Energy spectra of fractional quantum Hall systems in the presence of a valence hole

ArkadiuszW´ojs

DepartmentofPhysics,UniversityofTennessee,Knoxville,Tennessee37996

InstituteofPhysics,WroclawUniversityofTechnology,Wroclaw50-370,Poland

arXiv:cond-mat/0006505v1 [cond-mat.mes-hall] 30 Jun 2000JohnJ.Quinn

DepartmentofPhysics,UniversityofTennessee,Knoxville,Tennessee37996

Theenergyspectrumofatwo-dimensionalelectrongas(2DEG)inthefractionalquantumHallregimeinteractingwithanopticallyinjectedvalencebandholeisstudiedasafunctionofthefillingfactorνandtheseparationdbetweentheelectronandholelayers.Theresponseofthe2DEGtotheholechangesabruptlyatdoftheorderofthemagneticlengthλ.Atd<λ,theholebindselectronstoformneutral(X)orcharged(X−)excitons,andthephotoluminescence(PL)spectrumprobesthelifetimesandbindingenergiesofthesestatesratherthantheoriginalcorrelationsofthe2DEG.The“dressedexciton”picture(inwhichtheinteractionbetweenanexcitonandthe2DEGwasproposedtomerelyenhancetheexcitonmass)isquestioned.Instead,thelowenergystatesareexplainedintermsofLaughlincorrelationsbetweentheconstituentfermions(electronsandX−’s)andtheformationoftwo-componentincompressiblefluidstatesintheelectron–holeplasma.Atd>2λ,theholebindsuptotwoLaughlinquasielectrons(QE)ofthe2DEGtoformfractionallychargedexcitonshQEn.Thepreviouslyfound“anyonexciton”hQE3isshowntobeunstableatanyvalueofd.ThecriticaldependenceofthestabilityofdifferenthQEncomplexesonthepresenceofQE’sinthe2DEGleadstotheobserveddiscontinuityofthePLspectrumatν=1.371.35.Ji,71.35.Ee,73.20.Dx

I.INTRODUCTION

Anumberofexperimental1–20andtheoretical21–37

studiesoftheopticalpropertiesofquasi-two-dimensional(2D)electronsystemsinhighmagneticfieldshavebeencarriedoutintherecentyears.Instructureswherebothconductionelectronsandvalenceholesareconfinedinthesame2Dlayer,suchassymmetricallydopedquan-tumwells(QW’s),thephotoluminescence(PL)spectrumofanelectrongas(2DEG)involvesneutralandchargedexcitoncomplexes(boundstatesofoneortwoelectronsandahole,X=e–handX−=2e–h).10–20,29–35TheX−canexistintheformofanumberofdifferentboundstates.Inzeroorlowmagneticfield(B≤2TinGaAs),

−

onlytheopticallyactivespin–singletXsoccurs.29,34,35AlthoughitispredictedtounbindintheB→∞limitasaconsequenceofthe“hiddensymmetry”ofane–h

−

systeminthelowestLL,21–23theXsisobservedinthePLspectraeveninthehighestfieldsavailableexperi-mentally(∼50TinGaAs).14AdifferentX−boundstateisformedinafinitemagneticfield:anon-radiative

−12,13

(“dark”)spin-tripletXtd.Incontrastwithanear-−23

lierprediction,theXtdremainsboundintheB→∞

−−

totheXtdlimit,30,31andthetransitionfromtheXs

groundstateisexpectedatB≈30T(inGaAs).34,35Atevenhigherfields,Laughlinincompressiblefluidstatesof

−

stronglyboundandlong-livedXtdfermionicquasiparti-32,33cleswerepredicted.Veryrecently,yetanotherbound−

Xstatehasbeendiscovered34inastrong(butfinite)magneticfield:aradiative(“bright”)excitedspin-triplet−−Xtb.TheXtbhasthesmallestbindingenergybutthelargestoscillatorstrengthofallX−states,anddomi-

natesthePLspectrumatveryhighmagneticfields.14ThePLspectraofsymmetricQW’sarenotveryusefulforstudyingthee–ecorrelationsinthe2DEG.Insuchsystems,the2DEGrespondssostronglytotheperturba-tioncreatedbyanopticallyinjectedholethattheorig-inalcorrelationsarelocally(inthevicinityofthehole)completelyreplacedbythee–hcorrelationsdescribinganXorX−boundstate.ThePLspectracontainingmoreinformationaboutthepropertiesofthe2DEGitselfareobtainedinbi-layersystems,wherethespatialsep-arationofelectronsandholesreducestheeffectsofe–hcorrelations.24Thebi-layersystemsarerealizedexper-imentallyinheterojunctionsandasymmetricallydopedwideQW’s,inwhichaperpendicularelectricfieldcausesseparationofelectronandhole2Dlayersbyafinitedis-tanced.Unlessdissmallerthanthemagneticlengthλ,thePLspectraofbi-layersystemsshownorecombina-tionfromX−states.Instead,theyshowanomalies1–7atthefillingfactorsν=13,atwhichLaughlinin-38

compressiblefluidstatesareformedinthe2DEG,andthefractionalquantumHall(FQH)effect39isobservedintransportexperiments.

Thebi-layere–hsystemcanbeviewedasanexampleofamoregeneraloneinwhichthe2DEGwithwellde-finedcorrelations(e.g.,Laughlincorrelationsatν=1

2DEGcanbetunedindependently,asinascanningtun-nelingmicroscope(STM).40Inanothersimilarsystem,achargedimpuritycanbelocatedatacontrolleddistancefromthe2DEG.18,41,42The2DEGhasitsowncharacter-isticlengthsandenergies,suchastheaveragedistance(∼̺−1/2=λ

󰀁3

and

2

3.

