lnq2`2 `jnmjei(nm)Ljnmj(n+mjnmj)=2r2

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2m!mn;q 2n!mn;(3.19)Parityisdeterminedfromtheexponentnm(whetheritisevenorodd).3.2Hartree-FockApproximationTobuildarealisticmodelof2DelectronsweneedtoincludetheCoulombinteractionamongthem.Sincetheinteractionisafour-fermionoperatorthereisnotmuchhopeforustodevelopanalyticresultsunlessweapproximateit.Thebest,andmostwidelyused,wayofdoingthat[38,39,52]isthroughtheHartree-FockapproximationwhichcapturesthenecessarylongandshortrangeeectsoftheCoulombinteraction.Additionally,aswehaveexplainedpreviously,weneedtoprojectthisoperatorontoagivenLandaulevel,inordertotakeintoconsiderationthepeculiardynamicsthatariseduetothepresenceofthehighmagneticeld.ThistaskisperformedsimplyintheLandaugauge,wherewecanderiveanalyticexpressionsforallthetermsinvolved.Westartbyprojectingthefour-fermionoperatoroftheCoulombinteractionbyusingtheresultofEq.(4.32).WhatwendisH=1 2Zd2rZd2r0y(r)y(r0)V(rr0)(r0)(r)=1 2Zd2q 42e2

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42e2 2Zd2q

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A;(4.1)where`isthemagneticlength,andNandAarethetotalnumberofelectronsandareaofthesample,respectively.Aswementionedpreviously,theelectronsthatbelongtothelledLandaulevelsareconsideredinert(theydon'tparticipateinthecrystallizationprocess)andatmosttheyprovidescreeningeectsfortheCoulombinteraction.Asaresult,itisusefultodistinguishbetweenthetotalllingfactor(pertainingtothewholesystem)andthepartialllingfactor(pertainingtotheactiveelectronsinthepartiallylledLandaulevel).DependinguponthetypeofM-electron39

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ABc;(4.3)wherenistheLandaulevelindex,NisthemacroscopicnumberofelectronsinthepartiallylledLandaulevel,andABc=p

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2XR6=R0U(RR0);(4.6)tosecondorderintheelectrondisplacementsaroundthelatticesitesR.OurbasictaskistodenetheelectroninteractionpotentialU(r),comingfromtheCoulombrepulsionamongelectronsbutmodiedduetothespecialdynamicsthehighmagneticeldintroduces,theHartree-Fockapproximation,andthequantumcorrectionsarisingfromthemicroscopicphysicsofthesystem.Havingaccomplishedthattask,itiseasytoshowthattheelasticenergyassociatedwithEq.(4.6)isoftheformEelastic=1 2XqU(q)XR6=R0eiq(RR0)u(R)u(R0)u(R)u(R);(4.7)whereisthetotalsampleareacomingfromtheFouriertransformofU(r).Wecanintroduceatthispoint,theFouriertransformofthediscretedisplacementeldsaccordingtothefollowingdenitionsu(R)=Zd2q

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2A2cXQZd2q 2A2cXQZd2q 2Zd2q 2q2Q2U0(Q)

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2ijkl@iuj@kul;(4.15)whereijklareelasticconstantswithonlyacertainnumberofthembeingindependentornon-zero(dependingonthegivensymmetryoftheelasticmedium).Forthetri-angularlatticecongurationtheaboveexpressionsimpliestoEelastic=1 2@iui@juj+ 2(+2)(@xux)2+(@yuy)2+2@xux@yuy+(@xuy)2+(@yux)2+2@xuy@yux:(4.16)Intheabove,theelasticconstantsandaretheLamecoecientsthataredeter-minedbythefollowingrelationsthatholdduetothetriangularlatticecongurationxyxy=yxyx=xyyx=yxxy=;(4.17)xxyy=yyxx=;(4.18)xxxx=yyyy=+2;(4.19)xxyy=xxxx2xyxy;(4.20)andtherestofthepossibleijkl'siszero.Theshearmodulusc66(associatedwiththeenergycostofsheardeformations)isgivenbyxyxy=andthebulkmodulusc1

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2qq~u(q)~u(q):(4.21)ComparingtheabovewiththeexpressionfortheelasticenergyofEq.(4.12),wendthattheelasticconstantsandthedynamicalmatrixarerelatedwiththedenitionqq=e(q):(4.22)Inotherwordsthebulkmoduluswillbedenedasc1=exx(qx^x)=q2x,andtheshearmoduluswillbegivenbyc66=exx(qy^y)=q2yorifweuseEq.(4.14)weenduptothegeneralexpressionsforthebulkandshearmodulic1= 2

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Figure4.1:Shearmodulus(inunitsofe2=4`3)asafunctionofpartialllingfactorforthelowestLandaulevelandfortheWC(leftpanel)and2eBC(rightpanel)usingtheorderparameterapproach.crystal[44]).WetreattheM-electronbubblesaspoint-likeparticlesuctuatingaroundtheirlatticesitepositionsandtheelectronsinsideabubblearetreatedasclassicalinteractingparticles.Thisallowsustodenethelocalllingfactor,inaccordancewithGoerbigetal.,as[39](r)=XR(rBjrRu(R)j);(4.25)where(r)istheHeavisidestepfunction,rBtheM-electronbubbleradius,andu(R)theM-electronbubbledisplacementaroundthelatticesiteR.Theabovechoiceoflocalllingfactorproducesacrudestep-likeapproximationforthedensityproleofthecrystallinestructure.Thedirectandreciprocallatticevectorsforthehexagonallatticesymmetryaredenedas[44]Rjj0=p a2jj0

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Figure4.2:Shearmodulus(inunitsofe2=4`3)asafunctionofpartialllingfactorforthen=1LandaulevelandfortheWC(leftpanel)and2eBC(rightpanel)usingtheorderparameterapproach.wherewehaveusedEq.(4.4)fortheM-electronBClatticeparameter.Fora2Dsampleoftotalareatheinteractionenergyassociatedwiththebubblecrystalcon-gurationintheHartree-Fockapproximationisgivenby[39]E=1 2 (2`2)2XqVHF(q)j(q)j2;(4.28)where,VHF(q)istheHartree-FockpotentialgivenbyEq.(3.24)and(q)istheFouriertransformofthelocalllingfactorwhichisfoundfromEq.(4.25)tobe(q)=4M`2

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Figure4.3:Shearmodulus(inunitsofe2=4`3)asafunctionofpartialllingfactorforthen=2LandaulevelandfortheWC(leftpanel)and2eBC(rightpanel)usingtheorderparameterapproach.thatforthebulkmoduluswendthetypicallongwavelengthsingularitycomingfromthersttermofEq.(4.23)ifwetakeintoaccounttheformofthepotentialenergyfromEq.(4.30).ThisbehaviorisinaccordancewithwellknownresultsfortheclassicalWignercrystal[44].WhatwehaveachievedsofarisproduceananalyticexpressionfortheelasticmoduliinthesemiclassicalHartree-Fockapproximationwheretheelectrongasistreatedaspointparticlesuctuatingaroundtheirlatticeequilibriumpositions.InFigs.(4.1-4.4)weplottheshearmodulusversuspartialllingfactorforLandaulevelsn=0;1;2;3andfortheisotropicWC(M=1)andtheisotropic2-electronperbubblecrystal(2e-BC)(M=2)cases,wheretheinteractionenergyisgivenbyEq.(4.30).WenoticethatinFig.(4.1)(whichcorrespondstoaWCinthelowestLandaulevel)wereproducewellknownresultsbyMakiandZotos,wheretheisotropicWCstatebecomesunstablearoundllingfactor'0:48[54].Additionally,wendthatforthen=2andn=3LandaulevelstheisotropicWCdestabilizesaround'0:24and'0:18respectively,buttheM=2isotropicBCcanliveupto'0:39and'0:31,respectively.Thisisinaccordancewithknownresults[38,39]wheretheWCbecomesunfavorableeventuallytothehierarchyofmanyelectronBC's.

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Figure4.4:Shearmodulus(inunitsofe2=4`3)asafunctionofpartialllingfactorforthen=3LandaulevelandfortheWC(leftpanel)and2eBC(rightpanel)usingtheorderparameterapproach.Theaboveclassicalapproachmanagestoreproducegeneralpropertiesforthe2Delectrongasandshowanindicationofstabilityinterplaybetweenthedierentstates,butitisunabletoadequatelycapturethequantumphysicsassociatedwiththecorrelatedelectronsystemandinparticularthefactthattheelectronwavefunctionsextendaconsiderabledistance(oftheorderofthemagneticlength)aroundtheirlatticesitepositions,whichradicallyalterstheirshortrangeinteractions(capturedbytheFockterminourmodel).Thissemiclassicalpictureignoresthatfactbyassumingastep-functiondensitypattern,localizingthepoint-likeelectronsaroundtheirequilibriumsites.Asaresult,anyfurtherattempttoinvestigatetheenergeticsofthedierentphaseswillnotbeaccurateenoughinreproducingrealisticallytheshortrangeinteractionsassociatedwithelectronwavefunctionoverlaps.Inwhatfollowsweattempttoimproveonthisapproximation.4.1.2MicroscopicApproachInordertoincorporatethequantumphysicsof2Delectronsmorefaithfullyinourmodelwehavetobuildamicroscopictheoryoftheelectronwavefunctionsandderive,toHartree-Focklevel,informationabouttheenergeticsandstabilityofthe

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2Zd2q

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Figure4.5:Shearmodulus(inunitsofe2=4`3)asafunctionofpartialllingfactorforthen=2LandaulevelandfortheWC(leftpanel)and2eBC(rightpanel)usingthemicroscopicapproach.WeshowinAppendixBtheFouriertransformsoftheaboveinteractionpotentialfordierentmvaluesandforthe2eBCcase.Theabovegeneralexpressioncanbeusedtondthecohesiveenergyassociatedwithsuchasystem,namelyEHF=1 2Xi6=jXm;m0Umm0(RiRj)+XiXm
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Figure4.6:Shearmodulus(inunitsofe2=4`3)asafunctionofpartialllingfactorforthen=3LandaulevelandfortheWC(leftpanel)and2eBC(rightpanel)usingthemicroscopicapproach.OurresultsfortheWCand2eBCinthen=2;3LandaulevelsareshowninFigs.(4.5-4.6).Wenoticethatthepartialllingfactorregionofstabilityforagivencaseremainedthesameasinoursemi-classicalresults,butapproachinghalf-llingthebehaviorhasbecomedierent.Noticethatbyincreasingthellingfactoramountstodecreasingthemagneticeldorincreasingthedensityofelectronsinthesystem(withtheeectofreducingthecrystallatticeparametervalue,ascanbeclearlyseenbyEq.(4.4)).Thisparameterchangerenderstheshortrangephysicsmorerelevantandtheireectsmorewellpronounced.Consequently,weseeforexampleinFig.(4.3)thatthen=2isotropicWCwillbecomereentrantclosetohalfllingaccordingtothesemi-classicalmodelbutinFig.(4.5)(whereshortrangephysicsisaccountedforbythemicroscopicmodel)thisneverhappens.Thisistobeexpectedsinceaswementionedearlier,crystallizationisimplementedbythedirectlongrangetermintheCoulombinteractionwhileconglomerationisduetotheshortrangeexchangeterm.Thesemi-classicalmodelfavors(byconstruction)theformer,whilethecurrentmicroscopicmodelattemptstoincorporatethequantumphysicsofwavefunctionoverlaps,whichaectdramaticallytheimportanceofthelatterterm.Anotherexhibitionofthis,is

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Figure4.7:InteractionpotentialU01(r)(inunitsofe2=4`)vs.r(inunitsof`).thecharacteristicnon-monotonicbehaviorthatisobservedintheshearmoduliwhichsignalstheonsetofdominanceoftheexchangeterm.4.1.3NewState:BubbleCrystalwithBasisAswementionedearliertheinternaldegreesoffreedominabubblehavebeenconsideredhigherordercorrectionstothephysicsinvestigatedinthiswork.Thismightnotnecessarilybetrue,sincewehavenotsystematicallyprobedonthosedegreesoffreedomduetothedicultysuchataskpresentswhentreatedinamicroscopiclevel.Nevertheless,asapreliminaryattemptofinvestigation,wecanoerthefol-lowingspecialcasewhichcanbeeasilyincorporatedintothecurrentmodel.Wecanfocusourattentionontheisotropic2eBCcase,andallowthetwoelectronstoassumeanitedistancefromoneanotherwithinthesamebubble.Thiscanserveasarudimentaryapproximationforinternalstructure.Thepossibilityofsuchastatearises,ifoneplotstheinteractionpotentialbetweenthesetwoelectronswithinabubble(U01(r)givenbyEq.(4.40)forthem=0,m0=1case).TheFouriertransformofU01(r)isshowninEq.(B.21)oftheAppendix.Surprisingly,thereisawellpronouncedlocalminimumataninter-electrondistanceofr0'1:48`(asshown

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2)r0^x;m=0;1;(4.42)whereRiaregivenbyEq.(4.26)andmdistinguishesbetweenthetwoelectrons.InordertostudythestabilityofthiscrystallinestructureweneedtoderiveanexpressionforthedynamicalmatrixstartingfromthecohesiveenergygivenbyEq.(4.41)andTaylor-expandingtosecondorderinthedisplacements.Asalways,wedefertotheAppendixallthecumbersomedetailsandpresentherethenalresultemm0(q)=1 2Xmm0emm0xx(q);(4.44)

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Figure4.8:Shearmodulus(inunitsofe2=4`3)asafunctionofpartialllingfactorforthen=2(leftpanel)andn=3(rightpanel)Landaulevelforthebubblecrystalwithbasis.insmallqandwereproducethesameresultofEq.(4.24)butwiththeinteractionpotentialU(q)givenbyU(q)=1 2Xmm0eiqxr0(mm0)Umm0(q)=VHF(q)eq2`2=21+cos(qxr0)1 21+cos(qxr0)q2`2+1 8q4`4:(4.45)IntheAppendixweshowanalyticexpressionsfortherstandsecondderivativeoftheabovepotential.WepresentourresultsinFig.(4.8)fortheshearmodulusoftheBCwithbasisforthen=2;3Landaulevels.Wenoticethatthecrystallinestructureappearsquitestable.Thisisarstindicationthattheinternaldegreesoffreedommightplayanimportantroleinthephysicsofthe2Delectronsystem.ThisBCwithbasisstateisnotasolutionoftheHartree-Fockequation(contrarytotherestoftheM-electronstates[38]).ItarisesasthevariationalsolutionthoughoftheHartree-FockHamiltoniansatisfying@Ecoh(r0)

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dtumi:(4.49)ThisequationcoversboththeWCand2eBCcasessincetheindicesm,m0distinguishbetweenelectronsinthesamebubble.FortheBCwithbasisonehastorepeattheprocedurefromthebeginningstartingfromEq.(4.41)butusingthelatticevectorsofEq.(4.42)onlytondthefollowingexpressionforthedynamicalmatrixemm0(RiRj+r0(mm0)^x)=mm0ijXm00Xkmm00(Rk+r0(mm00)^x)mm0(RiRj+r0(mm0)^x);(4.50)whereisgivenbyEq.(4.48),andasweseetheexpressionreducestothegeneraloneofEq.(4.48)inther0!0limit.TheequationofmotionforthetwoelectronsinthiskindofBCbecomesmd2 dtumi:(4.51)WeshowintheAppendixingreatdetailhowtosolvetheaboveequationsofmotionandderivethenormalmodesforallthecrystallinestructuresofinterest.WeshowourresultsforthethreedierentisotropiccrystallinecongurationsevaluatedontheirreducibleelementoftherstBrillouinzoneinFig.(4.9).Ingeneral,thereisagaplessmode(magnetophonons)forlongwavelengthexcitationsandthereisagappedmode(magnetoplasmons)(oftheorderof!c)associatedwithinter-Landau

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Figure4.9:Normalmodesforthetriangularcrystallinestructuresinthen=2LandaulevelandontheirreduciblerstBrillouinzoneelement.Topleftpanel:WCat=0:18.Toprightpanel:2eBCat=0:30.Bottompanel:BCwithbasisat=0:20.Theleftaxescorrespondtomagnetophonons(lowercurves)andtherightaxestomagnetoplasmons(uppercurves).Frequencymeasuresin!0=e2=4~`unitsandaisthelatticecrystalparameter.levelexcitations.FortheWCcasewereproducewellknownresults[56],andboththe2eBCandBCwithbasisshowsimilarstructureintheirmodes.Theirdegreesoffreedomaredoubled(duetothepresenceofanextraelectron)whichdoublestheirmagnetophononandmagnetoplasmonmodesaswell.WecommentintheAppendixingreatdetailonthegraphicalpeculiaritiesassociatedwithplottingelementsofaBrillouinzone.Finally,onecanevaluatethezeropointenergyassociatedwiththenormalmodesofthesestatesbypicking\special"pointsinsidetheirreduciblerstBrillouinzone

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2M4`~ 91;1 91;1 91;1 9;(4.53)q4!6:2 31;1 3p 92;1 95;1 9;(4.54)whereallreciprocallatticevectorsmeasurein=aunits.InTable(4.1)wepresentourresultsforthezeropointenergyassociatedwiththedierentisotropiccrystallinestructuresinvestigatedsofar.Theonesmarkedwithanasteriskindicatetheonsetofinstabilityforthegivenstructureandthecorrespondingllingfactor.4.2EnergeticsofIsotropicCrystallinePhasesAtthispointwewouldliketoconsidertheenergeticsofthestateswhosestabilitywecalculatedintheprevioussectionsandinparticularinvestigatethepossibilityof

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Figure4.10:CohesiveenergyfortheisotropicWC(red)and2eBC(blue)inunitsofe2=4`forthen=2Landaulevel(leftpanel)andthen=3Landaulevel(rightpanel).SolidlinescorrespondtobareCoulombinteraction.DashedlinescorrespondtoCoulombinteractionwherescreeningeectsareaccountedfor,anddottedlinescorrespondtothelattercasewherenitethicknesseectsareincludedaswell.electrons(=1)separatedbyemptyareas(=0)ofnitewidth.Thecohesiveenergyassociatedwiththisphasewillbeshownbelow.4.2.1CohesiveEnergyofModiedCoulombInteraction:ClassicalModelWenumericallycalculatethebubblecrystalcohesiveenergyperelectronbasedonEq.(4.28)(inunitsofe2=4`)andgivenbyEBCcoh=1 2`2 e2`XQVHF(Q)J21(rBQ) (rBQ)2;(4.56)whereasusual,Misthenumberofelectronsperbubble,thepartialllingfactorgivenbyEq.(4.2),andrBthebubbleradius.TheneutralizingbackgroundcancelsthesingularHartreeterminvolvedintheQ=0case,butthenonsingularFocktermismaintained(theweightfactorinvolvingJ1(x)isevaluatedattheQ=0limitanditiseasytoshowitgives1=2).ThegeneralM-electronBCweinvestigatehereincorporatestheWCcaseaswellandforthedielectricconstantweusethemodied

PAGE 68

`e2

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Figure4.11:Cohesiveenergyinunitsofe2=4`fortheisotropicWC,2eBC,BCwithbasis,andstripestatefordierentmodicationsoftheCoulombinteractionandforthen=2Landaulevelusingthemicroscopicmodel.Topleftpanel:bareCoulombinteraction.Toprightpanel:screenedCoulombinteractionwithnonitethicknesseects.Bottompanel:screenedCoulombinteractionwithnitethicknesseectsincluded.thecohesiveenergyperelectron(inunitsofe2=4`):EBCbcoh= `e2

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Figure4.12:Cohesiveenergyinunitsofe2=4`fortheisotropicWC,2eBC,BCwithbasis,andstripestatefordierentmodicationsoftheCoulombinteractionandforthen=2Landaulevelusingthemicroscopicmodel.Topleftpanel:bareCoulombinteraction.Toprightpanel:screenedCoulombinteractionwithnonitethicknesseects.Bottompanel:screenedCoulombinteractionwithnitethicknesseectsincluded.whereweusedthefactthatU01(q)=U10(q).Aswementionedearlierwewouldliketoevaluatethecohesiveenergyforthestripestateaswellandcompareitwiththerestofthecrystallinestructures.ThestripestatecohesiveenergyisconstructedbystartingfromEq.(4.28),andassumingforthelocalllingfactor[36,39](r)=Xj(a=2jxxjj);(4.60)

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e2XjVHF2 aSjsin2(j)

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2Zd2q

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Figure5.1:Groundstateeigenvalueenergiesinunitsofe2=4`asafunctionofthepartialllingfactorfortheHartree-Fockequation.Leftpanel:WCand2eBCcasesforthen=2Landaulevel.Rightpanel:WC,2eBCand3eBCcasesforthen=3Landaulevel.theireigenvaluesEinascendingorder.AsaresultthecohesiveenergyofthegeneralM-electronBCassumestheformEcoh=1 2MMX=1E;(5.5)wherethefactorof1/2compensatesforcountingeachpairofelectronstwiceinthegeneralHamiltoniangivenbyEq.(C.8)[60].TheaboveisageneralresultforanM-electronBCsoitcanbeeasilyappliedtoourcasesofinterestfortheWCandthe2eBC.Additionally,wecanstudythe3eBCaswell,whichaccordingtopreviousstudies[38,39]canbecomeenergeticallyfavorableforthen=2andaboveLandaulevelsatacertainrangeofpartialllingfactors.WeshowourresultsinFig.(5.1)andasweseetheyareidenticaltotheonesdevelopedearlierwithinthemoresimpliedmicroscopicmodel.ThisisavericationthatthemicroscopicmodelusedearlierfortheM-electronBCcongurationisasolutionoftheHartree-Fockequation,inagreementwithpreviousstudies[38]wherethesolutionofthetime-dependentHartree-FockequationproducesthehierarchyofM-electronBC's.Itshouldbenotedthatfortheaboveresultswehaveusedaminimumexpansion