Sinceournumericalresults

obtainedforfairlylargesystemscanserveasraw“ex-perimental”inputforfurthertheories,wediscusstheminsomedetailthelastsection.Theyagreewithallourpredictionsmadethroughoutthepaper.

Althoughinthepresentworkwestudyaveryideale–hsysteminthelowestLL,ourmostimportantconclu-sionsarequalitativeandthusapplywithoutchangetorealisticsystems.Toobtainabetterquantitativeagree-mentwithparticularexperiments,theeffectsduetotheLLmixing(lessimportantatd≥2λ)andfiniteQWwidthsmustbeincludedinastandardway(see,e.g.,Ref.34ford=0).Someofourconclusionsshouldalsoshedlightonthephysicsofotherrelatedsystems,suchastheSTM.Inparticular,thescreeningofapotentialofasharpelectrodebya2DEGisexpectedtoinvolve“real”electronswhenUislargeandDissmall,andLaugh-linquasiparticlesintheoppositecase.Anasymmetrybetweentheresponseofa2DEGtoapositivelyandneg-ativelychargedelectrodeisexpectedinthelattercase,becauseofverydifferentQE–QEandQH–QHinterac-tionsatshortrange.Letusalsoaddthattheproblematν=23becauseofthecharge-conjugationsymmetryinthelowestLL.

−Thepresentedidentificationofboundstates(X,X−,XQHn,andhQEn)ine–hsystemsatanarbitrarydandthestudyoftheirmutualinteractionsisnecessaryforthecorrectdescriptionofthePLfromthe2DEGintheFQHregime.Whilethecompletediscussionoftheopti-calpropertiesofallbounde–hstateswillbepresentedinafollowingpublication,46letusmentionthatthetrans-lationalinvarianceofa2DEGresultsinstrictopticalselectionrulesforboundstates(analogoustothosefor-biddingemissionfromanisolatedXtd−31–33

).Asaresult,h(the“uncorrelatedhole”state),hQE*(anexcitedstateofanh–QEpair),andhQE2aretheonlystableradiativestatesatlarged,whiletherecombinationofhQE(thegroundstateofanh–QEpair)orof(unstable)hQE3isforbidden.DifferentopticalpropertiesofdifferenthQEncomplexesandthecriticaldependenceoftheirstabilityonthepresenceof–QE’s7inthe2DEGexplainthediscon-tinuitiesobserved1inthePLatν=13.

2

II.MODEL

Weconsiderasysteminwhicha2DEGinastrongmagneticfieldBfillsafractionν<1ofthelowestLLofanarrowQW.Adilute2Dgasofvalence-bandholes(νh≪ν)isconfinedtoaparallellayer,separatedfromtheelectrononebyadistanced.Thewidthsofelec-tronandholelayersaresettozero(finitewidthscanbeincludedthroughappropriateform-factorsreducingtheeffective2Dinteractionmatrixelements34),andthemix-ingwithexcitedelectronandholeLL’sisneglected.Thesingle-particlestates|m󰀘inthelowestLLarelabeledbyorbitalangularmomentum,m=0,−1,−2,...fortheelectronsandmh=−m=0,1,2,...fortheholes.Sinceνh≪νandnoboundcomplexesinvolvingmorethanonehole(suchasbiexcitonsX2=2e–2h)occuratlargeB,theh–hcorrelationscanbeneglectedanditisenoughtostudytheinteractionofthe2DEGwithonlyonehole.Themany-electron–one-holeHamiltoniancanbewrittenas

󰀄󰀆󰀂††

††ehee

(1)H=cicjckclVijkl+cihjhkclVijkl,

ijkl

†

wherec†m(hm)andcm(hm)createandannihilateanelectron(hole)instate|m󰀘.BecauseofthelowestLLdegeneracy,Hincludesonlythee–eande–hinteractionswhosetwo-bodymatrixelementsVeeandVeharede-finedbytheintra-andinter-layer√Coulombpotentials,22Vee(r)=e/randVeh(r)=−e/

S.Thesingle-particlestatesarethe

eigenstatesofangularmomentuml≥Sanditsprojec-tionm,andarecalledmonopoleharmonics.Thesingle-particleenergiesfallinto(2l+1)-folddegenerateangularmomentumshells(LL’s).Thelowestshellhasl=Sandthus2SisameasureofthesystemsizethroughtheLLdegeneracy.Thechargedmany-bodye–hstatesformde-generatetotalangularmomentum(L)multiplets(LL’s)oftheirown.ThetotalangularmomentumprojectionLzlabelsdifferentstatesofthesamemultipletjustasKorMdidfordifferentstatesofthesameLLonaplane.

3

DifferentmultipletsarelabeledbyLjustasdifferentLL’sonaplanewerelabeledbyL.Thepairofopticalselec-tionrules,∆Lz=∆L=0(equivalentto∆M=∆K=0onaplane)resultsfromthefactthatanopticallyactiveexcitoncarriesnoangularmomentum,lX=0.