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Figure5.2:Leftpanel:ValuesofanisotropythatminimizethecohesiveenergiesforWCand2eBCfordierentvaluesof.Rightpanel:GroundstateenergiesassociatedwiththeminimizingvaluesofanisotropyforWCand2eBC.Wealsoplotthetraditionalstripestateforcomparison.basisdimension(Ns),disallowingtheelectronstoformhybridsbyconstrainingthemtolieontheirnon-interactinggroundstates,forallthecrystallinecasesstudied.Sothefactthewegetidenticalresultswiththesimpliedmicroscopicmodeldevelopedearlierisnotasurprisebutitservesasaconsistencycheckforthiskindofimprovedmethodbeforeweemployitforthemuchhardertaskwhichwediscussbelow.5.2IntroducingAnisotropyintotheCrystallineStatesAsweexplainedintheintroductionthecontemporarytheoreticalresultsforthe2Delectronsystemunderthepresenceofaperpendicularmagneticeldpredictthatclosetohalf-lling,thestripestate(chargedensitywave)becomesfavorabletoallthecrystallinephases.Thisstripestateisdescribedinthecontinuumorderparameterlanguage(discussedearlier)byintroducingthepartialllingfactorofEq.(4.60).Onewouldexpectthoughthatthetransitionfromacrystallinetoaliquidphasewouldbelessabrupt(especiallyatxedlowtemperatures)allowingforthecrystallinesystemtoexploreinternaldegreesoffreedombeforenallymeltingintoaliquid.Also,toacertainextent,onewouldexpectreversibility(nohysteresis)associatedwiththedecreaseorincreaseoftheappliedmagneticeldaroundtheregionthatthestripe

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1"a a2jj0

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Figure5.3:Reciprocallatticepointsfordierentvaluesofanisotropy.Weseehowthecrystalgraduallyevolvesachannel-likestructure.theWCand2eBC,whiletherightpanelshowsthegroundstatecohesiveenergiesassociatedwiththeseminimizingvaluesofanisotropy.Wealsoplotthetraditionalstripestatecohesiveenergy(givenbyEq.(4.61))andasweseeasurprisingresultemergessinceitdoesnotbecomeenergeticallyfavorableovertheanisotropicWCstateevenclosetohalf-lling.Infact,theanisotropicWCstateincreasesconsiderablyitsenergydierencefromtherestofthestates,ashalf-llingisapproached.Also,theoverallcohesiveenergieshavedroppedinvalue,comparedtotheisotropiconesfromFig.(5.1).Inviewoftheseresultsoneisobligedtoreconsiderthedenitionofstripesinthesecrystallinesystemsintermsofanisotropiccrystals,andinvestigatefurthertheeectofanisotropyinthesystem.Beforeweproceed,wewouldliketoshowhowanisotropydeformsthereciprocallatticevectorsbyplottinganitenumberofthemgivenbyEq.(5.7).OurresultsareshowninFig.(5.3),wherereciprocallatticevectorsassociatedwiththecohesive

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Figure5.4:SolutionsoftheHartree-Fockequationfortheanisotropictriangularlat-ticeconguration.Topleftpanel:ValuesofanisotropythatminimizethecohesiveenergiesforWC,2eBCand3eBCfordierentvaluesofandforthen=2Landaulevel.Toprightpanel:GroundstateenergiesassociatedwiththeminimizingvaluesofanisotropyforWC,2eBCand3eBC.Bottomleftpanel:Sameastopleftpanelbutforn=3Landaulevel.Bottomrightpanel:Sameastoprightpanelbutforn=3Landaulevel.energylatticesumandtheoverlapintegrals(givenbyEq.(C.13))areplottedforrepresentativevaluesofanisotropy.Weseeachannel-likestructureemergesconsistingofone-dimensionalperiodicchainsofelectronguidingcenters.Inotherwords,thecrystallinediscretenessofbrokentranslationalinvarianceisalwaysmaintainedinthisnovel\stripe"conguration.Thisiscontrarytothetraditionalstripestateproperties,wheretranslationalinvariancealongthestripesisrestored.Aswewillseebelow,thisisacrucialdierencethatradicallyalterstheelasticpropertiesofthesystem

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Figure5.5:Densityprolesontheunitcellforthen=2Landauleveland=0:11fortruncatedbasisdimensionalityofNs=9.Lengthunitisthemagneticlength.Topleftpanel:Isotropicstate.Toprightpanel:Anisotropic"=0:8caseusingonlythem=0non-interactingstate.Bottomleftpanel:Sameastoprightpanelbutforahybridofm=0,m=2non-interactingstates.Bottomrightpanel:OptimizedhybridstatesolutionoftheHartree-Fockequationfor"=0:8.WCcaseandthen=2Landaulevelforsimplicity.Weplaceoneelectrononeachofthethreesitesoftheunitcellandwritetheelectronicwavefunctionasanexpansionofthenon-interactingwavefunctionsm(r)givenbyEq.(3.18).Whatwendis(r)=Ns1Xm=0amm(r);(5.9)whereamarenormalizedtounitycoecientstruncatedinaHilbertspaceofNsdimensions.Itisusefultorewritetheaboveincartesiancoordinates.Forthespecic

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`2+a1 `er2=4`2+Ns3Xm=0am+2 `mLm2r2 1"a 1"a

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2c11;x(@xux)2+c11;y(@yuy)2+c66;x(@yux)2+c66;y(@xuy)2+2c11;xy(@xux)(@yuy)+2c66;xy(@yux)(@xuy):(5.15)Theelasticmodulic11;x,c11;yareassociatedwithuniformcompressionsalongthe^x,^ydirections,respectively.Todescribesheardeformationsalongthesamedirectionsweusec66;x,c66;y,respectively.Thecrosstermc11;xy,introducestheinteractionenergyassociatedwiththemixingofthecompressionmodesdirectedalong^xand^yandthesameappliesfortheshearmodemixingassociatedwithc66;xy.Intheisotropiccase,theaboveexpressionfortheelasticenergydensityassumestheformofEq.(4.16)butinthepresentcase,theonlysymmetrylefttoimposeaconstraintontheelasticconstantsisrotationalinvariance,whichimposesthefollowinginterrelationc66;xy=c66;x+c66;y

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Figure5.6:Shearmoduli(inunitsofe2=4`)fortheisotropicWCand2eBCforthen=2LandaulevelproducedusingtheMiranovicandKoganapproach[61].thetransitionpointtoanisotropicWCiscrossed(around'0:1)thec66ybecomesvanishinglysmall.ThisisbecausethiskindofsheardeformationisalongthedirectionofthechannelsshowninFig.(5.3)whichdoesnotcostanyenergy(acharacteristicpropertyofsmectics).Thestrikingdierencewithaconventionalsmecticisthatc66xbecomeszeroaswell.ThetermintheelasticenergydensityofEq.(5.15)associatedwiththatelasticconstantisreplacedbyabendingtermK(@2yux)2[62].ThisisduetothefundamentaldierencebetweentheconventionalstripestateandtheanisotropicWC:intheformer,translationalinvarianceisrestoredalongthedirectionofthestripesbutinthelatter,thisisnolongertrue.Theperiodicchannel-likestructurepersistsatanyvalueofanisotropyorllingfactor.Asaresult,adeformationoftheformu=u0y^xcorrespondstoarigidrotationforthestripestate(withnoenergycostassociatedwithit)butitcorrespondstoacompressionalongthedirectionofthechannelsfortheanisotropicWCcase,withaniteenergycostassociatedwithit.5.4AnalysisofExperimentalResultsWeareinapositionnowtodiscusstheexperimentalndingsshowninFig.(1.6).Accordingtoprevioustheoreticaltreatmentsofthedynamicalresponseofanisotropic

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Figure5.7:Leftpanel:Shearmodulusc66xforthegroundstateoftheanisotropicWCforthen=2Landaulevel.Rightpanel:Shearmodulusc66yforthesamecrystallinestructure.Noticethattheybothcoincideintherangebelow'0:1,wherethegroundstateistheisotropicWC,butitdoesnotclearlyshowinthegraphsduetothedierentscalesused.Theshearmodulimeasureine2=4`units.WCundermicrowaveirradiation[35],theresonancepinningfrequencyisgivenby!p'

PAGE 92

2m_u2e_uA(u) 2n0Zd2r0[ru(r)][r0u(r0)]e2 2m_u2T+1 2m_u2L+1 2m!c[_uTuL_uLuT]1 2m!2Lu2L1 2m!2Tu2T:(A.2)86

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2mn0!cuL;(A.3)pL=@L 2mn0!cuT:(A.4)Thenextstepinvolvesbuildingtheequationsofmotionassociatedwiththeseeldsthatrequiresusingthewell-knownformula@L dt@L 2!2c+!2T+!2Lq

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2!c[uTpLuLpT]+1 2m!2L+!2c 2m!2T+!2c

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2!c(!2c+!2T+2!2L)uL!2c+!2T 2!c(!2c+!2L+2!2T)uL!2c+!2L 2!c(!2c+!2T+2!2L)uL0!2c+!2T 2(!2+!2)!+!+[!2!2T!2c=2]uT0+!c!+ 2uL0;(A.22)

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2(!2+!2)!+!+[!2!2T!2c=2]uT0+!c!+ 2uL0;(A.23)BT=1 2(!2+!2)!![!2+!2T!2c=2]uT0!c! 2uL0;(A.24)B0T=1 2(!2+!2)!![!2+!2T!2c=2]uT0!c! 2uL0:(A.25)NoticethatAyT=AT,ByT=BT.Aftersometediouscalculationswecanprovetoourselvesusing[ui0(q);pj0(q0)]=i~ij(2)22(qq0)that[AT;BT]=[A0T;B0T]=[AT;B0T]=[A0T;BT]=0(A.26)[A0T;AT]=1=n1;[B0T;BT]=1=n2(A.27)n1=2mn0!+

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2m_u2A+1 2m_u2Be_uAA(uA)e_uBA(uB) 2n0Zd2r0[ruA(r)][r0uA(r0)]e2 2n0Zd2r0[ruB(r)][r0uB(r0)]e2 2K n0(uAuB)2:(A.45)TheeectoftheinterlayerCoulombinteractionisintroducedthroughtheshortrangeandthelongrangepartdescribedbythelasttwoterms,respectively.Decomposingthedisplacementeldsintotransverseandlongitudinalcomponents(aswedidforthesinglelayercase)wendL=n0Zd2q 2m_u2LA+1 2m_u2TA+1 2m!c[_uTAuLA_uLAuTA]1 2m!2Tu2TA1 2m!2Lu2LA+1 2m_u2LB+1 2m_u2TB+1 2m!c[_uTBuLB_uLBuTB]1 2m!2Tu2TB1 2m!2Lu2LBuLAuLB2e2n0 2K n0(uAuB)2:(A.46)

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2(2m)_v2L+1 2(2m)_v2T+1 2(2m)!c[_vTvL_vLvT]1 2(2m)(cTq)2v2T1 2(2m)(cLq)2+2e2n0 2(m 2(m 2(m 2(m mn0u2T1 2(m mn0+2e2n0 mn0=2T;(A.49)!2L!c2Lq2+2K mn0+2e2n0

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2!2c+2T+2Lq

PAGE 102

2!2c+O2T+O2Lq

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mn0;(A.72)2L'!2L+2K mn0;(A.73)O2T=!2T;(A.74)O2L'!2L+2e2(2n0)

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21edquLcyBcBZd2q 21edquL:(A.76)IfweusethedenitionsforthedisplacementeldsintermsoftheircreationandannihilationoperatorsrewritteninmorecompactformuL=if1ay1+if1a1if2ay2+if2a2;(A.77)vL=if3ay3+if3a3if4ay4+if4a4;(A.78)wheref1=!cs

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2~2(t);(A.92)

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i!jj2;(A.93)andN=1

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3LT1 12T

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2Zd2q 42e2 2Zd2q

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2Zd2q 2XpVF(p)h(p)i(p):(B.8)ThisisaHartree-likeformbutthepotentialtermassociatedwiththeCoulombinter-actionisnolongergivenbyEq.(B.2).ItisdenedinsteadasVF(p)=1

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42e2 43=2e2p 42e2 2q2`22;(B.12)VF(q)=1 43=2e2p 42e2 8q4`42;(B.14)VF(q)=1 43=2e2p 42e2 2q2`2+3 8q4`41 48q6`62;(B.16)VF(q)=1 43=2e2p

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21Xm=01Xm0=0Umm0(q);(B.18)andsinceU01(q)=U10(q)wendU(q)=1 2U00(q)+U11(q)+2U01(q);(B.19)withthefollowingdenitionsU00(q)=eq2`2=2VHF(q);(B.20)U01(q)=11 2q2`2eq2`2=2VHF(q);(B.21)U11(q)=11 2q2`22eq2`2=2VHF(q):(B.22)B.3BubblewithBasisDynamicalMatrixThedynamicalmatrixforthebubblewithbasiscrystallinestructureisevaluatedbystartingfromEq.(4.41)andexpandingtosecondorderinthedisplacements.We

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21 2Zd2q 21+cos(qxr0)q2`2+1 8q4`42q`2VHF(q)eq2`2=21+cos(qxr0)1 21+1 2cos(qxr0)q2`2+1 16q4`4;(B.25)U00(q)=V00HF(q)eq2`2=21+cos(qxr0)1 21+cos(qxr0)q2`2+1 8q4`44q`2V0HF(q)eq2`2=21+cos(qxr0)1 21+1 2cos(qxr0)q2`2+1 16q4`42`2VHF(q)eq2`2=21+cos(qxr0)1 25+7 2cos(qxr0)q2`2+1 4q4`413 4+cos(qxr0)1 16q6`6:(B.26)

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dtui:(B.28)Ifweintroduceforthedisplacementstheformui=Aei(qRi!t)wendthatthesystemofAx,Ayhasanitesolutionwhendet0B@e00xx!2e00xyi!c!e00xy+i!c!e00yy!21CA=0;(B.29)where!c=eB=mandwehavedenede00(q)=1

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a1;1

PAGE 120

2Zd2q 42e2

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2Zd2q 2Zd2rZd2r0n(r)eVHF(rr0)n(r0):(C.7)IfweusetheAnsatzofEq.(5.2)theaboveHamiltonianbecomesHHF=1 2XijMX;0=1Zd2rZd2r0j(rRi)j2eVHF(rr0)j0(r0Rj)j2;(C.8)andbyextremizingitwithrespectto(rRi)weobtaintheHartree-FockequationMX0=1XjZd2r0eVHF(rr0)j0(r0Rj)j2(r)=E(r);(C.9)where=1;;Mdistinguishesamongelectronsinsideabubbleanddierenteigenstatesaswell.UsingtheexpansionofEq.(5.3)forthequasiparticlestates,wecanprojecttheaboveequationontothenon-interactingelectronwavefunctionbasis.Additionally,usingtheorthonormalpropertiesofthelatter,wecanobtainfortheexpansioncoecientsCmthefollowingequationMX0=1Ns1Xm3=0Ns1Xm4=0Ns1Xm1=0Cm30Cm40Cm1XjZd2rZd2r0eVHF(rr0)'m3(r0Rj)'m4(r0Rj)'m1(r)'m2(r)=ECm2:(C.10)

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(M2l)!(M1M2+l)!(x)l


Permanent Link: http://ufdc.ufl.edu/UFE0009801/00001

Material Information

Title: Tunneling between two dimensional electron systems in a high magnetic field and crystalline phases of a two dimensional electron system in a magnetic field
Physical Description: vi, 124 p.
Language: English
Creator: Klironomos, Filippos ( Dissertant )
Dorsey, Alan T. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Physics thesis, Ph.D   ( local )
Dissertations, Academic -- UF -- Physics   ( local )

Notes

Abstract: We study the bilayer quantum Hall system in the incoherent regime. We model the two layers as correlated Wigner crystals due to the presence of interlayer interactions and take the continuum limit that treats the system as an elastic medium. Using this approach, we find an analytic solution for the collective modes of the system and calculate the tunneling current associated with external electrons coupled to these modes, reproducing experimental results. Investigation of the role of interlayer interactions into the response of the system reveals a dual nature: they introduce an excitation gap in the collective modes and also soften the effect of intralayer interactions. We further study the collective states formed by the 2D electrons at low Landau levels by working from a semi-classical and microscopic perspective and evaluating the elastic moduli, normal modes, and zero-point and cohesive energies of the different crystalline structures. The effects of screening from filled Landau levels and finite thickness of the sample are found not to influence the overall interplay of the phases. When probing the internal degrees of the crystalline structures, the energy is lowered considerably (which signifies that these degrees have a prominent physical importance). Finally, the static Hartree-Fock equation for the triangular lattice symmetry subset is numerically solved and anisotropy effects are taken into consideration. The emerging picture is that the isotropic Wigner crystal is favored for small values of the partial filling factor but at higher values the system undergoes a first order transition to an anisotropic Wigner crystal never crossing any other crystalline state for the rest of the filling factor range. The anisotropic Wigner crystal shows a channel-like configuration for the guiding centers of the electrons but translation invariance along the channels is never restored. As a result we find that the anisotropic Wigner crystal is more favorable even from the traditional stripe state close to half-filling, and that shear deformations along the channels become cost-free due to the vanishing shear modulus along that direction.
Subject: anisotropic, electron, Hartree, quantum, shear, tunneling, wigner
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 130 pages.
General Note: Includes vita.
Thesis: Thesis (Ph.D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

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Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0009801:00001

Permanent Link: http://ufdc.ufl.edu/UFE0009801/00001

Material Information

Title: Tunneling between two dimensional electron systems in a high magnetic field and crystalline phases of a two dimensional electron system in a magnetic field
Physical Description: vi, 124 p.
Language: English
Creator: Klironomos, Filippos ( Dissertant )
Dorsey, Alan T. ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Physics thesis, Ph.D   ( local )
Dissertations, Academic -- UF -- Physics   ( local )

Notes

Abstract: We study the bilayer quantum Hall system in the incoherent regime. We model the two layers as correlated Wigner crystals due to the presence of interlayer interactions and take the continuum limit that treats the system as an elastic medium. Using this approach, we find an analytic solution for the collective modes of the system and calculate the tunneling current associated with external electrons coupled to these modes, reproducing experimental results. Investigation of the role of interlayer interactions into the response of the system reveals a dual nature: they introduce an excitation gap in the collective modes and also soften the effect of intralayer interactions. We further study the collective states formed by the 2D electrons at low Landau levels by working from a semi-classical and microscopic perspective and evaluating the elastic moduli, normal modes, and zero-point and cohesive energies of the different crystalline structures. The effects of screening from filled Landau levels and finite thickness of the sample are found not to influence the overall interplay of the phases. When probing the internal degrees of the crystalline structures, the energy is lowered considerably (which signifies that these degrees have a prominent physical importance). Finally, the static Hartree-Fock equation for the triangular lattice symmetry subset is numerically solved and anisotropy effects are taken into consideration. The emerging picture is that the isotropic Wigner crystal is favored for small values of the partial filling factor but at higher values the system undergoes a first order transition to an anisotropic Wigner crystal never crossing any other crystalline state for the rest of the filling factor range. The anisotropic Wigner crystal shows a channel-like configuration for the guiding centers of the electrons but translation invariance along the channels is never restored. As a result we find that the anisotropic Wigner crystal is more favorable even from the traditional stripe state close to half-filling, and that shear deformations along the channels become cost-free due to the vanishing shear modulus along that direction.
Subject: anisotropic, electron, Hartree, quantum, shear, tunneling, wigner
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 130 pages.
General Note: Includes vita.
Thesis: Thesis (Ph.D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0009801:00001


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TUNNELING BETWEEN TWO DIMENSIONAL ELECTRON SYSTEMS IN A
HIGH MAGNETIC FIELD AND CRYSTALLINE PHASES OF A TWO
DIMENSIONAL ELECTRON SYSTEM IN A MAGNETIC FIELD












By

FILIPPOS KLIRONOMOS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2005














ACKNOWLEDGMENTS

I would like to thank all of my friends and family who supported me through the

difficult years of research and all of my colleagues and professors in the Department of

Physics at the University of Florida who helped through the process as well. I would

like to specially thank my supervisor, Alan Dorsey, for his mentorship and support

and Mouneim Ettouhami for his contribution to this work and for showing me the

way independent research is conducted.














TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS . . . . . . ii

ABSTRACT . . . . . . . .. v

CHAPTER

1 INTRODUCTION TO THE QUANTUM HALL SYSTEM . . 1

1.1 History of the Quantum Hall Effect ................... 1
1.2 Bilv, r Quantum Hall Physics: Experiment and Theory ......... 3
1.3 Crystalline Phases of the 2D Electron System: Experiment and Theory 9

2 TUNNELING CURRENT OF COUPLED BILAYER WIGNER CRYSTALS 12

2.1 Single L -,--r Eigenm odes .......................... 12
2.2 Bil -,-r Eigenm odes . . . . . . . .. 15
2.3 Coupling the Bil-i, r to External Electrons: Tunneling Current ..... 18
2.3.1 Analytic Solution . . . . . . 23
2.3.2 Numerical Solution . . . . . . 29

3 QUANTUM HALL SYSTEM IN THE HARTREE-FOCK APPROXIMATION 32

3.1 Electron Dynamics in a Perpendicular Magnetic Field ......... 32
3.2 Hartree-Fock Approximation . . . . . . 37

4 ISOTROPIC CRYSTALLINE PHASES . . . 39

4.1 Stability Analysis of Isotropic M-electron Bubble Crystals . .... 39
4.1.1 Classical Order Parameter Approach ... . . 44
4.1.2 Microscopic Approach . . . . . 48
4.1.3 New State: Bubble Crystal with Basis . . ... 53
4.1.4 Normal Modes and Zero Point Energy . . ... 56
4.2 Energetics of Isotropic Crystalline Phases . . . ... 59
4.2.1 Cohesive Energy of Modified Coulomb Interaction: Classical
M odel .... . . . . . . 61
4.2.2 Cohesive Energy of Modified Coulomb Interaction: Microscopic
M odel . . . . . . . . 62











5 ANISOTROPIC CRYSTALLINE PHASES . .

5.1 Solving the Static Hartree-Fock Equation .....
5.2 Introducing Anisotropy into the Crystalline States
5.3 Elastic Properties of Anisotropic Crystals . .
5.4 Analysis of Experimental Results . . .

6 CONCLUSIONS . . . . .


. . . 66

. . . 66
. . . 69
. . . 77
. . . 80

. . . 83


APPENDIX


A BILAYER SYSTEM EIGENMODES . ...

A.1 Single L-,ivr Eigenmodes . . . .
A.2 Bil v-r Eigenmodes . . . . .
A.3 Tunneling Current . . . . .
A.3.1 Correlation Function . . .
A.3.2 Properties of the Correlation Function .

B ISOTROPIC CRYSTALS . . . .


B.1
B.2
B.3
B.4


Fock Term Calculation . . . .
Microscopic Potential . . . .
Bubble with Basis Dynamical Matrix . .
Normal Modes . . . . .


. . . 86

. . . 86
. . . 9 3
. . . 98
. . . 100
. . . 102

. . . 105

. . . 105
. . . 108
. . . 108
. . . 110


C BUILDING THE STATIC HARTREE-FOCK EQUATION .

REFERENCES . . . . . .


. . 114

. . 119


BIOGRAPHICAL SKETCH . . . . . 124















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

TUNNELING BETWEEN TWO DIMENSIONAL ELECTRON SYSTEMS IN A
HIGH MAGNETIC FIELD AND CRYSTALLINE PHASES OF A TWO
DIMENSIONAL ELECTRON SYSTEM IN A MAGNETIC FIELD

By

Filippos Klironomos

May 2005

Chi iii 'i,: Alan T. Dorsey
Major Department: Physics

We study the bilayer quantum Hall system in the incoherent regime. We

model the two 1l.. ri as correlated Wigner crystals due to the presence of interlayer

interactions and take the continuum limit that treats the system as an elastic medium.

Using this approach, we find an analytic solution for the collective modes of the system

and calculate the tunneling current associated with external electrons coupled to these

modes, reproducing experimental results. Investigation of the role of interlayer inter-

actions into the response of the system reveals a dual nature: they introduce an

excitation gap in the collective modes and also soften the effect of i-ntir-'w r inter-

actions.

We further study the collective states formed by the 2D electrons at low Landau

levels by working from a semi-classical and microscopic perspective and evaluating

the elastic moduli, normal modes, and zero-point and cohesive energies of the dif-

ferent < il -i11!ii. structures. The effects of screening from filled Landau levels and

finite thickness of the sample are found not to influence the overall interplay of the

phases. When probing the internal degrees of the crystalline structures, the energy









is lowered considerably (which signifies that these degrees have a prominent physical

importance).

Finally, the static Hartree-Fock equation for the triangular lattice symmetry

subset is numerically solved and anisotropy effects are taken into consideration. The

emerging picture is that the isotropic Wigner i *-I J1 is favored for small values of the

partial filling factor but at higher values the system undergoes a first order transition

to an anisotropic Wigner crystal never crossing any other crystalline state for the

rest of the filling factor range. The anisotropic Wigner crystal shows a channel-like

configuration for the guiding centers of the electrons but translation invariance along

the channels is never restored. As a result we find that the anisotropic Wigner crystal

is more favorable even from the traditional stripe state close to half-filling, and that

shear deformations along the channels become cost-free due to the vanishing shear

modulus along that direction.














CHAPTER 1
INTRODUCTION TO THE QUANTUM HALL SYSTEM

1.1 History of the Quantum Hall Effect

A hundred and one years after Edwin Hall discovered the Hall effect in 1879,

Klaus von Klitzing [1] discovered the Integer Quantum Hall Effect (IQHE), intro-

ducing the physics community to a new and remarkable class of condensed matter

phenomena. A disordered 2D electron system (2DES) at low temperatures and high

magnetic fields can exhibit sharp dips in the dissipative resistivity (pxx) and sharp

plateaus in the Hall resistivity (pxy) at certain values of the magnetic field B. These

plateaus happen at integer values of the quantum of resistivity h/e2. The sample

used by von Klitzing was a silicon metal oxide semiconductor field effect transistor.

Disorder is induced by the roughness of the insulator-semiconductor interface in these

structures. Two years later in 1982, Tsui et al. [2] performed the same experiment

but with a GaAs sample of higher mobility and at lower temperature and discovered

that the Hall resistivity piy can take fractional values of h/e2 which are of the form

p/q, where p is an odd integer and q can be even or odd. This was the discovery of

the Fractional Quantum Hall Effect (FQHE). In Fig. (1.1) we can see this remarkable

behavior.

Theoretical work has explained in a satisfying manner most of the pronounced

features of Fig. (1.1) where all of the hierarchy of quantum Hall states is shown [3].

The single particle gap that opens up in the bulk of the material and the charged edge

excitations give rise to the transport phenomena responsible for the IQHE. This gap is

attributed to the single particle localized states lying between the spread out (due to

disorder) Landau levels. At the edges of the sample, the confining potential distorts























10 20 30
MAGNETIC FIELD (Tesla)


Figure 1.1: Dissipative and Hall resistivities of a GaAs sample. Reprinted from
St6rmer, P,.I-:. ,.a B177, 401 (1992). Copyright (1992), with permission from Elsevier.


the Landau level splitting, giving rise to gapless single particle excitations. In the

FQHE, the physics is of a many body nature. The electrons form an incompressible

ground state with a gap (which is smaller than the IQHE) due to their interactions

that become dominant when their kinetic energy is quenched by the applied magnetic

field. The ground state can be accurately described (in the symmetric gauge) by

Laughlin's wave function [4]

N N
t *z, ZN)= (z z- e 4Z i (1.1)
i>j 1

where v = N is the filling factor for a single spin, N is the total number of electrons,

+ is the total flux penetrating the sample, 4o = h/e is the flux quantum and zi =

(xi + :,/ )/1B is the complex position of each electron, where 1B = h/eB is the

magnetic length. Going back to Fig. (1.1) we notice that the FQHE happens for

filling factors v = q/p where p is odd. Eq. (1.1) gives a first explanation for that,

assuming the spin degree of freedom of the electrons is frozen: Fermi statistics are

obeyed only when 1/v is odd. For the rest of the FQHE states different theories









conduction band conduction band

Si donor Si donor *


AlGaAs AlGaAs

GaAs GaAs
d




valence band valence band
Figure 1.2: Symbolic graph of a bilayer structure of interlayer distance d. The
electrons coming from the Si donors are trapped in the GaAs-AlGaAs interface and
form the 2D electron gas.

have been developed based on Laughlin-like wavefunctions [5, 6], microscopic field-

theoretical treatments [7], or composite fermion theory [8], which have been quite

successful.

1.2 Bilayer Quantum Hall Physics: Experiment and Theory

Highly interesting and intriguing physics arises if two 2DES are brought within

a nonzero separation distance d. Experimentally these structures can be grown by

molecular-beam epitaxy where two semiconductors (usually GaAs and doped AlGaAs)

form a quantum well at their interface (~100-1000 A wide) where electrons, coming

from Si-doped 1l-,- iS occupy, form the 2D electron gas. When an undoped AlGaAs

interface separates the two quantum wells by a distance d, then the ratio d/B becomes

a measure of the interlayer interaction strength. We show in Fig. (1.2) a schematic

graph of the vilv.i -r structure. What makes these heterostructures so interesting is

that they exhibit "forbidden" QH plateaus. In reported experiments by Suen et al. [9]

and Eisenstein et al. [10], the ilv.i -r system exhibits plateaus at total filling factors

VT = 1 and VT = 1/2. This is a "violation" of the odd denominator constraint for the









4.0

NO QHE

3.0 o o
0
-o
QHE
2.0
*
*


1.0
0.00 0.02 0.04 0.06 0.08 0.10
ASAS/(e 2/ B)

Figure 1.3: Phase diagram at VT = 1 of interlayer Coulomb interaction strength vs
single particle tunneling strength. Energy measures in units of the i-nitr-lv. r Coulomb
interaction. Solid symbols indicate samples showing QHE behavior, open symbols
denote those that do not. Reprinted inset of figure 1 with permission from S. Q.
Murphy, J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, and K. W. West, Phys. Rev.
Lett. 72, 728 (1994). Copyright (1994) by the American Physical Society.


FQHE. These new FQHE states are attributed to the extra degree of freedom (the

l-1 v-r index) each electron possesses; and to the fact that the spin-polarization of the

2DES is relaxed, leading to spin-textures [11]. Yoshioka, MacDonald and Girvin [12]

have proposed the so called T3,3,1 state [13] as a candidate ground state for a double

l1-,r FQHE at VT = This is a Laughlin-like state which introduces correlations

among the electrons in the two 1V,---iS and keeps them from occupying the same

position in the 2D planes, as if they were lying on the same plane. As experimental

data indicate, the bil.-I r structures have an interesting physical behavior that orig-

inates from the interplay of the in it-r- il- r Coulomb electron interaction (within the

same li -r) and the interlayer Coulomb electron interaction (between the two Il, rz).

A phase diagram for the QHE, experimentally produced by Murphy ct al. [14], is

shown in Fig. (1.3). The strength of interlayer Coulomb interaction, relative to the

i-nit-r i- .r one, is plotted as a function of the tunneling strength ASAs (measured in the

same units) that determines the energy difference of a single electron associated with







5


A) N,,.10.9 D) AN=5.4 1.2 V
N, 42 v = 1


B) NI (x200)

004
l Z5 0.4-
b. N) N 6.4
--^-------- 1 0\. 0

0.0 0.5 1.0 1.5 2.0
5 0 5 -5 0 5
Interlayer Voltage (mV) . (K)

Figure 1.4: Zero bias peak anomaly in the tunneling conductance of a .ili ,-r system.
Left panel: Tunneling conductance vs interlayer voltage V at VT 1. Right panel:
Temperature dependence of the zero bias tunneling conductance at VT 1 at high
and low densities. Reprinted figures 1 and 3 with permission from I. B. Spielman,
J. P. Eisenstein, L. N. Pfeiffer, and K. W. West Phys. Rev. Lett. 84, 5808 (2000).
Copyright (2000) by the American Physical Society.


symmetric or antisymmetric occupation of the '.ilv .. r quantum well system. What is

astonishing is that the QHE persists even when ASAs approaches zero, in other words

when tunneling is turned off. The phase boundary intersects the vertical axis at a

nonzero value d/lB -- 2. This signifies the onset of correlation effects between the two

2DES. On the other hand, when strong tunneling is present, electrons tunnel back and

forth rapidly, assuming symmetric states with respect to the two 1lV. r-. Correlations

are not important in this limit and the system behaves as a single-l -r 2DES, where

the electrons are confined in a wider quantum well. If d/lB becomes large though, the

antisymmetric (and more localized in individual wells) state becomes favorable again

d. -li ',' ii'-; ASAS and the QHE altogether. This -,-. -i a quantum phase transition

from an incompressible to a compressible state.

Further investigation into the bilayer structures has revealed direct evidence

of this interlayer coherence. In an experiment conducted by Spielman et al. [15]

measuring the tunneling conductance in GaAs/AlGal-_As double quantum well

heterostructures, a well pronounced feature appears at total carrier densities of NT <









5.4 x 101cm-2 at VT = 1. The two GaAs wells of width 180 A are separated by

a 99 A A10.gGao.1As barrier -1-V r. The low temperature (~40 mK) mobility of the

samples is 2.5 x 105 cm/Vs. At greater carrier concentrations or equivalently greater

d/lB ratio, this feature is completely absent due to the energy penalty associated with

the rapid injection or extraction of an electron into the strongly correlated electron

system, as we discussed above. What is interesting is that at filling factors where the

2DES must be thermodynamically compressible, it appears incompressible because

the charge defects acquire a large relaxation time scale at high magnetic field, due to

tunneling. Thermal fluctuations are expected to bridge the IQHE gap and similarly

destroy the FQHE state by producing more tunneling events. In this case the relax-

ation time of the charge defects that tunneling electrons create is very low at high

magnetic field. Figure (1.4) shows the resonance peak in the tunneling conductance

and its temperature and carrier density dependence. The height of the peak continues

to grow as the density is reduced and exceeds even the zero magnetic field tunneling

conductance peak (3 x 10-8 Q-1) by more than a factor of 10 [16]. It should be noted

that VT = 1 is held constant in all the traces, so the magnetic field is varied accordingly.

The appearance of this resonance peak -, r.-.- -I the existence of a soft collective mode

of the ril r system which enhances the electron's ability to tunnel. If we assign a

pseudospin quantum number to each electron, indicating the 1-. r index which it lies

in (up or down just like spin-1/2), then the ,-y-plane" pseudospin-ferromagnetic

state the bilayer system assumes is a symmetry breaking state. The Goldstone mode

associated with this broken symmetry, as predicted [17 20], could be the collective

mode identified above. Spielman et al. [21] directly observed this linearly dispersing

collective mode in the '-il- I r 2DES. The best candidate state describing the system's

ground state responsible for this high coherence peak is the Halperin '1,1,1 state. In

this state, an electron in one l--r is abv- opposite to a hole in the other 1-.Cr









in the same 1I--r in which the voltage drop measurements are taken. Figure (1.5)

shows these results and the crucial dependence the response of the 'il-i r system has

on d/lB At d/lB > 1.83 the interlayer coherence is gradually destroyed and the

iii i!" single-i -Vr behavior is restored.

Previous theoretical work has addressed the coherence peak feature the bilayer

system exhibits. Balents and Radzihovsky [23] studied it from the quantum Hall

ferromagnet point of view. They found rich variations of the tunneling conductance

as a function of bias voltage, tunneling strength, disorder, temperature, and an applied

magnetic field parallel to the I v-r-. They provided a scaling theory where disorder

effects, based on an idealized pure system with parallel side contacts, were discussed as

well. From the same viewpoint Ady Stern et al. [24] studied the finite Josephson effect

and argued, based on disorder effects, that it is not a true Josephson effect (infinite

tunneling conductance). They also predicted the rich characteristics the tunneling

conductance should exhibit and they attributed the finite peak in it to topological

defects in the order parameter m (pseudospin magnetization) caused by deviations of

the total density from VT 1. Finally, Fogler and Wilczek [25], studying the system as

a "classical" Josephson junction, applied perturbation theory and derived a tunneling

current formula showing the general characteristics described above.

In all the above theoretical approaches, phenomenological arguments and scaling

techniques attempt to shed some light on the qualitative physics of this interesting
`iliv- r phenomenon. Nevertheless, a coherent picture with some microscopic insight

to the characteristics of the system is still lacking. This is the direction we have

taken. We have tried to shed some light on the effect of the interlayer Coulomb

interaction and its importance to the collective physics of the liliv.vr system. We

have modeled this system as two Wigner (< i 1- according to the original work of

Johansson and Kinaret [26], based on the independent boson model; but we have

introduced a coupling between the 1 .,- r- as well, arising from the existence of the














03 04 05 06 07 0.8 0.9 1.0 '









4. 05 0.6
0.4A










0.5 10 15 20 25 3.0 4.10 4.15 4.20 4.25 4.30 4.35
f(GHz) v

Figure 1.6: Microwave resonance response of a quantum Hall system. Left panel:
Real part of crx vs frequency f for different filling factor values (offset for clarity).
The inset is reproduction of selective filling factor values and at an expanded scale.
Right panel: Peak frequency vs filling factor for the two resonances shown to coexist
on the left panel. Reprinted figures 1 and 3 with permission from R. M. Lewis, Yong
C!. i, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K. W. West Phys. Rev.
Lett. 93, 176808 (2004). Copyright (2004) by the American Physical Society.


interlayer interactions. Our approach has been rewarding because it has provided

insight into the dual role of the interlayer interactions We have studied only the

incoherent regime of the system, but this kind of systematic modeling has paved the

way for later attempts to include coherence and reproduce the fascinating I-V response

shown earlier.

1.3 Crystalline Phases of the 2D Electron System: Experiment and
Theory

The proceeding introduction into the physics of quantum Hall systems (whether

Real part or single vs freq) should have convinced the reader that systems like these have a

rich insei is of states or phases that can potentially manifest themselves, depending on

the different parameter values associated with such systems (such as disorder, carrier
the different parameter values associated with such systems (such as disorder, carrier









density, applied magnetic field or temperature). Two i '.i"r classes of experiments

can be conducted with the quantum Hall systems to explore the different phases they

can realize. One class consists of transport experiments (such as the ones presented

in the previous section) where features in the conductivity (or absence thereof) can

lead to conclusions about the different possible phases. This class is subdivided into

DC and AC transport experiments; which further specialize in capturing different

characteristics of the system (such as the pinning threshold [27] or reentrant insulating

states around given IQHE or FQHE states [28] or even anisotropic behavior [29, 30]).

An insulating phase usually indicates crystallization in the system. The other large

class of experiments consists of resonance absorption experiments where the sample is

irradiated (usually microwave radiation) and from the resonance response it exhibits,

one can derive valuable information about the collective modes and the actual state

of the system [31-33]. The latter method is able to capture phase coexistances, since

(in principle) different phases will leave different traces in the absorption signal.

A typical microwave resonance experiment for a 2D electron system in the n = 2

Landau level is shown in Fig. (1.6). What we see is the microwave absorption response

of the system, traced in the real part of the longitudinal conductivity axx, for different

applied magnetic field values or filling factors. The traces are displaced for clarity.

According to data i, iJ1 ,i-i on these measurements [33], the resonance curve can be

fitted by two Lorentzians indicating a two-phase coexistence. This is also shown in

Fig. (1.6) where the peak frequencies of these two phases are plotted for different

filling factor values. Disorder and pinning 1 i', a crucial role in the collective response

of this system [34, 35] since the pinned domains around impurities resonate to the

external stimulus the alternating electric field provides. On the other hand, the effect

of disorder and the extent of pinning in the system is determined by the actual state

( i i,-i ,ii.w or liquid) that the system occupies. As a result for one to aspire to

describe the collective behavior of a quantum Hall system, one is forced first to study









the different phases such a system is capable of realizing (for the whole range of filling

factors) and then to develop a dynamic response theory for these states.

Quantum Hall system states have been studied extensively using the Hartree-

Fock approximation [36-39] or the density matrix renormalization group method

(DMRG) [40, 41]. The prevailing picture, also pertaining to the experimental results

shown in Fig. (1.6), is that at low filling factors, the system is in a Wigner crystal

(WC) state; and for filling factors close to half-filling, the traditional stripe state

(charge density wave) becomes favorable. At intermediate filling factors, a variety of

bubble < i -I J1 (BC) states emerge. A bubble < i i -I J1 is a Wigner crystal-like structure

(where instead of having one electron guiding center occupying a given site, there are

more; creating a hierarchy of M-electrons per bubble crystalline structures where M

is an integer [36, 42]). According to Hartree-Fock results, the last possible BC state

that can become favorable (as the filling factor is increased) is the M n+ 1 electrons

per bubble i il -I 1 for the n-th Landau level. On the contrary, this last BC state is

not observed using the DMRG technique and only up to M = n electrons per bubble

Si-I are realized. However, all of these studies focus on the density profile (order

parameter) of the system in an ad hoc way that does not explore the microscopic

physics involved; this is the direction we have pursued. We have attempted to shed

some light on the microscopic of the
tum Hall system and explore how the internal degrees of freedom react to external

stimuli such as magnetic field changes. This improves our understanding of the physics

involved and can serve as a stepping stone to describe the dynamic response of such

states in light of experimental results (such as those shown in Fig. (1.6)).














CHAPTER 2
TUNNELING CURRENT OF COUPLED BILAYER WIGNER CRYSTALS

2.1 Single Layer Eigenmodes

Our next task is to formulate a model of a bilayer 2D electron system where tun-

neling is a quantum mechanical process between the two Il. ri separated by a distance

d, in order to capture the incoherent behavior of the system. The bulk of the electrons

in such a system provides the collective modes that couple to an independent tunneling

electron. Implicitly assumed is that tunneling events are uncorrelated; and most of

all, the electrons involved in the tunneling processes are uncorrelated with the bulk

of the electrons comprising the hil. -r system. This is a justified assumption, since

a typical value of the tunneling current passing through the system is in nA, which

corresponds to one tunneling event every 10 ps; while a typical period of oscillation for

the collective modes of the 'il-i -r system is one order of magnitude lower. This means

that any local excitation caused by a tunneling event is dissipated away much faster

than the time it takes for another tunneling event to occur. Additionally, the collective

modes dissipate their own energy through the emission of lattice acoustic phonons

generated by the underlying substrate (GaAs) with propagating speeds of 5200 m/s,

much less than the collective mode phase speed. The typical thermal activation time

is 1 ns (which translates to 100 tunneling events throughout the sample). So the

small number of tunneling events (and the rapidity with which their local charge

defects relax) justifies our treatment of the tunneling electron as uncorrelated and

independent from the bulk.