Itisclearthatcertainpropertiesofa“strictly”spheri-calsystemdonotdescribetheinfiniteplanarsystemthatweintendtomodel.Forexample,ifunderstoodliterally,finiteseparationdbetweentheelectronandholesphereswouldleadtodifferentvaluesofthemagneticlengthinthetwolayers,andthusintroduceanasymmetrybetweenelectronandholeorbitals(eveninthelowestLL).WhilethiseffectdisappearsintheR→∞limit,itiselimi-natedbyformallycalculatingthematrixelementsoftheinteractionpotentialVeh(r)atanyvalueofdforelec-tronsandholesconfinedtoasphereofthesameradiusR.Thisprocedurejustifiestheuseofsphericalgeometryatarbitrarilylargelayerseparation(notonlyatd≪R).

III.BOUNDELECTRON–HOLESTATESINA

DILUTE2DEG

InordertounderstandPLfroma2DEGatarbitraryfillingfactorνandlayerseparationd,onemustfirstiden-tifytheboundcomplexesinwhichtheholes(minoritycharges)canoccur.Aftertheseboundcomplexesarefoundandunderstoodintermsofsuchsingle-particlequantitiesastotalchargeQ,bindingenergy∆,angu-larmomentuml,orPLoscillatorstrength(inverseop-ticallifetime)τ−1,aperturbation-typeanalysiscanbeusedtodetermineifthosecomplexesaretherelevant(or“true”)quasiparticles(TQP’s)ofaparticulare–hsys-tem,weaklyperturbedbyinteractionwithoneanotherandthesurrounding2DEG.Ifitisso,thelowenergystatescanbeunderstoodintermsoftheseTQP’sandtheirinteractions.ThePL(emission)probestheelec-tronsysteminthevicinityoftheannihilatedholeandthereforetheopticalpropertiesofTQP’sdeterminethe(lowtemperature)PLspectraofthesystem.

Thistypeofanalysishasbeenrecentlyappliedtothee–hsystemsatd=0inthelowestLL,32,33anditshowedthatthelowlyingstatescontainedallpossiblecombina-tionsofbounde–hcomplexes(excitonsXexcitonicionsX−

=e–hand

=nX–e)andexcesselectrons,in-teractingthroughn

effectivepseudopotentials.TheshortrangeofthesepseudopotentialsyieldsLaughlincorrela-tionsbetweenelectronsandexcitonicions,whichisolatethelatterfromthe2DEGandmakethemactlikewelldefinedTQP’swithoutinternaldynamics.Whenappliedtorealisticsymmetricallydoped(d=0)QW’satlargeBandlowdensity(ν<1

π/2e2/λistheexcitonenergyinlowestLLandP†=󰀃theEq.m(−1)mc†(2),amh†

“multiplicative”m(ontheHaldane’ssphere).Becauseof(MP)eigenstateofH(astatecontainingNXneutralexcitonswithmomentumzero)canbeconstructedbyapplicationofP†NXtimestoanyeigenstateoftheinteractingelec-trons.†TheexcitonscreatedorannihilatedwithoperatorsPandP(i.e.byabsorptionoremissionofaphoton)havethesameenergyEXwhichisindependentofotherelectronsorholespresent.ThenumberNXofsuch“de-coupled”excitonsisconservedbyH,onlythestateswithNX>0areradiative,andtheemission(absorption)gov-ernedbytheselectionrule∆NX=−1(+1)occursatthebareexcitonenergyEX.23

Somewhatsurprisingly,itturnsout28,30thatthe“to-tallymultiplicative”eigenstate

obtainedbyadding|ΨNhthe󰀘=Bose-condensed(P†)Nh|Ψ󰀘,

groundstate(3)

ofNX=Nhexcitonseachwithk=0tothegroundstate|Ψ󰀘ofexcessthecombinedN−Nhelectrons,isnotalwaysthegroundstateofe–hsystem.Thisresultsbecausetheinteractionofanexcitedexcitonicstate(i.e.onewithk=0)oftheBosecondensatewiththefluidofexcesselectronscanlowerthetotalenergybymorethanthecostofcreatingtheexcitedexcitonicstate.Typically,aMPstateP†0excitoninto|Φa󰀘statecreated|Φ󰀘byisopticalanexcitedinjectionstate,ofandakthe=absorptionisfollowedbyrelaxationtoadifferent(non-MP,i.e.non-radiative)groundstate.

TheconditionunderwhichthetotallyMPstateinEq.(3)23isthee–hgroundstatefollowsfromthemappingontothe↑–↓(spin-unpolarizedelectron)sys-tem,inwhich|ΨNhthemaximumspin.󰀘Sincecorrespondsν↑=1−toνtheand↑–ν↓↓state=νh,withandthe2DEGisspin-polarized(intheabsenceoftheZeemansplitting)onlyattheLaughlinfillings,theconditionforthetotallyMPe–hgroundstate|ΨNhν−ν−(2p+1)−1󰀘,

is

h=1(4)

withp=1,2,....Atallotherfillings(e.g.,ν−νh=

1

0.300.150.007E−EX (e2/λ)−∆ (e2/λ)Xtd−Xsd−X2−6ree (λ)52e-1h, 2S=40singlettriplet(a) d/λ=0.00(b) d/λ=0.500.080.004X=2e+1h, ∆=EX-E∆X−2=3e+2h, =EX−+EX-E−0.00Xtd−-0.05(a) binding energy0.00.51.01.52.00.00.5(b) e-e separation1.01.52.030.10d/λd/λ0.00E−EX (e2/λ)FIG.2.Thebindingenergy∆ofthetripletandsinglet

−−

chargedexcitonstates,XtdandXsd,andofthechargedbiex-−

citon,X2,asafunctionoflayerseparationd.EXistheexcitonenergyandλisthemagneticlength.