Let us start by introducing the single l-vr model in which the 2D electron

system is assumed to be in a Wigner ( i ,-I 1 state. This is not the true experimental









realization of the system for the filling factor considered [27, 28] but serves as an

accurate starting point if one wants to incorporate short range correlations among the

electrons present in the real, liquid-like state of the system. Additionally, we work in

the continuous approximation limit, treating the Wigner < i i-~I J1 as an elastic medium

and imposing a momentum cut-off qgo 2 2 o, where no is the l-v-r density. This way

we are able to capture the long wavelength physics of the correlated electron gas, and

retain some information of the short range correlations. Introducing the appropriate

Lam6 coefficients [43] A, p to describe elastic deformations of a configuration with

hexagonal symmetry (the triangular lattice is the Wigner < i -I ,1 ground state [44]), we

can write the following Lagrangian describing the system dynamics in the continuum

limit


L = no d2r 2 ln2 eu A(u) A(89U)2 (aUl + 1iur)2
2 2no 'I

+ no d2r'[V- u(r) ][V u(r')] r'l (2.1)


where u is the displacement field, c the dielectric constant of the host material (GaAs

in our case of study), and A(u) the applied vector potential. As seen in the last

term we have included an i-ntr-lv.- r Coulomb interaction term in the continuum

approximation where local charge variations are given by 6n/no = -V u. This is

correct in the absence of vacancies and interstitials. For the vector potential we choose

to work in the symmetric gauge so that A(u) = (-Buy/2, Bu1/2, 0) where B is the

applied magnetic field. In the absence of the perpendicular magnetic field, the normal

modes of the elastic system are the transverse and longitudinal acoustic modes where

their corresponding acoustic speeds are related to the elastic parameters according to


L = /(A + 2p)/mno, (2.2)

CT = ./mno. (2.3)









Since these are the normal modes of the system in the absence of the magnetic field,

we would like to decompose the displacement field in terms of them, and then Fourier-

transform Eq. (2.1) to obtain


L no Toit2 } r^T+ ^rL + -Mc [iUTUL -UfUT] -MU) 2 -2_ -mUT 2 (2.4)
J J(27)2 2 2 L 2 2 2 T T

where we use the real field property u*(q) = u(-q) and the convention |u|2 u(q)

u(-q). We see that the magnetic field enters in the dynamics only through the

third term which mixes the transverse and longitudinal modes as expected. For the

cyclotron frequency we have uc = eB/m while the longitudinal and transverse zero

magnetic field eigenfrequencies are respectively given by


S q2 + 2q, (2.5)
WL cq + 2Tn c

WT = CTq. (2.6)


The expected effect of the i-ntvr 1 r Coulomb interaction is to introduce incompres-

sibility (which is realized by the long wavelength divergence of the longitudinal mode

velocity). As a result, the quadratic term involving CL becomes negligibly small and

for all practical purposes [44, 45] we can set CL = 0; but for the sake of completeness,

we will retain it until the last moment. The analytic expression for the transverse

velocity is [44]

CT 0.0363 (2.7)
3emao

where ao is the Wigner J v-I ,1 lattice parameter; and we have assumed triangular

lattice configuration that seems to be the ground state. Appendix A details the

eigenmode calculation. Here, we present only the eigenfrequencies of the single -1v.-r

system

U) 2 t 2 + + L \/(c2 + cU4 + U))2 4U2 U. (2.8)









We notice that in the zero magnetic field limit (c = 0) the above modes decouple

into the pure longitudinal and transverse ones as expected. Also, in the high magnetic

field limit (i.e., to lowest order in 1/we) we obtain


+ -= + L (2.9)
2c
L_ = T (2.10)
Uwc

according to Kohn's theorem [46] which predicts cyclotron frequency absorption for

a translationally invariant system. We have recovered these plasmon modes in the

continuum limit (w+); but since we have assumed a Wigner crystal state for the

electronic system, we have also maintained gapless excitations (w_).

The above treatment completes the single 1-ivr study of the 2D electron system

treated in the continuum elastic limit. We can decompose the transverse and longitu-

dinal fields involved in Eq. (2.4) in terms of the eigenmodes of the system and couple

them to tunneling electrons, in the same way that phonons couple to electrons. That

is why the modes of Eq. (2.8) are called magnetophonons (w_) and magnetoplasmons

(a;), respectively.

2.2 Bilayer Eigenmodes

Having completed the single 1 -v.r treatment of a 2D electron system we can

turn on the -ilv,-r problem where two 2D electron systems are separated a finite

distance d from one another and interact through the interlayer Coulomb interaction.

For simplicity we can assume that the 1lv. rS- have the same density no (which can be

arranged experimentally) and write down the following Lagrangian


L LA+LB+no 2r (UA )2
2 no
no d 2/ [V uA(r)][" uB(r)] (2.11)
-- 7x')2+y ( y' y)2+ d2









where LA, LB are the independent single v1-vr Lagrangians similar to Eq. (2.4). The

term involving the K parameter (associated with the short range physics of the inter-

I-V,-r Coulomb interaction) is expected to arise when the two Wigner crystals prefer

to lock their positions (and move in-phase) by penalizing out-of-phase fluctuations.

Since we are working in the continuum long wavelength limit it is impossible to capture

that physics unless we explicitly add this extra term into the system dynamics. By

construction, K/no is a measure of the energy density per electron associated with

the short range correlations induced by the Coulomb interaction and can be assumed

to scale accordingly as
K e2 /47ed
= K -- (2.12)
no 7T12

where K is dimensionless and I = h/eB. The dimensionless parameter K can be

extracted from magnetophonon experimental measurements or theoretical calculations

associated with this rilv r system. For the second term (the long range part of

the Coulomb interaction) we have used the usual 3D form applied for the two 2D

electron systems and we have employ, ,1 (as in the intirylv.- r Coulomb interaction

case) the continuum linear approximation in order to describe local charge density

fluctuations. Diagonalizing this coupled '-iliv -r system involves introducing in-phase

and out-of-phase modes given by


1
UA V U, (2.13)
2
1
UB v + U, (2.14)
2


which turn out to be the eigenmodes of it. As a result, the two coupled single l1v- r

dynamics of Eq. (2.11) decompose to uncoupled "effective single 1 l,- i dynamics

of in-phase and out-of-phase nature. We started with a system of a total of four

modes, so we expect two in-phase and two out-of-phase modes as a result. Details

of the calculation are given in the Appendix. The result for the two out-of-phase









eigenfrequencies is


S=1 [K + 2 + Q2 /( + ? + )2 4Q (2.15)


where the "effective transverse and longitudinal acoustic mode frequencies are given

by


T2= c-q2 + o (2.16)
2 2 2 2K 2mno
Sc q2 + q(1 e qd). (2.17)
mno 2m -

Notice that the single l -.-r form of these results is preserved but the acoustic modes

have acquired a gap relating to the short range correlation physics introduced by the

K parameter. For the in-phase eigenfrequencies we obtain similarly


O = t + O + O + + O + O~)2 40 0 (2.18)


where the effective transverse and longitudinal acoustic mode frequencies are given

by


O= c q2, (2.19)

Sc q2 + 2 q( + e-qd). (2.20)


As is expected for the in-phase modes, there is no gap introduced by the short range

physics, since these type of modes respect the locked-in position of the two Wigner
(i,-I l1- Since we have solved the single 1Iv-r problem and have found analytic

expressions for the creation and annihilation operators of its eigenmodes, we can

apply those results to the "effective single 1 ,. il cases here, after we transform the

appropriate parameters involved. For example, in order to obtain analytic results









for the out-of-phase operator modes we have to perform the following changes to the

parameters of the single 1v-r case: m -- 2, 4 -- Q2, w% -- 2 For the in-phase

case, the changes become: m -> 2m, 4 -O w 0 Final results are presented

in full in the Appendix.

This completes the treatment of the -il r system. We now have analytic

expressions for the eigenmode operators of the coupled 'il r quantum Hall system

and a way to describe lattice field displacements in terms of those. What remains is

coupling those modes to tunneling electrons introducing the electron-magnetophonon

and electron-magnetoplasmon interaction. This will open up an excitation channel

for the injected tunneling electrons to dissipate their energy and for the bulk electrons

to relax the charge defect associated with the tunneling event. This is the topic of

the next section.

2.3 Coupling the Bilayer to External Electrons: Tunneling Current

Now that we have accomplished the task of calculating the eigenmodes of a
iil-i- r 2D electron system in the continuum approximation, we must complete the

picture by introducing a coupling of those modes to an independent tunneling electron

injected into the system through a steady current. To do that, we must distinguish

between the bulk electrons (and their operators) associated with the displacement

field u, and an independent electron tunneling from one 1-, r to the other. For the

latter, we will use ce, CA and ctB, CB as the creation and annihilation operators for the

two 1I.~. rs, respectively. Assuming for simplicity that the tunneling electron is at the

origin of li. r A, and couples through the unscreened Coulomb interaction to charge

density fluctuations of the Wigner crystals in both 1,-. rs A and B, we can express

the interaction energy associated with that coupling as follows


Hee 472 d2 2rA rA + 2rdrB (2.21)
47c |r rAl 47 rB dl








In the continuum linear approximation the charge fluctuations in the two Il.-. 1r will
be given by MnA = -noV UA and 6nB = -noV UB, respectively. Placing the
independent electron at the origin corresponds to ne(r) = (2)(r). Combining all of
the above the coupling term assumes the following form

e28 0d2 VA UA 62n fd2 BB UB

e20 d 2q 2d
iq 2 UA (q) UB(q) .(2.22)
47e j (27)2 q I

In other words, the coupling term in the i-.-Ii r system associated with an independent
electron injected into the bulk of either of the two quantum Hall systems has the form


coupling --CACA (27)2q LA2 LB)

-cq(UL + e- d A) .] (2.23)


Introducing at this point the in-phase and out-of-phase displacement fields given by
Eqs. (2.13, 2.14) we can transform the above to


oupng CAA (27 ( + d d)ULj

CBB 2 2 + e- )v + (1 e-)uL (2.24)


Using the analytic expressions, derived in the previous section, for the operator form
of the in-phase and out-of-phase modes we show in the Appendix that the above
coupling term can be written in terms of creation and annihilation operators of the
iil-,i r quantum Hall eigenmodes as

4 4
Coupling CACA i 1. {(f t a,) +C BCB {iZ .[. as) (2.25)
8" s 1 I s= 1a









where at and a8 are the corresponding creation and annihilation operators for the

bulk electrons and the coupling matrix elements are given by


( e-dq) f, s- 1,2, s, 1,2,
if. ,= < (2.26)
(1+ e- dq)f, s -3,4, 1. i, s -3,4,

Notice that we have included in the s summation the integration in q as well to avoid

cluttering the symbolism.

The above completes our treatment of the bilayer quantum Hall system since

we have an analytic expression for the eigenmodes of the system and the way these

modes couple to an independent electron injected in the bulk of the system. We

can write the following independent boson Hamiltonian, similar to the one used by

Johansson and Kinaret (JK) [26], to describe the bil-v r system energetic involved

in the tunneling processes


H -Ho + H+ + HT

rCA + iz (at as)j CtCA + CB + i i(a as)jCCB
S S

+ YhUata, + TCtCB + TCeCA. (2.27)


We have a system of two 2D electron gases under the presence of a perpendicular

magnetic field in the elastic continuum approximation, producing a collective mode

bath to which an external tunneling electron couples in order to dissipate its energy.

The same channel is used by the bulk electrons to "smooth-out" the local charge

defect created by tunneling events. These tunneling events are independent quantum

mechanical processes with finite tunneling matrix elements T, calculated in a similar

manner as JK report [47]. The collective mode operators at, a8 obey boson statistics

and cA(B) CA(B) obey fermion statistics. In the above CA and CB are the Madelung








energies of the two Wigner crystal lattices. We follow JK in evaluating the tunneling
current associated with this model. Their approach involves the application of Fermi's
Golden Rule that can be rewritten in the following form

C / d+O /
I(V) dievd ^ ([Hj(t), Hf+(O)])

C dt e2VI(t) e- (t) (2.28)


The correlation functions associated with this expression are given by


IT(t) = (H(t)H+(0)), (2.29)

I:(t) (H+(t)Hj(0)), (2.30)


where the time-dependence is meant in the interaction picture representation. For the
calculation of the above correlation functions we use the same approach as JK. Since
the tunneling process is statistically independent from the collective mode propagation
it can be averaged independently. The statistical averaging involves the linked cluster
expansion method [48]. In this particular case there is only one independent link
associated with the exponential resummation. In the Appendix we show in more
detail how the calculation proceeds. We obtain for the correlation function


I(t)= v(1 v)T2C(t), (2.31)


where v (= (cCA) and 1 v = (cBct) and for the time-dependent part we get


C(t) exp {- 2( 2 L 2) -N)+ ( ^) (2.32)


where N, is the boson thermal occupation number for the magnetophonons and
magnetoplasmons. To obtain the form of Eq. (2.30) it suffices to interchange A and









B in Eq. (2.32). The experimental temperature range in the tunneling current is of

the order of 0.1 K~10-5 eV while the bias voltage is in the range of mV, so a zero

temperature calculation is appropriate which simplifies things considerably since the

bosonic occupation numbers Ns = 0. In addition, due to the high magnetic field, the

magnetoplasmon modes will have a large gap and will not contribute to the electron

coupling so we can drop them. Notice that we are still left with the in-phase magneto-

phonons along with the out-of-phase ones but as we can see from Eq. (2.32) they enter

as a difference in the correlation function and since they are exactly equal for both of

the l'zv-i [Eq. (2.26)] they cancel out. This is to be expected since a tunneling event is

associated with an out-of-phase motion of the two Wigner crystals. The one that the

tunneling electron leaves from will try to close the "hole" left behind while the other,

receiving the tunneling electron, will try to "open-up" and create an available posi-

tion for it. This corresponds to an out-of-phase motion. If we gather all of the above

together and switch to dimensionless units for the momentum integration (x = q/qo)

we find the following result for the time dependent correlation function


C(t) exp J df(x) (e-iw(x)t 1, (2.33)


where the weight function and the magnetophonon frequency can be approximated in

the high magnetic field limit as

cx x(1 e-7)2 1
f(x) W (
C qo +x2 + 2a + jx(1 -e-7) 2+ a
1 1
X [a + 3x(1 c-)]3/2 6 a (2.34)

a)x ) a V/+3x(1 e )Vx2+a. (2.35)
wc









In the above we have defined the parameters

no 2 2
c h c(2.36)
c 87hm e '
2K K 1 e2
a 2 22c ( qo) 2' (2.37)

P- --- o- (2.38)
2emcTqo

7 =dqo, (2.39)

6= ( (2.40)


and have taken CL = 0. The parameter a is dimensionless and gives a measure of

the magnetophonon gap. For the second equation associated with it we have used

the definition of Eq. (2.12). The /3 parameter is dimensionless as well and does

not depend on a0o if the dependence on it from CT is taken into consideration using

Eq. (2.7). The parameter 7 gives a measure of the relative strength of the intrylv. r

and interlayer Coulomb interaction. In order to proceed with the derivation of the

correlation function we can differentiate Eq. (2.33) and take the Fourier transform to

obtain the following equation


wC(U) = dxf (x)w(x)C(w u(x)). (2.41)


As we show in the Appendix, this correlation function is zero for w < 0.

The above integral equation for the correlation function is very hard to solve

exactly. In the following subsection we show how we can derive important information

to build an Ansatz solution for the I-V response of the bil-r system.

2.3.1 Analytic Solution

In order to try and approximate an analytic solution for the integral equation

of the correlation function given by Eq. (2.41) it is important to derive as much








information as possible from it. What turns out to be particularly useful is the
derivation of the .,-i~:, l, I tic behavior for large frequency values (large bias). In that
case we can expand C(u uw(x)) in u(x) and obtain to lowest order a first order
differential equation with the solution


C(w) exp r- 21)j, (2.42)


where


0/1
ci j dxf(x)w(x)
U d4 1x x(1 %-x)2 1 1 (243)
Jo 6 + X + 2a + Ox(1 e-%) a + Ox(1 e-') 6 a 2'



0/1
C2 dxf(x)w2(x)

L3 1 x(t_ y)2 2 t + a
=c( dx1 .x (2.44)
4 q o d + X 2 + 2a + Ox(1 e-') a + 3x(1 e-_) a X2 a .


We see that there is exponential suppression in the tunneling current for very large bias
values, something to be expected since the system is unable to cope with the large
inflow of energy the tunneling electrons carry and need to dissipate. Any attempt
to dissipate these large amounts of energy creates large number of magnetophonons
which in turn cause large quantum fluctuations in the system potentially destabilizing
it.
We are in a position now to investigate different correspondence limits associated
with Eq. (2.43). We are interested in the limiting behavior of the above .,-vmptotic
solution for different 1-v-r separation values d. For the case where the two 1v.,.riS
are far apart and can be considered uncorrelated (d > ao), we can ignore the ex-
ponentials in the integrand of Eqs. (2.43-2.44) and it turns out that cl ~ 1/ao and









/C c~ 1/a B are the corresponding limits. This is the same scaling behavior JK
produce using phenomenological arguments. In the opposite limit, where the inter-
l--r separation is much smaller than the intra-electron distance (d < ao) we can

expand the exponentials in the integrand of Eqs. (2.43-2.44) and find ci ~ d2/a and
/2 ~ d/a B. This limit is absent in the JK model. In this regime correlation effects
become important. Coherence significantly modifies the actual behavior of the system
but this is expected only in the region close to zero-bias [15]. For the remaining bias
voltage region, correlation effects have the prominent role and the Coulomb barrier
peak survives but is "red-shifted" significantly [15]. As we see our model is able to
reproduce such a limiting behavior.
The above .,- ii il .1 ic expansion provides a useful starting point to apply a trial
solution of Eq. (2.41) by assuming a power-law combined with a Gaussian exponential
behavior for the correlation function according to


C(w) = Nwe -2. (2.45)


The above Ansatz captures the essential .,-,iii! .ltic behavior and if the parameters are
evaluated self-consistently it should qualitatively reproduce a solution. In particular,
if we multiply-differentiate Eq. (2.33) we end up to the following moment equations
(associated with C(u)) that we can use to derive values for N, r and A:


j dwC(uw) = 27, (2.46)
2rOO
j dwC(w) =27Cl, (2.47)

j dw2C() = 2(c2 + ). (2.48)
JoO








For the above equations the following general integral becomes useful

1 ur-y2 1 r+2 (2.49)


Also, it is convenient to switch frequency (w) to bias voltage (V) (measuring in mV) by
introducing the change w = eV/lOOOh. That way the argument of C(Uj) will measure
in mV while the corresponding Ansatz parameter A in the exponent of Eq. (2.45)
will acquire the form A A (10) 2. Using the general integral result above we can
perform the integration in Eqs. (2.46-2.48) and obtain the results

1 (e r+l \ 1
N( A- 21 27, (2.50)
2 1OOOh 2 )
N ) A- 2 +) 27cl, (2.51)
2 1OOOh 2 '
1"e r+3 +3 2 2
N Oh 2 r 2 2 (c2 + c). (2.52)


We can divide Eq. (2.51) and Eq. (2.52) by Eq. (2.50) to get


C 2 F ( r+2 )
A ( 1000 h) [ 4i ] (

A=( e 2)PT 3 (2.54)


or equating the two

p2(r+2) (r+1)
r3) + /c' (2.55)


we get a self-consistent equation for r that we can solve numerically. The usual range
of r for the magnetic field values considered is 1/2 < r < 2. This can be regarded as
an estimate for the low bias current power-low behavior. Having r at hand we can
go back and evaluate the rest of the parameters. As a final result we find that the














S/ ..11 T
6 -
975T
I \ "
UO
4, 4-
| : 8.25T \ \





0 5 10 15 20 25
Interlayer Voltage (mV)

Figure 2.1: Tunneling current curves for different magnetic field values using the
moment expansion solution of Eq. (2.41). The legend shows the peak bias values
calculated by Eq. (2.57).


tunneling current correlation function assumes the form


C(V) O N)e V'reAV2, (2.56)



where V measures in mV. To obtain the peak bias value we need to find the root of

the first derivative of the above equation



Vo O- h (2.57)
e 2A'


where we have converted it to mV.

With the last piece of the puzzle in place we are ready to test our theory with

a realistic experimental setup. We choose the same system JK used as their reference

[49]. The riliv r sample area is S 0.0625mm2 and the single 1-,- r electron density

is no 1.6 x 1011 cm-2. The perpendicular magnetic field varies from 8 T to 13.75 T

and the Wigner i i-I 1 lattice parameter has the value ao 270 A which corresponds


















X
E 05-
d=115
04-

03-
d=145
02

01 d=85 -4

0 2 4 6 8 10 12 14 16 18 20
V (mV)

Figure 2.2: N., in i,1,. .1 tunneling current curves for different interlayer separation
distances d measured in A.


to the stable hexagonal lattice configuration (no = 2/V3a ) [44]. The double well

separation distance is d=175 A and the dielectric constant of GaAs is e=12.9co _1.14

x 10-11 F/m. The transverse sound velocity for the electron gas given by Eq. (2.7)

is CT 53552 m/s. As we mentioned earlier we have to take the longitudinal sound

velocity cL = 0 in order for our results to correspond correctly to the physics of the

experimental system. For the electron mass we use the electron effective mass value

in the GaAs background m 0.067me. For the K parameter we use Eq. (2.12) where

the value of K is extracted by time-dependent Hartree-Fock calculations investigating

the magnetophonon dispersion relation for the .il i v-r system, performed by C6t6 et

al.. They were able to provide us with a K 0.0085 value. Our I-V results based on

this model are shown in Fig. (2.1) with the corresponding peak bias values given by

Eq. (2.57).

At this point we are in a position to investigate the effect of the interlayer

Coulomb interaction in the 'il v.-r system. We notice that the strength of this inter-

action is controlled by the interlayer distance d which is introduced into the model in









two places. One is in the tunneling matrix elements T in the independent boson model

Hamiltonian (with an exponentially suppressive behavior) and the other is through

the long range part of the interlayer Coulomb interaction term. Since we are inter-

ested only in the latter, we will normalize the tunneling current for different interlayer

separation values d. What we expect to reproduce is the experimental behavior shown

by Eisenstein et al. [50] where the peak bias values are "red-shifted" by an amount

proportional to e2/ed as the interlayer spacing is reduced. This behavior is due to

the attraction between the hole left behind in a tunneling event and the tunneling

electron itself which is of the order of e2/ed. In other words the creation of excitons

associated with tunneling events are expected to "soften" the effect of their nt-ril-v, r

Coulomb interaction and consequently lower the energy barrier imposed to tunneling.

In Fig. (2.2) we show our results for the normalized tunneling current solution for

different interlayer separation distances d measured in A. As we see our model is able

to capture this important physical behavior of the system.

As a result of our theoretical analysis we can conclude that the effect of the

interlayer interactions in the bilayer system is two-fold. First, in the short range

physics it introduces a gap in the long wavelength excitations that contributes to

the small bias suppression of the tunneling current. And second, in the long range

physics it "softens" the effect of the int-r liv- r Coulomb interactions through the

excitonic creation associated with tunneling events and as a result it "red-shifts" the

tunneling current peak bias values.