Xsd(c) d/λ=0.751618202224161820−0.00(d) d/λ=1.002224FIG.1.Theenergyspectra(energyEvs.angularmo-mentumL)ofthe2e–hsystemonaHaldanespherewiththeLandauleveldegeneracyof2S+1=41,fordifferentvaluesofthelayerseparationd.Theopenandfullcirclesdistinguishstateswithsingletandtripletelectronspinconfigurations.EXistheexcitonenergyandλisthemagneticlength.

B.ChargedExcitonStates

LLAnexampleofanon-MPe–hgroundstateisthe

−30−

“dark”spin-tripletchargeexciton(Xtd).TheXtdistheonlybound2e–hstateinthelowestLLatd=0.Itisthemoststablee–hcomplexatνh≤2ν,butitsbindingenergydecreasesatd>0whenthee–hat-traction(atshortrange)becomessmallerthanthee–erepulsion.Thedependenceofthe2e–henergyspec-trumondisshowninFig.1.Thespectraarecalcu-latedinthesphericalgeometryfortheLLdegeneracyof2S+1=41.TheenergyismeasuredfromtheexcitonenergyEX,sothatfortheboundstates(thestatesbe-lowthedashedlines)itisthenegativeoftheX−bindingenergy,∆X−=EX−E.Openandfullsymbolsdistin-guishsingletandtripletelectronspinconfigurations,andeachstatewithL>0representsadegeneratemultipletwith|Lz|≤L.TheZeemanenergyofthesingletstatesisnotincluded.TheangularmomentumLcalculatedonaspheretranslatesintotheangularmomentumquan-tumnumbersonaplaneinsuchway34,52thateachLLatL=0,−1,−2,...(containingstateswithK=0,1,2,...,i.e.withM=L−K=L,L−1,L−2,...)isrepresentedbyamultipletatL=S+L.Thus,thelowenergymultipletsinFig.1atL=20,19,and18representtheplanarLL’satM≤L=0,M≤L=−1,andM≤L=−2,respectively.

Itisimportanttorealizethattherecombinationofan

−

isolatedXtdatd=0isforbiddenbecauseoftwoinde-pendentsymmetries.32–34The∆NX=−1selectionrule

5

resultingfromthehiddensymmetry,whichallowsrecom-binationfromapairofMPstatesatL=SandE=EX

only,isliftedatd>0.However,thetranslationalsym-metryyieldingconservationofLandLz(onaplane,MandK)holdsatanyvalueofd.BecausetheelectronleftinthelowestLLafterrecombinationhasl=S(L=0),onlythose2e–hmultipletsatL=S(L=0)arera-diative.TheyaremarkedwithshadedrectanglesinallframesofFig.1.InlargersystemscontainingmorethanasingleX−,thetranslationalsymmetryisbrokenbycol-−

lisions,andweakXtdrecombinationbecomespossible.

−

TheXtdbindingenergy∆X−,calculatedbyextrapola-td

tionofdataobtainedfor2S≤60,isabout0.052e2/λatd=0(veryclosetothevalueobtainedearlierbyPalaciosetal.31intheplanargeometry).Asexpected,∆X−de-−

creaseswithincreasingseparationuptod≈λ,whenXtdunbinds.Somewhatsurprisingly,anewboundmultiplet,

−

asingletXsdatL=S−2(L=−2),occursatfinited.Itsbinding∆X−reachesmaximumofabout0.013e2/λat−

isanon-radiative(“dark”)stateandd≈0.8λ.TheXsd

shouldbedistinguishedfromtheradiativesingletstate−XsatL=S(L=0),whichistheX−groundstate

−

atlowmagneticfields(andsmalld).TheXsdisa2e–h

−

analogofthesingletDstate(twoelectronsboundtoadistantdonorimpurity)withthesameL=−2.AseriesoftransitionsbetweensingletandtripletD−stateswithincreasing|L|havebeenfoundwhenthedistancebetweentheimpurityandtheelectronlayerwereincreased.42

−

BoundstatesoflargerexcitonicionsXn=nX+earealsopossibleatsmalld.Theyallhavecompletelypo-larizedelectronandholespins,andtheirbindingenergy,

−=EX+E−∆Xn−E,decreaseswithincreasingsizeX

n−1

sd

td

−−−(n).ThedependenceofXtd,Xsd,andX2bindingener-gies(calculatedat2S=60)onseparationdisshowninFig.2(a).AsitwasdiscussedinSec.II,finite-sizecal-culationsgivegoodapproximationto2e–henergiesonlyforthebound(finite-size)states.Whilethebindingen-ergiesarecorrectatthevaluesofdforwhich∆>0,theyshouldasymptoticallyapproachzeroford→∞insteadofcrossingitasinFig.2(a).Theaveragee–edistance

ree=

󰀁

3.

Thefulldotsmarkthemulti-pletsobtainedintheexactdiagonalizationoftheNe–hsystemandtheopencirclesmarktheMPstates(withanlX=0excitondecoupledfromtheN−1electronfluid).InFig.3(acd),theN−1electronsinthelowestenergyMPstateatL=0formtheLaughlinν=1

angularmomentaLoftheQHe–QEX−pairinthelowestenergystatesofthese(N−2)e–X−systemsareobtainedbyaddinglQEeandlQEX−oftwodistinguishablepar-ticles.Theresultis:L=1,2,...,N−3.Indeed,themultipletsatthesevaluesofLformthelowestbandofnon-MPstatesinFig.3(acd),separatedfromhigherstatesbydashedlines.ThedependenceofenergyonLwithinthesebandscanbeinterpretedastheQHe–QEX−pseudopotential,anditsincreasewithLmeansthatitisattractive(forapairofoppositecharges,Lincreaseswithincreasingaverageseparation).TheL=1groundstatesinFig.3(acd)shouldbethereforeunderstoodasanexci-tonicboundstatesofaQHe–QEX−pairintheLaughline–X−fluid.Inthisstate,aLaughlinQHtypeexcitationofcharge+1

3.Asimilaranalysisfor∗∗

Fig.3(b)givesle=3andlX−=2,yieldingtwoQHe’seachwithlQHe=3andoneQEX−withlQEX−=2.TheallowedvaluesofLforsuchthreeparticlesare:12,22,33,42,52,6,and7,exactlyasfoundforthelowestnon-MPstatesinFig.3(b).

Thestrongestindicationthatthelowestenergybandsofnon-MPstatesinFig.3containanX−interactingwithexcesselectronscomesfromdirectcomparisonofex-actNe–henergies(dots)withtheapproximateenergiesofthe(N−2)e–X−chargeconfiguration(pluses).The(N−2)e–X−energiesarecalculatedusinganeffectivee–X−pseudopotentialandtheX−bindingenergy.SincetheresultsdependonunknowndetailsofVeX−(duetothedensity-dependentpolarizationoftheX−intheelec-tricfieldofelectrons),wemakea(rough)approximationandinsteadofVeX−usethepseudopotentialoftwodis-tinguishablepointchargeswithangularmomentaleandlX−.TheobtainedspectraarequiteclosetotheoriginalonesandallcontainthelowlyingbandsaspredictedbytheCFmodel.AmuchbetterfitisobtainedforVeX−including(N-dependent)polarizationeffects.

ItisapparentthatonlytwotypesofstatesexhausttheentirelowenergyspectrashowninFig.3:theMPstatescontainingadecoupledlX=0excitonandthenon-MPstatescontaininganX−.Noneofthelowenergystatescanbeunderstoodintermsofanexcited(lX=0)excitoninteractingwiththeexcessN−1electrons.Inparticu-lar,thebandsoflowestenergystatesatL=1,2,...,N−3inFig.3(acd)donotdescribedispersionofaso-called“dressedexciton”X∗(chargeneutralexcitonwithanenhancedmassduetothecouplingtoQE–QHpairex-citationsoftheLaughlinν=1

state.

The“dressedexciton”pictureissimplywrongindescrib-ingthenatureoftheTQPofthesystem.Forexample,theX∗haszerochargeandcontinuousenergyspectruminsteadofQ=−eandLandauquantizedorbitsofanX−.ThereasonwhythesuggestiveideaofanX∗doesnotworkisthatthecouplingofak=0exciton(whichhasanonzeroin-planeelectricdipolemomentµ∝k)toelectronsistoostrongtobetreatedperturbatively.

3

B.SmallLayerSeparation

TheknowledgeofthenatureoftheTQP’sofanysys-temisessentialforunderstandingitsresponsetoanex-ternalperturbation.SinceanX∗isexpectedtobehavedifferentlythananX−whenelectronandholelayersareseparated,theincorrectassumptionofthe“dressedexciton”pictureatd=0mustresultinincorrectinter-pretationofthee–hstatesatd>0aswell.

Atasmalllayerseparationd<λ,allbounde–hstatesacquireasmallelectricdipolemomentµ,whichisproportionaltodandorientedperpendiculartheelec-tronandholeplanes.Thesedipolemomentsresultinarepulsivedipole–dipoleinteractionbetweene–hcom-plexes,whichisproportionaltod2/r3atdistancer≫d.Whiletheelectron–dipolee–XrepulsionisthereasonforthedecreaseofthebindingenergyofanisolatedX−at0IntherangeofdvaluesforwhichtheX−isbound,theX−dipolemomentincreasesitstotalrepulsionwithelectronsandotherX−’s.Itispossiblethatthisin-creasedrepulsioncouldenhancetheexcitationgapofanincompressiblefluide–X−state.ExamplesofdifferentbehaviorofthegapareshowninFig.4.InFig.4(a),the9e–hgroundstateatd=0.5λisthe7e–X−statewith[3*2]correlations(mX−X−isundefinedforonlyoneX−).InthegeneralizedCFpicture,thisstatecontainsoneQEX−withlQE=2andafilledshellofelectronCF’s.InFig.4(b),the8e–2hgroundstateatd=0.5λisthe4e–2X−incompressiblestate[332].InFig.4(c),the6e–3hgroundstateatd=0.3λistheLaughlinν=1

5

stateofX−’sincreasessignificantlyuptod=0.7λ.

7

5.822.5.802.625.782.605.76λ)2.582/E (ee2/5.74( 2.56λE)5.72(a) 9e-1h(b) 8e-2h2.7e-X− [3*2]2S=20− [332]2S=135.70d/λ=0.e-2Xd/λ=0.52.5201234560123456-2.36(d) excitation gap-2.380.03-2.40λ)/2eE*-2.42( − E0.02-2.44E (e2/7e-X−-2.46(c) 6e-3hλ)4e-2X−ν=1/5 of 3X−2S=123X−-2.48d/λ=0.30.0101234560.00.20.40.60.81.0Ld/λFIG.4.(abc)Theenergyspectra(energyEasafunc-tionofangularmomentumL)ofelectron–holesystemswithLaughline–X−correlations:(a)7e–X−groundstatewith[3*2]correlationsina9e–hsystematthelayerseparationd=0.5λ;(b)4e–2X−incompressiblegroundstate[332]ina8e–2hsystematd=0.5λ;(c)Laughlinν=1

r2sureoftherangeDisanaveragee–heh

(0)+d2.Amea-distancerehintheexcitonstatewhoseenergyishalfofthebindingenergy.InFig.5(b)weplotthenormalizedexcitonpseudopo-tentialsasafunctionofwavevectork=L/R.Sincerehisproportional56tok,andthevaluek1/2forwhichVeh(k1/2)=−1