2.3.2 Numerical Solution

We have numerically integrated the integral equation for the correlation function

in two different v--v-, first by a direct integration of the integral equation, and then by

introducing the density of states (similar to the JK method). Both methods give the

same results of course so we will present the latter one only. We can write Eq. (2.41)














.**1
S 6-
a :/ 9.75 T \

0 /
I ~ 8.25 T \ '




0 5 10 15 20 25
Interlayer Voltage (mV)
Figure 2.3: Tunneling current curves for different magnetic field values produced by
numerically integrating Eq. (2.41). The legend shows the peak bias values obtained
with this approach. They are in strong agreement with the analytic results.

in the following dimensionless form


zC(z) dxf(x) C(z ()0(z )), (2.58)
Jo 7o 7o 7o

where we have introduced the parameter o = to convert the frequency argument

of the correlation function into mV. Before we proceed we should notice that the

magnetophonon frequency is bounded in a region or < '(x) < a2 which means that

the density of states is non zero only in that range. The values of a1 and a2 are given

by substituting x = 0 and x 1 into Eq. (2.35) respectively. The upper bound a2

appears due to the momentum cutoff we have introduced. The resulting form of the

correlation function integral equation is


(1 =l'nIy)C(z- y), at < Z < a2,
C(z) Z (2.59)
fl _r(y)C(z y), > a2,







31

where the definition for the density of states is


g(y) = Y f i}g) (2.60)
( f (x)
o dx (y)

and x(y) is the root of the equation u(x) = 7oy. In this approach one has to "jump-

start" the algorithm with an assumption for the low bias points. We use a linear

approximation since we can show that for z > a1 values C(z) ~ z-a1. Our numerical

solution is presented in Fig. (2.3). As it is clearly shown the qualitative behavior of

our analytic solution is verified and the peak bias values are similar as well.














CHAPTER 3
QUANTUM HALL SYSTEM IN THE HARTREE-FOCK APPROXIMATION

3.1 Electron Dynamics in a Perpendicular Magnetic Field

We would like to focus our attention here on a single 1vr quantum Hall

system and try and shed some light on the microscopic physics involved in this highly

correlated electronic system. We would like to investigate the competition between

different crystalline states and their stability for different applied magnetic field values.

This kind of work requires some attention to be paid to the microscopic involved in

such a system. The quantum nature of the electrons incorporated in the physics of

wavefunction overlaps and associated with the electron-electron Coulomb interaction

has to be considered, in order to investigate, as accurately as possible, the energetic

and stability of the different phases associated with the quantum Hall system.

We start this work by finding the non-interacting electronic wavefunction in the

presence of a perpendicular magnetic field. For that task, we introduce the Landau

gauge A = (-By, 0, 0) and write down Schr6dinger's equation for the 2D electron,

given by

2( eBy)2 + P (x, y)= E (x,y). (3.1)

Since in this gauge choice, the magnetic field does not fully couple the two directions, a

plane wave solution is expected in one of them (x-direction) resulting in a wavefunction

decoupling of the form: j(x, y) = erkx x(y). This kind of decoupling produces a

displaced harmonic oscillator equation for the y-direction given by


[ + mwf (y )2] ) hc (n+ t (y). (3.2)
2m 2 2 2









In the above, we have defined Y = kjS2 as the y-coordinate center of mass (C\ 1)

variable, V/h /B as the magnetic length, and c = eB/m is the cyclotron

frequency. The normalized solution for the displaced harmonic oscillator is given

by
1, 2 (y-Y
0(. y) 1e '-), (3.3)
T 1/4f1/2 2>!

where n is the so called Landau level index, associated with kinetic energy excitations

of the non-interacting electrons, and H,(x) is the usual Hermite polynomials of order

n. The effect of the applied magnetic field is to quench the kinetic energy of the

electrons in the 2D system, which results (in the real system) in an enhancement of

the role of interactions among electrons. This can become prominent at high magnetic

field values, where kinetic energy excitations (of the order of ha;) might exceed the

thermal energy range (of the order of kBT), and as a result become inaccessible.

This is the magnetic field range that the kinetic energy becomes irrelevant, and only

inter-electron interactions affect the energetic of the system and introduce a large

class of hierarchical states, as the magnetic field is varied. The degeneracy associated

with the plane wave eigenstates in the x-direction allows a macroscopic number of

electrons (fermionic particles) to occupy the same kinetic energy eigenstate (even in

the non-interacting limit). The spin degree of freedom is assumed to be frozen at

these high magnetic field values. For a system with finite length Lx in the x-direction

the degeneracy g can be found to be

L -ko 0 _4
xw = dkiv = dY= (3.4)
27 o 27f2 0 27f2 o0

where Q is the total area of the system, and 0 = h/e is the flux quantum (associated

with the quantum Hall system which is twice the value of the superconducting flux

quantum). The quantum mechanical operator expressions for the C\ I coordinates X,

Y (that enter into the dynamics of the electrons) can be derived from their classical









counterparts, and are found to be


X = (3.5)
mwjc
Y=y- (3.6)


where the dynamical moment 7r, Ty are given by the following expressions in the

Landau gauge


= ih [xH]H px -eBy, (3.7)

7Y =my [y,H] py. (3.8)


Combining the above definitions together we can derive the C \ coordinate forms in

terms of the usual quantum mechanical operators:


X = (3.9)

S P= (3.10)


We see the effect of the magnetic field and the Landau gauge choice partially mix

the dynamics of the two directions. The C\ I coordinates are constants of the motion

since they commute with the non-interacting Hamiltonian H, introduced in Eq. (3.1),

something to be expected since the cyclotron motion does not drift. Additionally, they

are conjugates since [X, Y] = i2. The dynamical moment are conjugates as well

since [7x, y] = -ih2/ W and they additionally obey [X,7k r] [Y, y] = 0. In other

words, the C\ I coordinate operators along with the dynamical momentum operators

represent different parts of the degrees of freedom of the electrons. One can use these

four operators to define appropriate creation and annihilation operators (associated

with these degrees of freedom) to fully describe the electronic field. Additionally, from









the following commutation relation


[I,7] = [x,p eBy] = ih, (3.11)


in the limit of high magnetic field we find


[x,y]= -2, (3.12)


which implies that high magnetic fields radically change the electron dynamics. In

that limit the position operators (that usually commute with one another) become

conjugates. What this entails, is that special care needs to be taken when we define

physical observables if we want to correctly incorporate the physics of high magnetic

fields in such a system [51]. The method that has been developed to address this,

involves the projection of all physical observables onto given Landau levels [52]. In

principle, a subset of Landau levels needs to be retained for general magnetic field

values. However, for the case where inter-Landau level excitations are not important

(high magnetic fields) we can restrict the projection space onto only one Landau level.

The mechanism to project onto a given Landau level involves the restriction into a

subset of the available Hilbert space of the wavefunction basis used to define the

electronic field operator. The non-interacting wavefunction basis (given by Eq. (3.3))

is usually used to construct the electronic field operator. Projecting onto the n-th

Landau level we find

S(r) (r)c.,y, (3.13)
Y

where c ,y, c,,y are the creation and annihilation operators associated with the non-

interacting eigenstates. All physical observables involve the above electronic operator.

One can show that the electron density operator n(q) when written in terms of the

projected onto the n-th Landau level operator p(q) acquires a structure factor as can









be shown from


n(q)= i2, '(r, (r)e-, qr

= f L, q --iqy Y
Y
Y


F,(q)p(q),


Cn,yCn,Y+q.g2


(3.14)


where L,(x) is the n-th order Laguerre polynomial. These structure factor has the

form
q2,2 q202
F,(q) e 4 L (3.15)

and the analytic expression for the projected density operator is given by


p(q) e-iqY /2. (3.16)
Y

The following commutation relation holds for these projected density operators [53]


[p(q), p(k)]


-2i sin (q2 k)2)p(q + k).


(3.17)


The Landau gauge is useful in introducing the physics of electrons in high

magnetic fields but the non-interacting electron wavefunctions associated with such

a basis are not very simple to use due to the presence of the continuous quan-

tum number associated with the CM\1 coordinate position. A much more useful

basis of non-interacting electrons arises out of the symmetric gauge choice: A =

(-By/2, Bx/2, 0). In this basis the good quantum number becomes the z-component

of angular momentum and the wavefunction form is given by [38]


( r) (nm (r \ i(n-m)OL n-ml ( -r2/2
' (n+m-| n-m|)/2 2f2
f f^ (Z r


(3.18)









where n is the Landau level index, m the z-component angular momentum index, and

L'(x) are the associated Laguerre polynomials. The normalization constant is given

by


Cnnm 2 (3.19)
27rn!

Parity is determined from the exponent n m (whether it is even or odd).

3.2 Hartree-Fock Approximation

To build a realistic model of 2D electrons we need to include the Coulomb

interaction among them. Since the interaction is a four-fermion operator there is

not much hope for us to develop analytic results unless we approximate it. The

best, and most widely used, way of doing that [38, 39,52] is through the Hartree-

Fock approximation which captures the necessary long and short range effects of

the Coulomb interaction. Additionally, as we have explained previously, we need to

project this operator onto a given Landau level, in order to take into consideration the

peculiar dynamics that arise due to the presence of the high magnetic field. This task

is performed simply in the Landau gauge, where we can derive analytic expressions

for all the terms involved. We start by projecting the four-fermion operator of the

Coulomb interaction by using the result of Eq. (4.32). What we find is


H = d2t d2r'rj(r) j(r')V(r r') (r') (r)

1 fd2q 1 27re2 (23.2
2q t 27 -2 [F.(q)]2p (q)p(-q), (3.20)
2 (27)2 47e q


where c is the background dielectric constant. At this point we treat the 2D electron

system as ideal by ignoring the finite thickness in the third direction, which is present

in a real system. Additionally, we do not include screening effects arising from the

presence of electrons in the filled Landau levels. Later on we will be able to relax









that constraint and investigate a more realistic model and conclude on the validity

of this simple approach. The Hartree-Fock approximation consists of pairing the four

fermion operators in groups of two, averaging on one of the groups as follows


P(q)p(-q) ( e- ,Y_ t,,Y+ 2) ,y_. ,:
Y,Y

S(cY-qt2/2Cny'- -'/2)CnY'+qy2/2CY+n '/2 (3.21)


What this entails is that the interaction potential VHF = VH + VF is composed of

two parts, the Hartree part associated with long range physics (classical Coulomb

interaction)


VH (q) 272 [F(q)]2, (3.22)


and the Fock (exchange) part associated with short range physics, and as we show in

the Appendix is of the form

OO
Vp(q) dxxVH(x/f Jo('.'), (3.23)


where Jo(x) is the zeroth order Bessel function and the x integration is dimensionless.

In the Appendix we provide analytic expressions for both of the terms above for the

n = 0, 1, 2, 3 Landau levels. The final expression for the energy associated with the

projected Coulomb interaction of a 2D electron gas in the Hartree-Fock approximation

becomes

HHF (2 2VHF(q)(p(-q))p(q). (3.24)

This will be our starting point for treating the 2D electron system and investigating

the energetic and the stability of the different quantum states associated with it.














CHAPTER 4
ISOTROPIC CRYSTALLINE PHASES

4.1 Stability Analysis of Isotropic M-electron Bubble Crystals

We would like to investigate the stability of the different crystalline states the

2D electron gas is capable of realizing at higher Landau levels. As we mentioned in

the introduction the crystalline states can be characterized in general as M-electron

bubble crystals. These < i-l -I 1- have the same structure (and triangular symmetry)

as the Wigner
M electrons; and according to previous theoretical Hartree-Fock investigations these

M-electron bubble <( ,-I 1 succeed each other in increasing M order as we approach

half-filling in a given Landau level [36-39]. The last state to win the energetic race

is the charge density wave state (CDW), termed stripe state, that is realized close to

half-filling.

Before we proceed with the stability calculation let us investigate how the dif-

ferent parameters of the < i ,-I ,liii. structures are interrelated. The total filling factor

of the system is given by

v 2-71- (4.1)


where f is the magnetic length, and N and A are the total number of electrons and

area of the sample, respectively. As we mentioned previously, the electrons that

belong to the filled Landau levels are considered inert (they don't participate in the

< i, -1 11i. 0i,,ii process) and at most they provide screening effects for the Coulomb

interaction. As a result, it is useful to distinguish between the total filling factor

(pertaining to the whole system) and the partial filling factor (pertaining to the active

electrons in the partially filled Landau level). Depending up on the type of M-electron









BC configuration of the system, the partial filling factor is defined as

N*
v* = 2= 2--- (4.2)

2 ,-- (4.3)
ABC


where n is the Landau level index, N* is the macroscopic number of electrons in the

partially filled Landau level, and A = V3/2at is the M-electron BC unit cell area

(aB is the lattice constant and as usual we assume triangular lattice configuration).

A typical M-electron bubble has a radius rB. Since the local filling factor on each

bubble is one, while the density is M/TrrB, if we apply Eq. (4.2) we are lead to the

relation rB = 2M for the bubble radius. Additionally, applying Eq. (4.3) for the

WC case (M = 1) we find that aB = avM, where a is the WC lattice constant. One

can consider Eq. (4.3) as an alternative definition for the unit cell area, which traced

back to the lattice constant, produces the useful result


aB M (4.4)


Finally, it is easy to find the ratio between the M-electron bubble radius and the

lattice parameter to be

r 3 (4.5)
aB 27

The above definitions will prove useful since they allow us to fix the sample density (as

is the case in a real sample) and determine changes in the lattice configuration, when

the applied magnetic field is varied. In order to avoid cluttering the symbolism too

much we will drop the specific BC subscript from the above definitions and introduce

it only when necessary.

In order to investigate the stability of these structures we need to calculate the

shear modulus associated with any given M-electron bubble crystal and discover the









region where it becomes zero, which signifies the onset of instability. The shear moduli

are evaluated by expanding the cohesive energy given by the general formula


Ecoh = U(R-R'), (4.6)


to second order in the electron displacements around the lattice sites R. Our basic

task is to define the electron interaction potential U(r), coming from the Coulomb

repulsion among electrons but modified due to the special dynamics the high magnetic

field introduces, the Hartree-Fock approximation, and the quantum corrections arising

from the microscopic physics of the system. Having accomplished that task, it is easy

to show that the elastic energy associated with Eq. (4.6) is of the form

Elastic 2 t U(q) Y Ciq'(R-R1) [(R)u(R')- U,(R) u(R)], (4.7)

q R#R'

where Q2 is the total sample area coming from the Fourier transform of U(r). We

can introduce at this point, the Fourier transform of the discrete displacement fields

according to the following definitions


U (R) (q)eiqR (4.8)

U,(q)= Ac > u(R)e-iq'R. (4.9)
R

The discrete and continuous transformations are mixed, and one needs to be careful

with the units. For that reason, we introduced Ac (the unit cell area). This maintains

the proper units for io,(q), which according to Eq. (4.8) are L3. If we substitute the

above in Eq. (4.7) and use the following definition


iq.R (q Q) Q ,,Q, (4.10)
R c c








we find that the general elastic energy expression becomes


Eelastic =A JU(Q + q)(Qa + q,)(Q + ) U(Q)Qa}

x u (q)ua(Q' q)
Q/

J ( q {U(Q + q)(Q, + q,)(Q + %q) U(Q)QaQ}

x ia(q)Ui3(-q). (4.11)


In other words we have brought the elastic energy equation into the general form
given by
Eelastic 2j ( (q)3 (q)q)(q), (4.12)

where ,"(q) = ui(-q) and 03o(q) is the dynamical matrix defined as


t(q) = >Q {U(Q + q)(Q, + q)(Qa + qa) U(Q)QaQa}. (4.13)


In the high magnetic field limit (and at low temperatures) the electronic wavefunction
extent (of the order of ) can be assumed to be much smaller than the lattice parameter
of any given crystalline structure, and as a result we can expand the dynamical matrix
given by Eq. (4.13) up to second order around q = 0. Additionally, by assuming
isotropic interactions (which is true for the Coulomb interaction) we end up to the
following result for the dynamical matrix


{ U r(Q) ( + '+ U'(Q)

1 F q
+2 QQ q] Q U
2C QL40 Q



-iqJQ3 (4.t14)









where the divergent term Q = 0 is removed from the sum if the properties of the

positive background are taken into consideration. We should notice at this point
that the non-singular Fock term associated with U(Q = 0) is maintained in the sums,

since the positive background cancels only the singular Hartree term. This is observed

throughout this work. According to the classical theory of elasticity [43], the elastic

energy density of any 2D medium is given by the general formula


Selastic = -A ., ," UkUl, (4.15)


where Aijk are elastic constants with only a certain number of them being independent

or non-zero (depending on the given symmetry of the elastic medium). For the tri-

angular lattice configuration the above expression simplifies to


+elastic IAUiUUUj + ( aiUj + jUi
2 L
1 (A + 2p) [(0au)2 + (9yUy)2] + 2A\u9 yuy

+ Pu [(aY)2 + (9YU)2 + 2,9uy9yu,] (4.16)


In the above, the elastic constants A and p are the Lam6 coefficients that are deter-

mined by the following relations that hold due to the triangular lattice configuration


Axyxy A-yxyx A-xyy Ayxxy p, (4.1 7)

Axxyy Ayyxx A, (4.1 8)

xxxx yyyy A + 2p/, (4.19)

Axxyy Axxxx -2"Axyxy, (4.20)


and the rest of the possible Aijk's is zero. The shear modulus c66 (associated with

the energy cost of shear deformations) is given by Ayxy = p and the bulk modulus ci








(associated with the compressibility of the system) is given by Axxxx = A + 2p. Our
result for the dynamical matrix can be related with the above definition of the energy
density for the triangular configuration if we introduce the Fourier transforms of the
displacement fields and write

1
Elastic = 2Ab/3g lt3Q/,(q)i *(q). (4.21)


Comparing the above with the expression for the elastic energy of Eq. (4.12), we find
that the elastic constants and the dynamical matrix are related with the definition


\avq^ v 3(q). (4.22)


In other words the bulk modulus will be defined as c1 = xz(qzx)/qx, and the shear
modulus will be given by c66 'xx(qy)/q2 or if we use Eq. (4.14) we end up to the
general expressions for the bulk and shear moduli


c 2fM (q)

+ ( ) [U(Q) + U'(Q) + Q U"(Q) U'(Q) (4.23)

c 266 2 U'(Q) + QU (IQ) -u'(Q) (4.24)


In the above expressions we have used Eq. (4.4) for the M-electron BC lattice constant.
These are the most general results one can find by Taylor expanding Eq. (4.6) to second
order in the displacements and assuming an isotropic interaction potential.

4.1.1 Classical Order Parameter Approach

In this approach we treat the 2D electron system as an isotropic M-electron BC
of triangular symmetry (which is proved to be the stable ground state for the Wigner









S 10 x 10-3

3- 8-

2- 6-

1 0 4-

0 2

-1 0C

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
v v

Figure 4.1: Shear modulus (in units of e2/4 c3) as a function of partial filling factor
v* for the lowest Landau level and for the WC (left panel) and 2e BC (right panel)
using the order parameter approach.


(iv-l I1 [44]). We treat the M-electron bubbles as point-like particles fluctuating

around their lattice site positions and the electrons inside a bubble are treated as

classical interacting particles. This allows us to define the local filling factor, in

accordance with Goerbig et al., as [39]


v*(r) (rB |r R u(R)|), (4.25)
R

where 0(r) is the Heaviside step function, rB the M-electron bubble radius, and u(R)

the M-electron bubble displacement around the lattice site R. The above choice of

local filling factor produces a crude step-like approximation for the density profile of

the crystalline structure. The direct and reciprocal lattice vectors for the hexagonal

lattice symmetry are defined as [44]


Rjj' y + j, (4.26)

Q 27 2j- (j'
Q a / j' (4.27)










X103 x 10-3

6
2
5
4


-1 2
1
-2-1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
v v

Figure 4.2: Shear modulus (in units of e2/47 c3) as a function of partial filling factor
v* for the n = 1 Landau level and for the WC (left panel) and 2e BC (right panel)
using the order parameter approach.


where we have used Eq. (4.4) for the M-electron BC lattice parameter. For a 2D

sample of total area Q the interaction energy associated with the bubble Ji ~1I 1 con-

figuration in the Hartree-Fock approximation is given by [39]



E (22 VHF(q) A(q)2, (4.28)
2 (7 q


where, VHF(q) is the Hartree-Fock potential given by Eq. (3.24) and A(q) is the

Fourier transform of the local filling factor which is found from Eq. (4.25) to be



A(q) M2 J (qrB) e-iq(R+(R)) (4.29)
Q2 qrB R


Ji(x) is the first order Bessel function. If we substitute the above in Eq. (4.28) it is

easy to show that it assumes the general form of Eq. (4.6) where U(q) is given by



U(q) VHF(q) ( 2MJ(qB) 2. (4.30)



We can investigate the stability of this structure by calculating the shear modulus,

given by Eq. (4.24), for different magnetic field values, or partial filling factors Notice










X 10-3 x 10-3

4-
1



-2
-2 -
-3
-4 -4-

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
V V

Figure 4.3: Shear modulus (in units of e2/47c 3) as a function of partial filling factor
v* for the n = 2 Landau level and for the WC (left panel) and 2e BC (right panel)
using the order parameter approach.


that for the bulk modulus we find the typical long wavelength singularity coming from

the first term of Eq. (4.23) if we take into account the form of the potential energy from

Eq. (4.30). This behavior is in accordance with well known results for the classical

Wigner ( i-il I1 [44].