1.01.0(a)d/λ0.80.00.50.8)01.0(∆X0.61.50.6-V /2.0eh )d(k) / (0.40.4∆XX∆0.20.2(b)0.00.00.00.20.40.60.81.001234561/(1+d/λ)kλFIG.5.Thenormalizedbindingenergyofafreeexci-ton,∆X(d)/∆X(0),asafunctionof(1+d/λ)−1(a),andthe

normalizedelectron–holepseudopotentialsVeh(k)/(−∆X)asafunctionofwavevectork(b).distheseparationbetweenelectronandholelayers,andλisthemagneticlength.

describingtheperturbingpotentialsVUDwhichcanbeachievedinbi-layere–hsystemswithdifferentd.

LaughlinquasiparticleshavemorecomplicatedchargedensityprofilesthanelectronsorholesinthelowestLL.ThisinternalstructureisreflectedintheoscillationsoftheQEandQHpseudopotentialsatthevaluesofLcorre-spondingtosmallaverageseparationbetweentheQEorQHandthesecondparticle.Forexample,despiteLaugh-linquasiparticlesbeingchargeexcitations,neitherQE–QEnorQH–QHinteractionisgenerallyrepulsive.55,57Onthecontrary,theQE2molecule(thestatewithmax-imumL,i.e.minimumaverageQE–QEseparation)iseitherthegroundstateoraveryweaklyexcitedstateoftwoQE’s(thenumericalresultsforfinitesystemsarenotconclusive).55

InordertocalculatethepseudopotentialsVhQE(L)andVhQH(L)associatedwiththeinteractionbetweenLaugh-linquasiparticles(QEorQH)ofaν=1

3

fluid,andmultiplyingthedifferencebyǫ.Ifǫis

sufficientlylarge,thepseudopotentialscalculatedinthisway(andshowninFig.6(ab))donotdependonǫanddescribetheinteractionbetweentheholeoffullcharge+eandtheLaughlinquasiparticle.

AsimilarprocedurehasbeenusedtocalculatethepseudopotentialsVeQH(L)andVeQE(L)oftheinteraction

(a) h-QE (7e-1h, 2S=17)(b) h-QH (7e-1h, 2S=19)0.40.0d/λ=0.0d/λ=0.5λ)2/Hd/λ=1.0eQ(εd/λ=2.0 V+EEQQd/λ=3.0hε*E0.0-0.4Qh0.00(e) QE-QE-0.020.4(d) e-QH (7e-1e, 2S=19)QE-0.0420.01357/λ)2e( V0.02(f) QH-QHQH20.00H0.0Qe-0.02(c) e-QE (7e-1e, 2S=17)1357-0.45671011126710111213LLFIG.6.Thepseudopotentials(pairenergyVasafunc-tionofpairangularmomentumL)oftheinteractionbetweenquasiparticles(quasielectronQEandquasiholeQH)oftheseven-electronLaughlinν=1

rateestimateoftheh–QEpseudopotentialparametersatthetwosmallestvaluesofL,i.e.thebindingenergy∆ofthehQEandhQE*complexeswiththesmallestandthenextsmallestaverageh–QEseparation(thehQE*com-plexisimportantindiscussion46ofPL)givesthecurvesplottedinFig.8.Theinteractionofthe2DEGatν≈1

3,

becausetheQE’sthatcanbe

boundtoaholeexistonlyatν>1

thatatd<1.5λtheenergyofh–QE(and3.Fig.8

showse–QH)attractionexceedsεQE+εQHandtheQE–QHpairsarespontaneouslycreatedinthe2DEGtoscreenthehole(orelectron)chargeatanyvalueofν≈1

3)

withenergyEhe,∆heisdefinedasadifference

betweenEheandthestateinwhichtheholeiscompletelydecoupledfromallNelectrons(whichat2S=3(N−2)formastatewiththreeLaughlinQE’s).Notethat∆heisnotequivalenttothebindingenergyofafreeexciton(itisnotequaltothee–hattractionbutalsoincludestheenergyneededtoremoveanelectronfromtheLaughlinstatesothatitcanbeboundtothehole).

ThehQE2isthemoststronglyboundFCXinentirerangeofd(atleastuptod=10λ),andhenceitis

0.00.0λ))H2/eεQ(+ VEεQ2(23EEQQhh-0.5(a) h-QE2 (7e-1h, 2S=16)(b) h-QE3 (7e-1h, 2S=15)-0.50.5(c) e-QE2 (7e-1e, 2S=16)(d) e-QE3 (7e-1e, 2S=15)0.5d/λ=0.0)Hd/λ=0.5εQ/λ)2d/λ=1.0+EeQ( Vd/λ=2.0ε(3d/λ=3.00.00.046810124681012LLFIG.7.Thepseudopotentials(pairenergyVversuspairangularmomentumL)oftheinteractionbetweenmoleculesconsistingoftwoorthreequasielectrons(QE2andQE3)oftheLaughlinν=1

ItcanbeseeninFig.8that∆3.

hQE<λ,andtwoQE–QHpairsarespontaneously2>εQE+εQHatdcreatedinthe2DEGtoformhQE2evenatν<1

3e)

are

morestablecomplexesthanhQE2.Thetransitionfromfractionalto“normal”excitonphaseoccursatd≈1.5λ,thatisatthethecrossingof∆hQE2and∆heinFig.8(theshadedrectanglemarksthe“normal”excitonphase).

VII.NUMERICALENERGYSPECTRA

UsingamodifiedLanczosalgorithm,wehavebeenabletoquicklyandexactlydiagonalizeHamiltoniansofdi-mensionsupto∼106.ThisallowedcalculationofenergyandPLspectraofNe–hsystemswithN≤9andatthevaluesof2Supto3(N−1),correspondingtotheholeinteractingwiththeLaughlinν=1

0.20hQEεhQE*QE+εQHhQEhQE2he30.15λ)2/e0.10(∆ 2(εQE+εQH)0.050.00012d/λ345FIG.8.Thebindingenergy∆offractionallychargedex-citonshQEnasafunctionoflayerseparationd,calculatedforthe8e–hsystemwithafixednumberofLaughlinquasi-particlesinthe8eelectronsystem(ǫ≫1;seetext).λisthemagneticlength.Thehestatecontainsanexcitonandorig-inatesfromthemultiplicativestateatd=0.Intheshadedpartofthegraph,thehehasthelargestbindingenergyandthehQEncomplexesdonotform.

state,l∗sevenelectronsfillcompletely−theirCFshelloftron”e=S−(N−2)=3andtheX(QEAtd≈X−2λ),withthel∗XX−−=l∗becomesa“quasielec-unbindse−1=and2.

the7e–X−fluidundergoesreconstruction.SincefourQE’softhenineelectronLaughlinν=1

compared−

3,

−toQ=−1forafreeX),andstabilizestheXQHstateuptoatleastd=2λ.

ThebandofstatesmarkedwithdashedlinescontainsthehQE2(moststableofallFCX’s)interactingwiththethirdQE(notethathereQEdenotesquasielectronoftheLaughlinν=1

2

toobtain

lQE2≡2lQE−1=6andthen.Finally,addingaddingtoitlh=21

2lhQEasiftheywerecompletelydistinguishableparticles2tolQE

gives|lhQE2−lQE|≤L≤lhQE2+lQE,or1≤L≤8.However,becausehQE2containstwoQE’swhicharefermionsin-distinguishablefromthethirdQE,theexclusionprincipleforbidsthestateswithL=1and2(L3QE≤15

5.249e + h−8e + X0−+X + QH+−hQE2 + QE−+5.70E (e2/λ)he−XQH5.066.002S=21(a) d/λ=0.0(b) d/λ=0.55.506.24E (e2/λ)5.866.42(c) d/λ=1.0(d) d/λ=1.56.126.98E (e2/λ)6.3402466.90(e) d/λ=2.0810120hQE3246(f) d/λ=4.081012FIG.10.Theenergyspectra(energyEvs.angularmo-mentumL)ofthe9e–hsystemcalculatedonaHaldanespherewithmonopolestrength2S=21fordifferentlayersepara-tionsd.Theopencirclesmarkthemultiplicativestatesatd=0.λisthemagneticlength.

LLaddingittolh=

21

,andthegroundstateishQE3

atlhQE3≡|lh−lQE3|=3.Atevenmuchlargerd,thegroundstateconsistofthethree-QEgroundstate(whichforN=9at2S=21occursatL3QE=5

2

−1)asX−QH2,

thetransitionfromonestatetotheotheriscontinuous.

2(N

12

4.909e− + h+X− + 3QH+5.348e− + X0hQE+ + QH+2)2/λe( E2S=23X−QH34.78(a) d/λ=0.0(b) d/λ=0.55.225.885.h+ + QE−)2/λe( E5.(c) d/λ=1.0hQE(d) d/λ=1.56.065.766.56λ)2/e( E5.94(e) d/λ=2.0(f) d/λ=4.06.46024681012024681012LLFIG.12.Theenergyspectra(energyEvs.angularmo-mentumL)ofthe9e–hsystemcalculatedonaHaldanespherewithmonopolestrength2S=23fordifferentlayersepara-tionsd.Theopencirclesmarkthemultiplicativestatesatd=0.λisthemagneticlength.

ItisapparentfromthedependenceofPLintensity46ondthatitoccursaboutd≈1.66λ.

2S=23(Fig.−12):Atsmalld,thelowenergystatescontainanXinteractingwiththreeQH’softheLaugh-lin[3*2]stateofthe7e–X−fluid.ThegeneralizedCF

pictureusesl∗

eandpredictsL≡=S1,−2(N,3−6

,2)...=9,13forthisband.Indeed,2,

4atleastatsmallL,theseX−–3QHstatescanbeidenti-fiedinFig.12(ab).TheangularmomentumX−QH3resultsfromaddingthreelQH∗ofabound

toobtainll3=3lQH−3tol∗XhQH∗

likelyX−

QH3=|33isthelowest−statelX−

|at=L7.=Althoughe7inFig.12(a),most−

ithashigherenergythanotherstatesandthusitisun-stable(duetotheshortrangeofQH–QHrepulsion;55seealsoFig.6(f)fortheQH–QHpseudopotentialinaseven-electronsystem).

Atd>λ,theX−unbindsandtheX−–3QHbandundergoesreconstruction.Atd≈λ,twocompetinglowenergybandsoccurinthespectrainFig.12(bcd).Onedescribestheholewithlh≡S=23

similartothat2interactingthroughapseu-dopotentialinFig.6(a).ThisbandhasL≥|lh−lQEand8)arehQE|=and7,andhQE*.thelowestThesecondtwostatesband(atinvolvesL=7anadditionalQE–QHpairanddescribesthehQE2withlhQE2≡|lh−lQE2|=|lh−(2lQE−1)|=7

2.