What we have achieved so far is produce an analytic expression for the elastic

moduli in the semiclassical Hartree-Fock approximation where the electron gas is

treated as point particles fluctuating around their lattice equilibrium positions. In

Figs. (4.1 4.4) we plot the shear modulus versus partial filling factor v* for Landau

levels n = 0, 1, 2, 3 and for the isotropic WC (M = 1) and the isotropic 2-electron

per bubble (< i ,1 (2e-BC) (M = 2) cases, where the interaction energy is given by

Eq. (4.30). We notice that in Fig. (4.1) (which corresponds to a WC in the lowest

Landau level) we reproduce well known results by Maki and Zotos, where the isotropic

WC state becomes unstable around filling factor v* 0.48 [54]. Additionally, we find

that for the n = 2 and n = 3 Landau levels the isotropic WC destabilizes around

v* ~ 0.24 and v* ~ 0.18 respectively, but the M = 2 isotropic BC can live up to

v* 0.39 and v* 0.31, respectively. This is in accordance with known results [38, 39]

where the WC becomes unfavorable eventually to the hierarchy of many electron BC's.







48

x 10-3 x 10-3
6
4-
4 2
22 0


0 -2 -

-2 -4-

-4 -6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
V V

Figure 4.4: Shear modulus (in units of e2/47f3) as a function of partial filling factor
v* for the n = 3 Landau level and for the WC (left panel) and 2e BC (right panel)
using the order parameter approach.


The above classical approach manages to reproduce general properties for the

2D electron gas and show an indication of stability interplay between the different

states, but it is unable to adequately capture the quantum physics associated with the

correlated electron system and in particular the fact that the electron wavefunctions

extend a considerable distance (of the order of the magnetic length) around their

lattice site positions, which radically alters their short range interactions (captured

by the Fock term in our model). This semiclassical picture ignores that fact by

assuming a step-function density pattern, localizing the point-like electrons around

their equilibrium sites. As a result, any further attempt to investigate the energetic of

the different phases will not be accurate enough in reproducing realistically the short

range interactions associated with electron wavefunction overlaps. In what follows we

attempt to improve on this approximation.

4.1.2 Microscopic Approach

In order to incorporate the quantum physics of 2D electrons more faithfully in

our model we have to build a microscopic theory of the electron wavefunctions and

derive, to Hartree-Fock level, information about the energetic and stability of the









system. For the microscopic theory we will use the non-interacting electron wave-

functions in the symmetric gauge given by Eq. (3.18). The difference with the semi-

classical approach applied earlier, is in the Ansatz for the local charge density which

we can improve by assuming that for the general M-electron isotropic BC the real

space approximation of it becomes [36, 38, 55]

M-1
n,(r) E '' (r- Ri)l2. (4.31)
i m= 0

In other words, we assume that the electrons in the M-electron isotropic BC con-

figuration are in their non-interacting eigenstates (characterized by the z-component

of angular momentum quantum number m, and the Landau level index n) and by

Pauli's exclusion principle are forbidden to occupy identical states. The spin degree of

freedom is assumed to be frozen by the high magnetic field and does not contribute.

For strictly perpendicular magnetic fields this is accurate, but the existence of an

in-plane component will change that, since it will couple with the electronic spin and

force it to become relevant. This microscopic approximation is better than the semi-

classical one, since the important short range physics (coming from the electronic

wavefunction overlaps) is taken into consideration. The Fourier transformation of the

projected electronic density is defined according to Eq. (3.14) as


p (q) q 2- L(q / (4.32)
Ce-q22/4Ln(q2f2/2) '


and the generalized Hartree-Fock cohesive energy similar to Eq. (4.28) assumes the

form
EH t q 2
EH 1 2 (22 VH(q) (4.33)

Since we are interested in the local density per bubble, it will prove useful to separate

the lattice summation from the density by defining the projected density at a given









bubble as pn(q) -= Z nm(q) so that


pn(q) p(q) -q. (4.34)


and the corresponding electron density at a given bubble nn(q) is given by an equation

similar to Eq. (4.32), namely

M-1
hfn(q) / dr'. (r) 2-e qr (4.35)
mO 0

The projected density for a given Landau level and given angular momentum m

assumes the form

n(q) f dr'. (r) (236qr
S-q22/4L,(q2f2/2) (4.36)

If we perform the above integration we find identical results for both n = 2 and n = 3

cases (independent of n) rendering the n index unnecessary, and allowing us to drop

it whenever possible to simplify the notation. Below we list the results we find for the

projected electron densities per bubble (for the two Landau levels n =2, 3) and for

the first three angular momentum cases


po(q) e-24, (4.37)

p(q) ( 2) 24, (4.38)

p2(q) q2f2 + q -)q 22 /4. (4.39)


Using the above analytic expressions we find the following result (depending on

angular momentum index m) for the 2D interaction of the electrons


U jmm,(r) (272 pnm(q)VHF (q) pm nq)eiq) (4.40)









x 10 x 10-
3 -



2-








5 3


Figure 4.5: Shear modulus (in units of e2/47c3) as a function of partial filling factor
v* for the n = 2 Landau level and for the WC (left panel) and 2e BC (right panel)
using the microscopic approach.


We show in Appendix B the Fourier transforms of the above interaction potential for

different m values and for the 2e BC case. The above general expression can be used

to find the cohesive energy associated with such a system, namely



EHF mm' Rj) + U (0), (4.41)
i4j T7,T7' i Tm

which has the general form of Eq. (4.6), besides the i i i, !" term associated with

the interaction of electrons within the same bubble. This term does not contribute to

the elastic properties of the system, since these degrees of freedom are associated with

deformations of the internal structure of the bubble, which we consider higher order

corrections in this kind of elastic approximation that we apply to the electronic system.

Additionally, we see that the above expression does not allow for self-interactions

among the electrons that lie on the same bubble.

The modified interaction potential defined in Eq. (4.40) incorporates quantum

effects among electrons and presents a better approximation to the bare Hartree-Fock

interaction since it averages on the spatial effect of the presence of other electrons.

Using this interaction potential we can reevaluate the shear modulus from Eq. (4.24).









x 10 x 10-












005 01 015 02 025 03 035 4 045 5 0 005 01 015 22 025 03 035 04 045 05
V V


Figure 4.6: Shear modulus (in units of e2/4wc3) as a function of partial filling factor
v* for the n = 3 Landau level and for the WC (left panel) and 2e BC (right panel)
using the microscopic approach.


Our results for the WC and 2e BC in the n = 2, 3 Landau levels are shown in Figs. (4.5-

4.6). We notice that the partial filling factor region of stability for a given case

remained the same as in our semi-classical results, but approaching half-filling the

behavior has become different. Notice that by increasing the filling factor amounts

to decreasing the magnetic field or increasing the density of electrons in the system

(with the effect of reducing the crystal lattice parameter value, as can be clearly seen

by Eq. (4.4)). This parameter change renders the short range physics more relevant

and their effects more well pronounced. Consequently, we see for example in Fig. (4.3)

that the n = 2 isotropic WC will become reentrant close to half filling according to the

semi-classical model but in Fig. (4.5) (where short range physics is accounted for by the

microscopic model) this never happens. This is to be expected since as we mentioned

earlier, i< l i,11i. Ii, ii is implemented by the direct long range term in the Coulomb

interaction while conglomeration is due to the short range exchange term. The semi-

classical model favors (by construction) the former, while the current microscopic

model attempts to incorporate the quantum physics of wavefunction overlaps, which

affect dramatically the importance of the latter term. Another exhibition of this, is
















0 021 -
0 019
0017
0015
0 013-
0011-
0 009 -
0 007-
0 005 -
0 2 4 6 8 10 12 14 16 18 20 22 24
r

Figure 4.7: Interaction potential Uoi(r) (in units of e2/4wc) vs. r (in units of f).


the characteristic non-monotonic behavior that is observed in the shear moduli which

signals the onset of dominance of the exchange term.


4.1.3 New State: Bubble Crystal with Basis


As we mentioned earlier the internal degrees of freedom in a bubble have been

considered higher order corrections to the physics investigated in this work. This

might not necessarily be true, since we have not systematically probed on those degrees

of freedom due to the difficulty such a task presents when treated in a microscopic

level. Nevertheless, as a preliminary attempt of investigation, we can offer the fol-

lowing special case which can be easily incorporated into the current model. We

can focus our attention on the isotropic 2e BC case, and allow the two electrons to

assume a finite distance from one another within the same bubble. This can serve

as a rudimentary approximation for internal structure. The possibility of such a

state arises, if one plots the interaction potential between these two electrons within

a bubble (Uoi(r) given by Eq. (4.40) for the m = 0, m' = 1 case). The Fourier

transform of Uoi(r) is shown in Eq. (B.21) of the Appendix. Surprisingly, there is a

well pronounced local minimum at an inter-electron distance of ro 1.48 (as shown









in Fig. (4.7)) that -I'-'-, -I the possibility of bubble deformations mediated by the

minimization of the electronic repulsion. In other words the two electrons in the

bubble might adjust their guiding center coordinates at this optimum distance ro in

order to minimize their repulsion. The difficulty of describing a general lattice with

basis of this sort is considerable (and out of scope for this work) so we would prefer

to investigate a limiting case that constraints the electrons to displace their guiding

centers along one direction only. This limiting case should be able to indicate if these

internal degrees of freedom have a prominent role in the physics of the 2D electron

system.

The way we can implement a finite distance between the two electrons in a

bubble is by redefining the lattice vectors associated with them as


rm, = Ri + (m )rox, m =0,1, (4.42)
2

where Ri are given by Eq. (4.26) and m distinguishes between the two electrons. In

order to study the stability of this < i-i 1li iii structure we need to derive an expression

for the dynamical matrix starting from the cohesive energy given by Eq. (4.41) and

Taylor-expanding to second order in the displacements. As alv-,v, we defer to the

Appendix all the cumbersome details and present here the final result


A2) =- eiQro( in') me')Un,(Q + q)(Qa + qj(Qf + q3)
C Q

UTmm(Q)QaQaj. (4.43)


Notice that in the limit ro 0 we reproduce the usual result of Eq. (4.13). For

the shear modulus calculation, we follow the usual procedure of expanding the total

dynamical matrix
S(q)1 (4.44)
(q ()









3x10 x.10











005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05
V V

Figure 4.8: Shear modulus (in units of e2/47c3) as a function of partial filling factor
v* for the n = 2 (left panel) and n = 3 (right panel) Landau level for the bubble
< i -I J1 with basis.


in small q and we reproduce the same result of Eq. (4.24) but with the interaction

potential U(q) given by


U(q) = eiq.ro(m-mUmm'(q)


= VHFEq)e- q2/2 1 + cos(,o) 1 + cos(qro))q22 + q 4] (4.45)
L-q2 + cos(q8ro) 2 (


In the Appendix we show analytic expressions for the first and second derivative of

the above potential.

We present our results in Fig. (4.8) for the shear modulus of the BC with basis

for the n = 2, 3 Landau levels. We notice that the (il -I 11ii. structure appears

quite stable. This is a first indication that the internal degrees of freedom might

p1 i,- an important role in the physics of the 2D electron system. This BC with basis

state is not a solution of the Hartree-Fock equation (contrary to the rest of the M-

electron states [38]). It arises as the variational solution though of the Hartree-Fock

Hamiltonian satisfying
aE0h(r 0, (4.46)
Or









where ro 1.48f. In other words, this BC with basis state is the best approximation

we can build at this point that probes the internal degrees of freedom associated with

bubble deformations, and minimizes at the same time the Hartree-Fock energy. Below

we evaluate the normal modes associated with all the above < i-I ,ii:,,w structures as

a final test of stability.

4.1.4 Normal Modes and Zero Point Energy

In order to study more systematically the stability of a < -i 1 ,iii,. configuration

we need to calculate the normal modes (for different filling factors) showing that

they have real dispersive nature instead of a diffusive one, which is characteristic of

instabilities. This kind of calculation is based on the elastic matrices (evaluated earlier

for the different crystalline states) since it investigates the dynamics of an electron

under the presence of the perpendicular magnetic field and in the vicinity of elastic

forces coming from the rest of the electrons in the system. Our most accurate model

is the microscopic one so we would like to use the elastic matrices associated with it

to perform the normal mode calculation.

The elastic force associated with electronic interactions is given by the real

space dynamical matrix which starting from the cohesive energy formula Eq. (4.41)

and expanding to second order in the displacements is found to be

Z Z'(Ri) "-(R,) omm' (4.47)



where

67 (r) 0aT U~Tn, (r), (4.48)

and the interaction potential of the different electrons inside a bubble is given by

Eq. (4.40). Notice that another path of approach could be to Fourier transform

Eq. (4.13) back to real space but there is a multiplicative constant associated with that









as we comment in the Appendix. Having written the dynamical matrix in real space

the equation of motion for an electron of mass m and charge e > 0 in the presence of

a perpendicular magnetic field B and elastic forces associated with interactions from

the rest of the electrons becomes


m dt2 E a3 (Ri Rj) lj eBa (4.49)


This equation covers both the WC and 2e BC cases since the indices m, m' distinguish

between electrons in the same bubble. For the BC with basis one has to repeat the

procedure from the beginning starting from Eq. (4.41) but using the lattice vectors

of Eq. (4.42) only to find the following expression for the dynamical matrix


KIf'(R, Rj + ro(m m')x) >II >I ,w (Rk + ro(m m")x)
T" k

Q '(R( Rj + ro(m mx), (4.50)


where
one of Eq. (4.48) in the ro -- 0 limit. The equation of motion for the two electrons in

this kind of BC becomes


md fu 4 .'(R R, + ro(m m')x)u eB (4.51)
jm'

We show in the Appendix in great detail how to solve the above equations of motion

and derive the normal modes for all the (
We show our results for the three different isotropic (< i-ii configurations

evaluated on the irreducible element of the first Brillouin zone in Fig. (4.9). In general,

there is a gapless mode (magnetophonons) for long wavelength excitations and there is

a gapped mode (magnetoplasmons) (of the order of wc) associated with inter-Landau
















00717 . 727 0072 074

0 0627 --0 6567 0063 0 71
.: .
0 0538 -0 5926 0054 068
S0448 0 528 0 45 0 6s

0o0358 ... -0 4645 0036 .. .-062

0 0269 ..' 4 004 0027 -059

00179- -0 3363 0018 -056

0 009- -0 2723 0009 -053

0 2082 00 5
rq/(f/a) rq/(f/a)
0 09 O 8

0 081 077

0 072.20. 0 74

0and 063a is the lattice paramet 07

0 054 states by picking points inside the irreducible first Brillouin zon68

0 045 0 65-

0 036 -0 62

0 027 -0 59

00/8 -056

0 009- -053

1 q/(ff/a) I


Figure 4.9: Normal modes for the triangular crystalline structures in the n 2

Landau level and on the irreducible first Brillouin zone element. Top left panel: WC

at v 0.18. Top right panel: 2e BC at 0.30. Bottom panel: BC with basis at
v* 0.20. The left axes correspond to magnetophonons (lower curves) and the right

axes to magnetoplasmons (upper curves). Frequency measures in UJ0 e2 /47rWM units

and a is the lattice ( i v-I ,J parameter.



level excitations. For the WC case we reproduce well known results [56], and both


the 2e BC and BC with basis show similar structure in their modes. Their degrees


of freedom are doubled (due to the presence of an extra electron) which doubles their


magnetophonon and magnetoplasmon modes as well. We comment in the Appendix


in great detail on the graphical peculiarities associated with plotting elements of a


Brillouin zone.


Finally, one can evaluate the zero point energy associated with the normal modes


of these states by picking -I.,- : X!" points inside the irreducible first Brillouin zone










Table 4.1: Zero point energy of WC, 2e BC and BC with basis (BCb) for different
partial filling factor values v* within their range of stability.
v* Ewc EBC EBcb
0.05 0.359785* 0.360259 0.360394*
0.10 0.362771 0.357701 0.358184
0.15 0.372628 0 :",7-il 0.359024
0.20 0.375746 0.360150 0.361797
0.25 0.362931* 0.362661 0.361640*


element that have the largest weight and then writing the zero point energy (in units

of e2 /4wc) as [57]
1 4wcth Nmodes 6
EZP 2M e2 E aiLj(qi), (4.52)
j= 1 i 1

where Nmodes is the number of modes for the given crystalline structure (two for WC,

and four for the 2e BC) while ai is the corresponding weight of the given special

point. Following Cunningham [57] the special points for the WC hexagonal lattice

configuration and their corresponding weights are


21 3 2 1 4, 1 8 1, -3 (4.53)


q4-6: {(,(1, ), (2, ), 2(5, )} a4, 6 -2, (4.54)


where all reciprocal lattice vectors measure in 7/a units.

In Table (4.1) we present our results for the zero point energy associated with the

different isotropic crystalline structures investigated so far. The ones marked with an

asterisk indicate the onset of instability for the given structure and the corresponding

filling factor.

4.2 Energetics of Isotropic Crystalline Phases

At this point we would like to consider the energetic of the states whose stability

we calculated in the previous sections and in particular investigate the possibility of









the inert filled Landau levels altering the energy interplay between different crystalline

phases. Also, another point of concern is the finite thickness of any real sample of 2D

electrons and how the extent of the wavefunctions in the z-direction can potentially

influence the energetic of the system. We would like to investigate both of the models

developed earlier, the classical approach and the microscopic approach, by evaluating

the cohesive energies, for the different crystalline configurations, given by Eqs. (4.28,

4.41), respectively.

In order to incorporate finite thickness effects and screening from filled Landau

levels we follow the general path which consists of modifying the dielectric constant

so that it acquires a q-dependent structure of the following form [58, 59]


e(q) e 1 + ( J2 (qRc)) exp (Aqf), (4.55)
qaB

where c is the bare dielectric constant of the substrate material (for GaAs it is 12.9co),

Re = 2n + 1 is the cyclotron radius, aB the Bohr radius, Jo(x) the zeroth order

Bessel function, and A a finite-thickness parameter involving the finite z-direction

extend of the electron wavefunction. The result of the above modification is to add

an extra q-dependence on the Hartree term of the Coulomb interaction (given by

Eq. (3.22)) but overall the expression remains of the same form. On the contrary, for

the Fock term (given by Eq. (3.23)) the expressions are no longer analytic (as the ones

shown in Appendix B) due to the complicated structure of the integral involved. Below

we evaluate the cohesive energies associated with the two models (the semi-classical

and the microscopic approximation) and for the different crystalline phases investi-

gated so far. For the sake of completeness we will include in our energetic comparison

for the n = 2 and n = 3 Landau levels the classical stripe state (charge density wave)

which consists of continuous stripe areas of the sample that are completely filled by











a t .. 01-i 1 ..
(M1)bare (M1) bare
(M=2) bare (M=2) bare
(M=1)screened -- (M=1)screened
0 0 -- (M=2) screened 0 05 =2) screened
.005 \. (M=1) screened,finite thickness
...... .. (M=2) screened,finite thickness




- LU
-015 -0 15

-02 -02

-0 25 -0 25

0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05
V V

Figure 4.10: Cohesive energy for the isotropic WC (red) and 2e BC (blue) in units
of e2 /47c for the n 2 Landau level (left panel) and the n 3 Landau level (right
panel). Solid lines correspond to bare Coulomb interaction. Dashed lines correspond
to Coulomb interaction where screening effects are accounted for, and dotted lines
correspond to the latter case where finite thickness effects are included as well.


electrons (v* 1= ) separated by empty areas (* = 0) of finite width. The cohesive

energy associated with this phase will be shown below.


4.2.1 Cohesive Energy of Modified Coulomb Interaction: Classical Model


We numerically calculate the bubble crystal cohesive energy per electron based

on Eq. (4.28) (in units of e2/47rc) and given by



nC 1 v* 47r 2 J (2 Q)
Ecoh 1 VHF 4Q JrBQ) (4.56)
h 27f2 M e2 (rBQ)2



where as usual, M is the number of electrons per bubble, v* the partial filling factor

given by Eq. (4.2), and rB the bubble radius. The neutralizing background cancels

the singular Hartree term involved in the Q = 0 case, but the nonsingular Fock

term is maintained (the weight factor involving Ji(x) is evaluated at the Q = 0 limit

and it is easy to show it gives 1/2). The general M-electron BC we investigate here

incorporates the WC case as well and for the dielectric constant we use the modified









expression given by Eq. (4.55) to incorporate finite thickness effects and screening

from the filled Landau levels.

Our results are shown in Fig. (4.10) for the n = 2 and n = 3 Landau levels,

respectively. We see that the screening effects from the inert Landau levels along with

the finite thickness effect from the electron wavefunction extend in the z-direction only

shift the associated cohesive energy scale, but do not alter the interplay between the

phases. At approximately the same partial filling factor (v* _- 0.22 for n = 2 and

v* ~ 0.17 for n = 3) the 2e BC < i-l I becomes more favorable compared with the

WC, irrespective of the type of modification applied to the Coulomb interaction. As a

result we can conclude that ignoring these corrections in our model we are not missing

out on important physics besides some quantitative adjustments.

4.2.2 Cohesive Energy of Modified Coulomb Interaction: Microscopic
Model

The microscopic model we developed earlier provides a much more accurate

approximation for the energetic interplay between the different crystalline states.

We evaluate the cohesive energy per electron (in units of e2/47cf) for a general M-

electron BC state by starting with Eq. (4.41) and using the Fourier transform of

Umm,(r) (given by Eq. (4.40)) to find


co -f e2 y / : U (Q) (4.57)
Q

As we mentioned in the classical approximation, for the summation in Q we retain

the Q = 0 term only for the Fock part. The above expression incorporates the M 1

WC case as well.

For the BC with basis we have to start again from Eq. (4.41) and use the

lattice vectors associated with this structure (given by Eq. (4.42)). We then Fourier

transform and use the by now familiar Eq. (4.10) to find the following expression for












S 0Wigner crystal l


0~ ",-0 02



-0 1 -006

LIJ 015 LU -008

02 01

-025 --- --- --~- -0 12-

-03 -0 14-

-0 35 -0 16
0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05
V V
gner crystal
0 bubble rO=00


-002- 1


-0 04




-008-





0 005 01 015 02 025 03 035 04 045 05
V

Figure 4.11: Cohesive energy in units of e2/47c for the isotropic WC, 2e BC, BC
with basis, and stripe state for different modifications of the Coulomb interaction
and for the n = 2 Landau level using the microscopic model. Top left panel: bare
Coulomb interaction. Top right panel: screened Coulomb interaction with no finite
thickness effects. Bottom panel: screened Coulomb interaction with finite thickness
effects included.


the cohesive energy per electron (in units of e2/4K7E):




E BCb )iQ)ro(m-m') (4.58)
cob-.YY Z U 1mm1(Q ) (4.58)
mm' Q


The above needs to be applied for the specific 2e BC with basis case we have developed


in this work. The expression simplifies to




Eohb Uoo(Q) + U11(Q) + 2 cos(Qxro) Uoi(Q)], (4.59)

4 C


-- wg... clysal
*"bubble r0_0 0


0 05k








64



015- 0

0 -
005-
-0 02






0- -
S-025 -0











LU" LL\
-M006










-0 02 08








0 005 01 015 02 025 03 035 04 045 05
(V



















Figure 4.12: Cohesive energy in units of e2/ 4w for the isotropic WC, 2e BC, BC
with basis, and stripe state for different modifications of the Coulomb interaction
and for the n = 2 Landau level using the microscopic model. Top left panel: bare
Coulomb interaction. Top right panel: screened Coulomb interaction with no finite
thickness effects. Bottom panel: screened Coulomb interaction with finite thickness
effects included.


where we used the fact that Uoi(q) = Uio(q).

As we mentioned earlier we would like to evaluate the cohesive energy for the

stripe state as well and compare it with the rest of the crystalline structures. The

stripe state cohesive energy is constructed by starting from Eq. (4.28), and assuming

for the local filling factor [36, 39]



v*(r02 0(a/2 -x x (4.60)
v* (r) 0 (a/2 Ix xjl) (4.60)









where a = v*as is the width of one stripe (determined by the partial filling factor),

while xj = jas, and as is the stripe periodicity. By Fourier transforming the above

we find that the cohesive energy of the stripe phase (in units of e2 /47c) is of the form

S 1 e H 27W sin2(TQ *j)
coh w2 *e2 > v Ka) .2 (4.61)


The optimal stripe periodicity is obtained by minimizing the above with respect to

as. Following Goerbig et al. [39], we use as = 2.76Rc for n = 2 and as = 2.74Rc for

n = 3. As ahv-i the Q = 0 term is retained in the Fock part of the potential and

the limit of the weight factor (at j = 0 it gives 7v*) is taken.

We present our results for the bare Coulomb interaction, and for modifications

associated with finite thickness and screening from filled Landau levels in Figs. (4.11-

4.12). Our conclusions from the classical approximation analysis remain intact for

both Landau levels, which leads us to the safe generalization that finite thickness

effects, and the inert Landau levels can be ignored in any further investigation of

the energetic among the (< i- i -11iiw. phases. However, another surprising result has

emerged. The BC with basis state has become the undisputed winner in the energetic

interplay with a very distinct energy difference from any other state for a wide range

of partial filling factors. This is another strong indication that the internal degrees

of freedom pl i,- a crucial role in the physics of the system. What this result infers

is that different (
one investigated here) might p1l -v a strong role into the physics of the quantum Hall

system. This can be a future direction for our research to further investigate the

possibility of structural transitions in these kinds of systems. Finally, we should draw

attention to the conventional stripe state winning over the conventional WC and 2e

BC states when half-filling is approached, something that is to be expected according

to previous investigations [36, 38, 39].














CHAPTER 5
ANISOTROPIC CRYSTALLINE PHASES

5.1 Solving the Static Hartree-Fock Equation

In our previous treatment of the 2D electron system we approximated the

Coulomb interaction in the Hartree-Fock level and used an Ansatz for the electron

density (being either a classical order parameter or a microscopic approximation based

on non-interacting electron wavefunctions). We would like to progress further in that

direction in faithfully capturing the electronic density characteristics of the system

by solving for the eigenfunctions of the Hartree-Fock equation and finding the quasi-

particle states associated with the 2D electrons. This constitutes an improvement on

the microscopic model and allows us to systematically study the different ( i i ,1-

line phases in the set of lattice symmetry associated with the triangular lattice

configuration. We plan to include anisotropy into the i-i iii., structures and

investigate how the energetic are affected.

Our starting point is the Hartree-Fock Hamiltonian (similar to Eq. (3.24)) that

is used in studies of the 2D electron system under the presence of a high perpendicular

magnetic field and given by [36, 38, 39]


1 f d2q 2
HHF VHF(q) n(q) 2, (5.1)


where n(q) is the electronic density and VHF(q) is the modified Hartree-Fock inter-

action potential given by Eq. (C.5). To avoid overloading our presentation of the

model we will place all the cumbersome definitions and calculations in Appendix

C and retain only the essential ones necessary for the presentation. We define the









electronic quasiparticle density for an M-electron BC according to

M
n(r) E (r R) 2 (5.2)
i a=1

where, ,(r) is the a-eigenstate of the above Hartree-Fock Hamiltonian. Notice

that this is a similar definition to Eq. (4.31) only we use the more accurate quasi-

particle wavefunctions instead of the non-interacting electron wavefunctions. Of

course, we have no prior knowledge of the former (and in fact we need to find them

self-consistently) so it is necessary to approximate their form by expanding them on

the latter basis of non-interacting wavefunctions y(r) according to

Ns-l
*' ,(r) = Cmam(r), (5.3)
m=0

where, N, is the dimensionality of the truncated Hilbert space used. We show in

the Appendix in detail how extremizing the above Hamiltonian with respect to the

quasiparticle wavefunctions and then projecting the result onto the non-interacting

particle wavefunction basis we obtain the following eigenvalue equation for the M-

electrons associated with each bubble in the BC

N.,-
> Gm 2mCm EaCmra, (5.4)
rn1 0

where Cma are the expansion coefficients associated with Eq. (5.3), GmimT is a 2nd

rank tensor given by Eq. (C.12) in the Appendix, while gmTnmn3m4 is a 4th rank tensor

associated with the overlap integrals of the quasiparticles and given by Eq. (C.13) in

the Appendix. We numerically diagonalize the above equation until our solutions

for the expansion coefficients converge within a 10-4 accuracy. We comment in the

Appendix on the details of the algorithm. Once the algorithm has converged, we

place each of the M electrons of a bubble on the quasiparticle states associated with








68



0 01


0I 05
0 05 \


-0 05
-0 15
LU..... LU1 o \


-02 -015-

-025- 0------ -'-02-'-

0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05
V* v

Figure 5.1: Ground state eigenvalue energies in units of e2 /47c as a function of the
partial filling factor v* for the Hartree-Fock equation. Left panel: WC and 2e BC
cases for the n = 2 Landau level. Right panel: WC, 2e BC and 3e BC cases for the
n = 3 Landau level.


their eigenvalues E, in ascending order. As a result the cohesive energy of the general

M-electron BC assumes the form



Ecoh= 1 E, (5.5)



where the factor of 1/2 compensates for counting each pair of electrons twice in the

general Hamiltonian given by Eq. (C.8) [60].

The above is a general result for an M-electron BC so it can be easily applied to

our cases of interest for the WC and the 2e BC. Additionally, we can study the 3e BC

as well, which according to previous studies [38, 39] can become energetically favorable

for the n = 2 and above Landau levels at a certain range of partial filling factors. We

show our results in Fig. (5.1) and as we see they are identical to the ones developed

earlier within the more simplified microscopic model. This is a verification that the

microscopic model used earlier for the M-electron BC configuration is a solution of

the Hartree-Fock equation, in agreement with previous studies [38] where the solution

of the time-dependent Hartree-Fock equation produces the hierarchy of M-electron

BC's. It should be noted that for the above results we have used a minimum expansion








69


08 02

07- -005 -
06-- -01
05 015
04- -02-

03- -025-
02- -0 3 -



0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05
V* V*

Figure 5.2: Left panel: Values of anisotropy that minimize the cohesive energies
for WC and 2e BC for different values of v*. Right panel: Ground state energies
associated with the minimizing values of anisotropy for WC and 2e BC. We also plot
the traditional stripe state for comparison.


basis dimension (Ns), disallowing the electrons to form hybrids by constraining them

to lie on their non-interacting ground states, for all the < i-i 111ii,, cases studied. So

the fact the we get identical results with the simplified microscopic model developed

earlier is not a surprise but it serves as a consistency check for this kind of improved

method before we employ it for the much harder task which we discuss below.


5.2 Introducing Anisotropy into the Crystalline States


As we explained in the introduction the contemporary theoretical results for the

2D electron system under the presence of a perpendicular magnetic field predict that

close to half-filling, the stripe state (charge density wave) becomes favorable to all the

(< i iii.,,i phases. This stripe state is described in the continuum order parameter

language (discussed earlier) by introducing the partial filling factor of Eq. (4.60). One

would expect though that the transition from a crystalline to a liquid phase would be

less abrupt (especially at fixed low temperatures) allowing for the < l,,-I ,11,i, system

to explore internal degrees of freedom before finally melting into a liquid. Also, to

a certain extent, one would expect reversibility (no hysteresis) associated with the

decrease or increase of the applied magnetic field around the region that the stripe









state becomes favorable. All of the above point to the importance of the internal

degrees of freedom associated with the crystalline phases, which for the subset of

triangular symmetry discussed in this work, translates into introducing anisotropy

in the (i-i iI Iii, configurations. Escaping out of this subset will allow us in the

future to investigate a more complete group of structural phase transitions which will

potentially involve reorientation of the unit cell due to local straining forces arising

from the electron correlations.

In order to incorporate anisotropy within the triangular lattice symmetry we

have to redefine the lattice vectors given by Eqs. (4.26-4.27) by introducing the

anisotropy parameter E (not to be confused with the dielectric constant e) which is

zero for no anisotropy and one for complete anisotropy. The new direct and reciprocal

lattice definitions become


R3 t' 1 2 v)) (5.6)

Q1T \( a (5.7)


where the M-electron BC lattice parameter is still given by


a M (5.8)


Incorporating the above new definitions into our code is straight forward and one has

to pick different anisotropy values to investigate (for given partial filling factor) which

one minimizes the cohesive energy of the given < i ,~I iii., structure.

We show our results in Fig. (5.2) for the simplified microscopic case, where we

do not allow inter-electron excitations within the WC and 2e BC by choosing the

expansion basis dimension to equal the number of electrons per bubble. The left

panel shows the specific minimizing values of anisotropy for given value of v* for




























. . i . . L . . .
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-25 -20 -15 -10 -5 o 5 10 15 20 25



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.. . . . . . . . . . . . . . . ..
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-25 -20 -15 -10 -5 o 5 lo 15 20 25
2=0 5


. . ..-.- ..- j . . . . L-- .- .- -- .-.- .-.-.- .- .-- -.-
. . . . . . . . . . . . ..---` `-` `-` ``- -`- -`- --` `-
`--` `-` `-` ``- -`- -`- -`` `-`- -`- -`` `-` `-` `-- -`- -`- -`-- -`- -`- -`` `-` `-` `-- -`-

. . . . . . . . . . . . ..---` `-` `-` ``- -`- -`- --` `-
`--` `-` `-` ``- -`- -`- -`` `-`- -`- -`` `-` `-` `-- -`- -`- -`-- -`- -`- -`` `-` `-` `-- -`-

...............................................---``-``-```--`--`---``-
. --. . . . . . . ..`` `-` `-` `-- -`- -`- -`-- -`- -`- -`` `-` `-` `-- -`-

. . . . . . . . . . . . ..---` `-` `-` ``- -`- -`- --` `-
`--` `-` `-` ``- -`- -`- -`` `-`- -`- -`` `-` `-` `-- -`- -`- -`-- -`- -`- -`` `-` `-` `-- -`-
. . . . . . . . . . . . ..---` `-` `-` ``- -`- -`- --` `-

`--` `-` `-` ``- -`- -`- -`` `-`- -`- -`` `-` `-` `-- -`- -`- -`-- -`- -`- -`` `-` `-` `-- -`-
. . . . . . . ..`- `-` `-` `-` -`- -`- -`- --` `-` `-` ``- -`- -`- --` `-












.. . . . ..................................................
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.................................................................................

..................................................................................

.."" '" ". "". "". "".". ". "".." """". ". "'". ""."."".""" ..".""" ." "."'"." '" "."".""".""" .." ."
.. .. .. ... .. ... .. .. ... .. .. ... .. ... .. .. ... .. .. ... .. ..
.. .. .. .. .. .. .. .. .. .. .. .. .. ...=O.. 7 5.. .. .


Figure 5.3: Reciprocal lattice points for different values of anisotropy. We see how



the









the WC and 2e BC, while the right panel shows the ground state cohesive energies






associated with these minimizing values of anisotropy. We also plot the traditional






stripe state cohesive energy (given by Eq. (4.61)) and as we see a surprising result






emerges since it does not become energetically favorable over the anisotropic WC state






even close to half-filling. In fact, the anisotropic WC state increases considerably its






energy difference from the rest of the states, as half-filling is approached. Also, the






overall cohesive energies have dropped in value, compared to the isotropic ones from






Fig. (5.1). In view of these results one is obliged to reconsider the definition of stripes






in these < i i iii. systems in terms of anisotropic X< i 1- and investigate further






the effect of anisotropy in the system.






Before we proceed, we would like to show how anisotropy deforms the reciprocal






lattice vectors by plotting a finite number of them given by Eq. (5.7). Our results






are shown in Fig. (5.3), where reciprocal lattice vectors associated with the cohesive













09 -WC WC
2o BC 'BC


0B 02
0 8 -0 1 -


07 -02- o







06 -0 3-
05 -
-04-



-0 6 -
028
-0 7 -
0 1


0 005 01 5 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05





07 -02-




S-03- e
-0 - -
04- ;' m

031 -06-

02-
-' 0 7 -\


0 005 01 0 15 02 025 03 035 04 045 05 005 0 1 05 2 025 03 35 04 045 0
V* V*

Figure 5.4: Solutions of the Hartree-Fock equation for the anisotropic triangular lat-

tice configuration. Top left panel: Values of anisotropy that minimize the cohesive

energies for WC, 2e BC and 3e BC for different values of v* and for the n = 2 Landau

level. Top right panel: Ground state energies associated with the minimizing values

of anisotropy for WC, 2e BC and 3e BC. Bottom left panel: Same as top left panel

but for n = 3 Landau level. Bottom right panel: Same as top right panel but for

n = 3 Landau level.



energy lattice sum and the overlap integrals (given by Eq. (C.13)) are plotted for


representative values of anisotropy. We see a channel-like structure emerges consisting


of one-dimensional periodic chains of electron guiding centers. In other words, the


< i *l -1iiw., discreteness of broken translational invariance is alv--,- maintained in this


novel "stripe" configuration. This is contrary to the traditional stripe state properties,


where translational invariance along the stripes is restored. As we will see below,


this is a crucial difference that radically alters the elastic properties of the system









as well. We should also mention here that in the Hartree-Fock approximation, the

electronic wavefunction overlap is greatly favored by the Fock term. When anisotropy

is introduced into the system the electronic guiding center lattice points are brought

into closer proximity enhancing wavefunction overlaps and thus improving on the

effect of the Fock term. That is the reason why the overall cohesive energy values are

reduced compared to the isotropic ones.

One should also mention at this point that a hidden degree of freedom emerges

in view of this analysis. When the electronic guiding centers are brought within

proximity, the dimensionality of the truncated space used as a basis to describe the

electron wavefunctions becomes crucially important. The reason is best understood

through the interplay of the Hartree and Fock terms. The more available states exist

for electrons within a bubble to occupy and form hybrids, the more the Hartree term is

optimized since the electronic charge will be spread out the most. On the other hand,

the Fock term, according to our analysis above, is optimized as well since electronic

overlaps will be inevitable (but less concentrated) within a bubble. Additionally, the

more non-interacting electron wavefunctions (eigenfunctions of the z-component of

angular momentum) are used in the expansion of Eq. (5.3) the more they will extend

around the BC sites causing "seo, ,i,1 i y inter-bubble overlaps, enhancing even more

the effect of the Fock term. This is of course a computational by-product that one

cannot get rid off unless one goes beyond the Hartree-Fock approximation which is

beyond the scope of this work. For practical purposes, if a reasonable number of

states is used in the truncated quasiparticle wavefunction space, and convergence is

assured through the algorithm, that should be enough to capture the essential physics

of the system. We should also notice at this point that the z-component of angular

momentum eigenstates change their parity with m, meaning that an electron within

a bubble will only form same parity hybrid states, using either even or odd values of

m. This is verified numerically independent of the dimensionality of the basis space.









As a result, for the WC case for example, if one wants the electron to form a hybrid

state using five non-interacting electron states one needs to use a dimensionality of

N, = 9. For the 2e BC one needs to add the extra odd parity state and use N 10.

We show our results for the n = 2 and n = 3 Landau levels in Fig. (5.4) where

we have used appropriate dimensionality for the different crystalline structures so

that a total number of five non-interacting electron wavefunctions participate in the

hybrids. For the M = 3 states though, we have not increased to N 15 since

from our experience that state never becomes energetically favorable anyway. As it

is clearly shown in these results, the 2e BC becomes irrelevant as well, contrary to

our Fig. (5.2) results, and the anisotropic WC seems to be the natural way that the

internal degrees of freedom in the crystalline system optimize the ground state as the

applied magnetic field is changed. C!i ,i: i5i the truncated quasiparticle wavefunction

space does not seem to alter the energetic between the states but as we mentioned

earlier all the cohesive energies suffer a downward shift due to the dominance of the

negative Fock term into the numerics.

Another point of interest is on the transitions from isotropic to anisotropic

WC that appear to be of first order and taking place around v* 0.1 for both

Landau levels. These transitions appear in the BC cases as well but are not as

strong. In light of our discussion earlier about the dominance of the Fock term in the

short range physics, along with the existence of the hidden degree of freedom (the

truncated quasiparticle wavefunction dimensionality) this is to be expected. Going

from the isotropic state to an anisotropic one radically affects the short-range physics

which is controlled by the value of anisotropy in the crystalline structures. In the

real system, a similar behavior can be expected as well for similar reasons, but the

degree of freedom associated with the truncated space dimensionality is absent, or

better stated, optimized. This can become plausible if one prints the actual electron

wavefunctions on a unit cell. In order to do that we can focus our attention on the























-0
-6 -4 -2 0 2 4 6 8 10 12 14 16 18 20
x/I


8 8











-6 -6
-6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20
x/I x/I

Figure 5.5: Density profiles on the unit cell for the n = 2 Landau level and v* = 0.11
for truncated basis dimensionality of Ns = 9. Length unit is the magnetic length.
Top left panel: Isotropic state. Top right panel: Anisotropic F = 0.8 case using only
the m = 0 non-interacting state. Bottom left panel: Same as top right panel but for
a hybrid of m = 0, m = 2 non-interacting states. Bottom right panel: Optimized
hybrid state solution of the Hartree-Fock equation for E = 0.8.


WC case and the n = 2 Landau level for simplicity. We place one electron on each of

the three sites of the unit cell and write the electronic wavefunction as an expansion

of the non-interacting wavefunctions 0m(r) given by Eq. (3.18). What we find is


Ns- 1
(r) (r), (5.9)
m=o0

where am are normalized to unity coefficients truncated in a Hilbert space of N,

dimensions. It is useful to rewrite the above in cartesian coordinates. For the specific








case of the WC considered here one can easily show that the above written analytically
in cartesian coordinates becomes


b v(x, y) a r22x+iY2+ a, ( x+y2) (x+ ey 2/2
N 3 a m x 2 \
+ am+2 Y L( 2 2(5.10)
o v2- ( + 2)!f f 22

In order to build the density profile on the unit cell we need to define the appropriate
lattice vectors (coming from Eq. (5.6) and associated with a unit cell placed at the
origin) as


R1 (0, 0), (5.11)
3 3a(,2(1- E).
R2 1 02 5.2)
R 1 -2 ( '(
R3 1= 2 1 ), (5.13)


and then numerically evaluate for any different set of a,'s we want the probability
density given by


Itot(r) 2 (r R) 2 + i(r R2) 2 + | _(r R3) 2. (5.14)


We show our results for the above density profile in Fig. (5.5) for different anisotropy
values and hybrid states. Hybridization and the Hilbert space dimensionality have
a dramatic effect in shaping the electron wavefunctions and consequently affecting
their interactions. On the first two graphs, the electrons are prevented to form hybrid
states, and are constrained on the m = 0 states. On the third graph, the electrons
are allowed to from m = 0, m = 2 hybrids and on the last graph, we have used the
optimization result from our code for the actual hybrid state solution of the Hartree-









Fock equation for the given values of v* and anisotropy. We notice that the latter is

a radically different density profile from all the rest.

5.3 Elastic Properties of Anisotropic Crystals

We can proceed further into studying the stability of the
This might pose a difficulty since (as we discussed earlier) the short range correlations

that exist in these anisotropic (< i --I ,! force electrons to extend their wavefunctions

at multiples of the magnetic length and render any perturbative expansion around

fixed equilibrium positions invalid or inadequate. One has to employ a better method

that avoids Taylor-expanding around the real electronic displacements. This can be

achieved by following the Miranovi6 and Kogan approach [61]. We start by writing

the general elastic energy density associated with a 2D anisotropic < i v-I as


el = Cll,(tu)2 + C,y(yUy)2 + C66, (OyU)2 + C66,y ( y )2 + 2cil,9y(.9u 9)( 2

+ 2c66,xy yU xUy) (5.15)


The elastic moduli ci,x, cn,y are associated with uniform compressions along the x:, y

directions, respectively. To describe shear deformations along the same directions we

use C66,x, C66,y, respectively. The cross term c1,xy, introduces the interaction energy

associated with the mixing of the compression modes directed along x and y and the

same applies for the shear mode mixing associated with c66,xy. In the isotropic case,

the above expression for the elastic energy density assumes the form of Eq. (4.16)

but in the present case, the only symmetry left to impose a constraint on the elastic

constants is rotational invariance, which imposes the following interrelation


C66,xy C66, + C66,y (5.16)
2









This can be easily proved if one applies a uniform rotation u = -",,i, uy = uox

(where u0 is a dimensionless constant) and demands invariance of the elastic energy.

The method that Miranovit and Kogan have developed is based on these kinds of

uniform deformations in the direct lattice that are traced back into deformations of

the reciprocal lattice vectors. That way, one avoids expanding around direct lattice

site fluctuations in order to calculate the elastic properties of different crystalline

structures. The Miranovi6 and Kogan method relates a general uniform deformation

of the form

U = :, (5.17)

where the coefficient Ua,3 is a dimensionless constant, to uniform reciprocal lattice

deformations, which to first order in Ua,s can be written as [61]


Qa = Qa U3,aQ3. (5.18)


We have defined Q and Q to represent the deformed and undeformed reciprocal lattice

vectors, respectively. Evaluating the elastic energy on the deformed reciprocal lattice

vector set and subtracting the value associated with the undeformed lattice, suffices

to reproduce the elastic constant associated with the given deformation. The general

expression becomes

c 2 [E(Q) EZ(Q)]. (5.19)

For a shear deformation along the x direction given by ux = /, Uy = 0, the cor-

responding deformation in the reciprocal lattice vectors becomes


Qx = QX, (5.20)

Qy = Qy uoQ,. (5.21)









For a shear along the y direction given by uy = uox, ux = 0, the corresponding

deformation in the reciprocal lattice vectors becomes


Qy = Qy, (5.22)

Qx = Q1 uoQy. (5.23)


A uniform rotation given by u = u,,, U, uox, induces the deformation


Qx = QX uoQy, (5.24)

Q Q + ,,, (5.25)


and finally, a squash deformation given by ux uox, u = -",,/, induces the reciprocal

lattice vector deformation


QX (1 uo)QX, (5.26)

Qy ( + uo)Qy. (5.27)


In all of the above u0 is a small dimensionless constant. In this work we are interested

in the shear deformations associated with the anisotropic crystals investigated, so we
only focus our attention on C66x and c66y.

We present our results for the shear moduli of the isotropic WC and 2e BC for

the n = 2 Landau level in Fig. (5.6). They serve as a benchmark for this method
of approach since our older results of Fig. (4.5) are reproduced (involving Taylor-

expansion on the lattice displacements). Our new results, pertaining to the anisotropic
(< i iii. structures studied earlier, are presented in Fig. (5.7) where we plot c66x

and c66y for the ground state of the anisotropic WC for the n = 2 Landau level. For

the partial filling factor values where the isotropic WC is favorable, the two shear
moduli are equal (it does not show in the figure due to different scales used) but when











012- /
01 -
0 08 -
o o06 -
004 -



-002 -
-004-
-006-



Figure 5.6: Shear moduli (in units of e2 /4Ec) for the isotropic WC and 2e BC for
the n = 2 Landau level produced using the Miranovic and Kogan approach [61].


the transition point to anisotropic WC is crossed (around v* 0.1) the c66y becomes

vanishingly small. This is because this kind of shear deformation is along the direction

of the channels shown in Fig. (5.3) which does not cost any energy (a characteristic

property of smectics).

The striking difference with a conventional smectic is that C66, becomes zero

as well. The term in the elastic energy density of Eq. (5.15) associated with that

elastic constant is replaced by a bending term K(0 u2)2 [62]. This is due to the

fundamental difference between the conventional stripe state and the anisotropic WC:

in the former, translational invariance is restored along the direction of the stripes but

in the latter, this is no longer true. The periodic channel-like structure persists at any

value of anisotropy or filling factor. As a result, a deformation of the form u = cinx

corresponds to a rigid rotation for the stripe state (with no energy cost associated

with it) but it corresponds to a compression along the direction of the channels for

the anisotropic WC case, with a finite energy cost associated with it.


5.4 Analysis of Experimental Results


We are in a position now to discuss the experimental findings shown in Fig. (1.6).

According to previous theoretical treatments of the dynamical response of an isotropic

















iO 025 '
02 0
0 015 0


0 005 -
005 01 015 02 025 03 035 04 045 05 0 005 01 0 15 02 025 03 035 04 045 05
V* V*

Figure 5.7: Left panel: Shear modulus C66, for the ground state of the anisotropic WC
for the n = 2 Landau level. Right panel: Shear modulus c66y for the same crystalline
structure. Notice that they both coincide in the range below v* 0.1, where the
ground state is the isotropic WC, but it does not clearly show in the graphs due to
the different scales used. The shear moduli measure in e2 /4cf units.


WC under microwave irradiation [35], the resonance pinning frequency is given by



>p -- (5.28)
PmLc


where, pm = m/(7a2) is the mass density, uc the cyclotron frequency, and E is the

quasiparticle self-energy associated with the presence of disorder into the system and

given by

A ^, (5.29)
('( ,I so

where, A is the variance of the random pinning potential, and o is the smallest

correlation length between (the magnetic length), and Qd (the disorder correlation

length). If we assume that the strong magnetic field imposes < d then we end up

with the following result for the resonance pinning frequency



A ~ MC6 (5.30)









This result is generalized for the M-electron BC case and we specifically show the

dependence on the partial filling factor that comes from the shear modulus c66 (the

dependence on is weak for n > 2). In light of the experimental findings of Fig. (1.6)

(right panel), one expects that the shear modulus associated with the isotropic WC

state for the n = 2 Landau level will increase until around v* 0.19 where it should

begin to decrease until the crystal becomes unstable. This is the exact behavior we

find for the isotropic WC shown in Figs. (4.3, 4.5). On the other hand, the second

coexisting phase shown in Fig. (1.6) starting around v* 0.15 follows the opposite

trend, if compared with the M-electron BC shear modulus. For example, the 2e BC

shear modulus is shown to go up and then decrease in Figs. (4.3, 4.5) for the region

of interest (0.15 < v* < 0.35), while according to Eq. (5.30) and the experimental

results of Fig. (1.6) it should have the opposite behavior. According to our energetic

analysis the anisotropic WC becomes favorable around v* 0.1, which is reasonably

close to the region that this coexisting phase reveals itself. All of the above point to

the conclusion that this second peak appearing in the experimental data is not due

to any isotropic M-electron BC but to an anisotropic crystal (WC or 2e BC). This

seems to be supported by the results of Li et al. [63] for the AC response of a quantum

Hall smectic which resemble the experimental ones. We expect the same conclusion

to hold true for the n = 3 Landau level as well.














CHAPTER 6
CONCLUSIONS

We have studied different aspects of the quantum Hall system in high magnetic

fields. At first we considered a bilayer system and studied the tunneling current

characteristics associated with it in the incoherent regime. We found that the inter-

IV,- interactions modify the tunneling current in two v--,v-. At first, due to the short

range part of those interactions, the collective modes of the ril-ivr system become

gapped. This leads to a suppression at low bias values of the tunneling current.

Secondly, we found that the long range part of the interlayer interactions soften the

effect of the Coulomb interaction among electrons in the same l-ivr and as a result

they shift the tunneling current curve to lower bias values. This is attributed to the

excitonic attraction that a tunneling event creates between the tunneling electron and

the hole that is left behind resulting into an overall reduction of the energy associated

with such a process.

Our study was analytical and systematic and was able to capture different

properties of the experimental system, such as scaling behavior of different para-

meters or tunneling current response to interlayer separation change. Although we

worked in the incoherent regime, we have set a foundation to develop this model fur-

ther, and incorporate coherence effects as well trying to reproduce the most recent

experimental results.

Further on, we studied the < i ii,. phases of the quantum Hall system by

analyzing their stability. We built our theory successively starting from contemporary

treatments of the problem using the classical order parameter approach, that averages

on the electronic density neglecting the important microscopic physics, and evolved

to the microscopic approach, that uses a more accurate Ansatz for the electronic









density which incorporates, to a certain degree, the microscopic physics involved in

the real system. We derived a general theory for the elastic moduli of such systems

and studied for specific Landau levels the stability of the < i -i 1 iii. structures finding

that for different ranges of the partial filling factor these structures are stable. We

also studied the normal modes associated with the above structures and in the process

probed the internal degrees of a bubble finding that these degrees of freedom pl i" an

important role into the physics of the system.

Additionally, we investigated on the effect of the filled Landau levels and the pos-

sibility that screening arising from them might contribute significantly into the physics

of the system and we also incorporated finite thickness effects, coming from the finite

extent of the real system in the third direction. What we found is that although the

actual cohesive energies shift in value, the interrelation among the different crystalline

states remained the same, and consequently concluded that by omitting the inert filled

Landau levels and finite thickness effects we are not missing out on important physics.

Finally, we further improved on our model by solving the static Hartree-Fock

equation associated with an electron in these quantum Hall systems and were able

that way to find the quasiparticle wavefunctions for different < i l-Ii i ,' structures.

In that part of the work, we were able to include anisotropy into the system and solve

for the ground state, finding that there is a first order transition between the isotropic

Wigner crystal and the anisotropic one that renders the latter the undisputed winner

in the energetic race. We found that the anisotropic Wigner crystal for strong values

of anisotropy resembles a smectic with much lower energy (for the whole range of

filling factors) than the traditional stripe state. Additionally, we showed that for the

anisotropic Wigner crystal, translational invariance is never restored along the smectic

direction but shear deformations along that direction cost negligible energy.

Our work was limited only in the subset of crystalline symmetry associated with

the triangular lattice but we have set the foundation for a more detailed study on the









possible structural transitions associated with reorientation and deformation of the

unit cell due to straining forces developing among the electrons in these systems.

This is part of our future direction, where we hope to find the ground states this

kind of system evolves into when the applied magnetic field is varied. These ground

states will provide to us realistic elastic constants, associated with the energy cost of

deformations, which we plan to couple to a dynamical response theory where disorder

effects and thermal noise will be incorporated as well. As a result we expect to be able

to reproduce the experimental findings of microwave resonance response associated

with these kind of systems.

Additionally, we would like to investigate the excitonic condensation problem

associated with the quantum Hall bilayer structures and attempt to shed some light

into the physics involved in such a system where coherence among electrons in both

I rv.-- i p, i, a dominant role. Having achieved the necessary understanding on that

state we would like to couple the modes associated with it to tunneling electrons and

reproduce that way the prominent zero bias coherence peak in the tunneling current

found in experiments.













APPENDIX A
BILAYER SYSTEM EIGENMODES

A.1 Single Layer Eigenmodes

Here we provide an analytic derivation for the diagonalization procedure of the
hili r quantum Hall system. We essentially repeat the procedure highlighted in the

main text, including all the details, for reasons of completeness and in order to provide
a coherent treatment of our theoretical model without having to reference formulas
at the beginning chapters, which will result in a somewhat disjoint presentation.
We start by introducing the single 1vr 2D system of electrons in the presence of
a perpendicular magnetic field in the continuum elastic approximation. For a system
of density no, the dynamics are described by the following Lagrangian


L = no d2r r 2 efu A(u) A(Au)2 _- a (m'u, + OIU")2
JL 2 2no 'I
+ no d2r[V. u(r)][V'. u(r) 2 (A.1)
+2 nr[0 u(r')] 47lr r'l |


where the i-ntir -1 r Coulomb interaction is treated in the continuum linear approx-
imation (charge fluctuations are given by 6n/no = -V u). This approximation
is correct in the absence of vacancies and interstitials. For the vector potential we
choose to work in the symmetric gauge so that A(u) = (-Buy/2, Bu1/2, 0) and B
is the applied magnetic field. We decompose the displacement field u into transverse

(UT = z (iq x u)/q) and longitudinal (UL = (iq u)/q) components after we Fourier-
transform Eq. (A.1) to find

1 d 2 2q2 1 2 2 / \
L no Tnit T Tni2 + MLL+^ c [UL -'LULr- -MU) 2 2 _-MU)2 2 m (A.2)









The fact that the displacement field is real introduces the property: u*(q) = u(-q).

Additionally, for simplicity we introduced the compact symbolism: u2 = u(q) u(-q).

The process of finding the eigenmodes of the above Lagrangian consists in identifying

the canonical moment of the transverse and longitudinal displacement fields. It is

easy to show that the corresponding results are

a 1
PT O nomrT( + tmnoWtUL, (A.3)
NUT 2
8 1
PL O nomTL -mnoucUT. (A.4)
OUL 2

The next step involves building the equations of motion associated with these fields

that requires using the well-known formula

Sd V 0, i T, L. (A.5)
Oui dt Oti O(Vuj)

As a result we find the following two coupled dynamical equations


UT + ciUL + TUT = 0, (A.6)

1UL cttT + LUL = 0. (A.7)


The presence of the magnetic field couples the two acoustic modes and one needs to

diagonalize the system of equations to find the new eigenmodes. It is easy to show

that the eigenfrequencies of such a system are given by


1 2 + U2 + U)2 + (2 + U( + U)22 4U] (A.8)


In the limit where the magnetic field is turned off we notice that the expected acoustic

modes emerge out. Having written the canonical moment of the displacement fields

we can switch to the Hamiltonian representation and write down for the Hamiltonian









of the 2D electron system


H no J(d {ULPL +UTPT L
n (o 2 2m 2 11 2
Sno j(2)2 2m + + WC[UTPL ULPTJ + 2m + ()UL

+ t (4 + ) U 2 (A.9)


Decoupling Eqs. (A.6, A.7) produces the following


L' + [w~ + a4 + w)2iui + 4WU2 = 0, i T,L. (A.10)


A general solution for both the displacement fields whose dynamics are described
by the above equation consists in identifying all four different components of them
corresponding to all four eigenmodes of the system. We write such a solution as


ui = Aiei+t + Ae-+t + Be-t + B'e-,-t, i = L,T, (A.11)


and using Eq. (A.6) or Eq. (A.7) we solve only for the four independent coefficients
we want to keep. In our case we have chosen the following


UT = AT'+t + A'e e-'+t+BreC t + B e-- t, (A.12)

UL = Z Are [-Aiw + AT-' 2 Bre-' + Be (A.13)
1+ -- UU L2 T

This is the complete analytic solution of the equations of motion for the transverse
and longitudinal part of the displacement field in the presence of a perpendicular
magnetic field.
Next, we canonically quantize the above fields by properly defining creation and
annihilation operators. For the sake of clarity and to avoid cluttering the symbolism









we will suppress no (which multiplies m) and include it only in the final results. To

canonically quantize we need to find the relation between all time derivatives of the

fields and their canonical variables. The equations of motion Eqs. (A.6, A.7) and the

canonical momentum equations Eqs. (A.3, A.4) combined together provide us with

these, given by


2 2
UL -PT (L + -)UL,
m 2
1 2 2
UT = c(c + T+ 2 )UL T +PT,
2 cPLM
1 /) p2 + U) 2 )u)2 p
2 m~T


(A.14)

(A.15)

(A.16)

(A.17)


We pick out the four non-redundant (out of the eight possible) equations for the

coefficients to end up with the 4x4 linear system:


AT + A' + BT + B'T UTo,

iw[AT A'] + iw _[BT -B= PO- Wu

-[A + A] _[BT + B'] = -po )ro,

i A 3 t
+ T T ~~2 CT+2Do


UC + UT PT
Pro,


where ULo, UTo and PLo, Pro are the displacements and canonical moment, respectively

evaluated at time t = 0. After solving the system, we obtain the following relations

between the field coefficients and the canonical variables


1

+ U+
2


2 2 L+ 2 2 2
U- _T c/-" + --PLo + --To


(A.22)


(A.18)

(A.19)

(A.20)

(A.21)


LO I









, 1 22 22 c .
-A- 2 T w/2]uT, + PLo I FPT

+ 2 U o (A.23)
2 "u


BT U) a) [U2 U2 U2
Ui )a)_ + T ,
w(2 +w})

2
2 J




B' 2) {_ [w w w /2]uTo -
,W(2 +- )U -
U)C (a)2 2
+ 1 -2 Lo .


cw-
UTc-
m


2
PLo


(A.24)



c_- (2 -_ 2 )

-PLo + i Po

(A.25)


Notice that A = AT, Bt = BT. After some tedious calculations we can prove to

ourselves using [uio(q),pji(q')] -= :, (27r)22 (q- q') that


[Ar, BTR] =[A, B ]

[A, AT]r 1/ni

2mn + u _
7 ='1


2mnnow- w
h 2
' n2= -S W7


We have reintroduced no in the formulas at this point. Also, the same q is assumed

in all the operators and we have omitted for clarity the 2D delta functions. Next, we

normalize the operators using the following new definitions


al(q)
AT(q) AT(q)
V1'!

a2(9)
BT (q) = BT(q)
V/'n2


at(q)

'n


(A.29)


(A.30)


.0



LL "
LL)


(A.26)

(A.27)

(A.28)









and that way we restore the usual commutation relation


[ai(q), at (q')] (27)22 (q q')ij. (A.31)


As a result, the final form of the annihilation eigenmode operators for the single 1 iv.r
quantum Hall system in the continuum elastic approximation where the effect of the
i-nit-r ,-i r Coulomb interaction is included (to linear order in the elastic displacement

field) is


a(q) )(-U2 + -U2 /2]u(q) + po(q)
2_ U2 W2 +2
+ I PTO (q) i +2 o (q) (A.32)




a2(q) / (U;27 U;2)(77 w2- [ [- P wf/2]ur0(q) "'7' PLcq)
no L +- loI
22 w2 w2 _+ 2(2
+ 2L- PTo (q) + ic 2ULo(q) (A.33)
nom 2

Going back to Eqs. (A.12, A.13) we can re-express them in terms of our new operators

t(q) w al(q) + t() +a2(q) -t
UT(q, t)= a" ei + e-i + a(qei + e- (A.34)
VU1 VU1! V/U2 V/U2
c(q )L+ at(q) +L L+ al (q )e= -+ c -(q )
c_ -i2ot a;c- a _q)
+ 2 a ) (A.35)
0- 0L Vm2

Our task has been completed because going back to the Hamiltonian of Eq. (A.9) and
substituting Eqs. (A.34, A.35) we get the diagonal form


H j fj2j [hdaq+a (q)al(q)+ _a(q)a2(q) + +2- (A.36)









We have found the eigenmodes and eigenfrequencies of the single 1-. r quantum

Hall system including the effect of the i-ntrvil.. v Coulomb interaction. It is interesting

to see how our solutions behave to different limiting cases. In the low magnetic field

limit (wc -+ 0) we discover that

2
wC2 2 L 2, (A.37)

L2 2 L 2 (A.38)
W WT 2 (A.38)


while Eqs. (A.32, A.33) become


ai(q, c -- 0) 1 -i Ult) + 1 Lo(q), (A.39)
2h V2mnoL

a2(q,c -- 0) 2-- UTo (q) + i / pro(q), (A.40)
r2 h V2mno hT

which is the entirely expected behavior of the system in zero magnetic field to decouple

into the original transverse and longitudinal eigenmodes. On the other hand, taking

the high magnetic field limit (wjc -+ oc) we discover expanding the square root in

Eq. (A.8) that


+ + + + O(wL 3), (A.41)
2wc

L o+0(w3), (A.42)
wLc


and using the above in Eqs. (A.32, A.33) we discover

Imno UTo ULo PLo + ipTo
aI (q, uc ) 2h 2 + 2mn-- +' (A.43)
V 2:fhT 2 +'2mnohu"c
0 flo~c WT .L 1
a2(q, c -- oc) 2--- UTo +1 ULo (A.44)
V 2hLLLT z2 2z









Notice that Eq. (A.44) is singular in uc. This is not a physical instability though,

because from Eq. (A.42) we see that the u_ mode it represents has died out in the

high magnetic field limit.

A.2 Bilayer Eigenmodes

Next, we consider the bilayer case for which we include the effect of the inter-
I-|r Coulomb interaction. The dynamics of the system is described by the following

Lagrangian


L = no jd2r mnfi + Mfu CIA A(uA) CUB A(uB)
A A 2 + 2 1
2 (A)2 + (L\LUL2A illr(amUlA + OIU2A)2 + am^UlB + OiUm"
2no

no d2r/[V 7 uA(r) [V UA(r')] --
2c |r r
no Jd2r'[V* uB(r)][V7. uB(r')] ---

no d22 [ A ( B ( ) ]
u(r) [V' uB(r/(x x')2(y y)2 + d2

SK (UA B 2 (A.45)
2 no

The effect of the interlayer Coulomb interaction is introduced through the short range

and the long range part described by the last two terms, respectively. Decomposing

the displacement fields into transverse and longitudinal components (as we did for the

single 1 vr case) we find




1 2 l .2 1 .2 1 1 2 2
2- m LA + 2t LB 2 + 2mUT + [..' ULB ULB UT] 2- M UT
1 2 2 2e2no0 qd_ K)2
L U LB UB LAULB qe (A B- U (A.46)
2 e 2 no








Next, we decouple the two single 1v.,.r Lagrangians by switching to the in-phase
v = (up + UA)/2 and out-of-phase u = uB UA modes that produces


L = no >(2m)i + (2m)i + (2m)uc[vTVL iLVT] (2m)(crq)2V

(2m) (Lq)2 + q(1 + -d)] v
+ l(Tn) + 2(T)i + t(T) [(T)-2 2K 2 2
2x)2 2 22 2 2 Mno
+ t( ) + ( )6 + ()cAUT ULUT] (2) (2rK) 2 -q 2

() (cLq)2 + + q(1 e-) U (A.47)


If we compare the structure of the in-phase and out-of-phase Lagrangians to the single
I,,--r one given by Eq. (A.2), we realize that we can define two "effectsil single 1'.r

dynamics whose eigenmodes and eigenfrequencies can be read off if we change the
definitions for the parameters involved in each case. For the out-of-phase modes we
need to apply the following changes


m _-, (A.48)
2K
w -[c q2 + o] = ), (A.49)
Tmno
S cq + + 27e2 q(1 -qd)] (A.50)


For the in-phase modes the parameters need to change according to


m -+ 2m, (A.51)

W4 --W = O, (A.52)

S- cq2 + 27e20 q(1 + e-qd)] 02 (A.53)


The cyclotron frequency a, stays unchanged in both of the cases. After performing
the above redefinitions for the parameters involved in the single 1-v-.-r eigenmode