Theangu-larmomentaLobtainedbyaddinglhQE2andlQHsatisfy|lhQE2−lQH“hard|≤Lcore”≤lhQEof2+lQH,i.e.0≤L≤7.Be-causeoftheVQE−QH(theQE–QHstateatL=1doesnotoccur45),thehQE2–QHstateatthehigh-estvalueofLisforbidden,andthehQE2–QHbandhasL=0,1,2,...,6.WeshowedinSec.VIthatcreationofanadditionalQE–QHpairtobindthesecondQEtohQEandformhQE2isenergeticallyfavorableatd≤λ(seethecrossingof∆hQE2and2(εQE+εQH)inFig.8).Indeed,inFig.12thehQEstatecrossesthehQE2–QHbandandbecomesthe9e–hgroundstateatd≈λ.

2S=24−(Fig.13):Atsmalld,theloweststatescon-tainanXinteractingwithfourQH’softhe[3*2]stateofthe7e–X−fluid.Thisbandisfairlybroadandoverlapswithhigherones,containingadditionalQE–QHpairs.IntheL≡S=12groundstateatverylarged,theholeisdecoupledfromtheν=1

0.050.040.030.020.010.00012 (a) excitation gap(b) binding energy0.120.10hhQEhQE*hQE2hQE3heE*−E (e2/λ)0.080.060.040.020.00∆ (e2/λ)345012345FIG.14.TheexcitationgapE∗−E(a),andbindingen-ergy∆(b)offractionallychargedexcitonshQEnasafunctionoflayerseparationd,calculatedforthe8e–hsystem.EXistheexcitonenergyandλisthemagneticlength.Thehestatecontainsanexcitonandoriginatesfromthemultiplica-tivestateatd=0.

d/λd/λestenergybandat3≤L≤11inFig.13(d)containsaboundhQEstatewithlhQE≡|lh−lQE|=7interactingwithaQH.Atd≈λ,theh–QEattractionbecomesevenstrongerandthehQE2statewithlhQE2≡|lh−lQE2|=3isformed.ThelowestbandinFig.13(c)describestheinteractionofanhQE2statewithtwoQH’s.Theaddi-tionoflhQE2andtwolQH’sgivesabandatL=0,1,23,...,10observedinFig.13(c).

InFig.14wepresentthedataregardingthestabilityofdifferentFCX’s,extractedfromthe8e–hspectra,similarthoseinFigs.9–13.WehavecheckedthatthecurvesplottedhereforN=8areveryclosetothoseobtainedforN=7or9,sothatallimportantpropertiesofanextendedsystemcanbeunderstoodfromarathersimple8e–hcomputation.Intwoframes,foreachhQEnweplot:(a)theexcitationgapE∗−EabovethehQEngroundstate,and(b)thebindingenergy∆.Theexcitationgapsareobtainedfromthespectraat2S=3(N−1)−ninwhichisolatedhQEncomplexesoccur.Thebindingenergy∆isdefinedinsuchwaythatEhQEn=EQEn+Vh−LS−∆,whereEhQEnistheenergyoftheNe–hsysteminstatehQEncalculatedat2S=3(N−1)−n,EQEnistheenergyoftheNesysteminstateQEncalculatedatthesame2S=3(N−1)−n,andVh−LSistheself-energyoftheholeinLaughlinν=1

ν=

1

Thecrossoverbetweenthe“integrally”and“fraction-ally”chargedexcitonphasesinane–hsystemcanbeviewedasachangeintheresponseofa2DEGtoamoregeneralperturbationpotentialVUDdefinedintermsofitscharacteristicenergy(U)andlength(D)scales.Ananal-ogoustransitionwilloccurinothersimilarsystems,inwhichthe2DEGisperturbedbyachargedimpurity41,42oranelectrode.However,adifferencebetweenthere-sponsetonegativelyandpositivelychargedprobesisex-pectedbecauseofverydifferentQE–QEandQH–QHin-teractionsatshortrange.

Ourresultsinvalidatetwosuggestiveconceptspro-posedtounderstandthenumericalNe–hspectraandtheobservedPLofa2DEG.First,incontrastwiththeworksofWangetal.,25andApalkovandRashba,26wehaveshownthatthe“dressedexciton”stateswithfinitemo-mentum(k=0)donotoccurinthelowenergyspectraofe–hsystemsatsmalld.Thecouplingofk=0excitonstothe2DEGistoostrongtobetreatedperturbatively,anddoesmorethanrenormalizationoftheexcitonmass.Rather,itcausesinstabilityofk=0excitonsandforma-tionofchargedexcitonsX−.Andsecond,wehaveshownincontrastwiththeworkofRashbaandPortnoi,27thatthecharge-neutral“anyonexcitons”hQE3arenotstableatanyvalueofd(theyarealsonon-radiative46).

ACKNOWLEDGMENT

3.

TheauthorsacknowledgepartialsupportbytheMate-rialsResearchProgramofBasicEnergySciences,USDe-partmentofEnergy,andthankK.S.Yi(PusanNationalUniversity,Korea)whoparticipatedintheearlystagesofthisstudy,andP.Hawrylak(NationalResearchCouncil,Canada)andM.Potemski(HighMagneticFieldLab-oratory,Grenoble,France)forhelpfuldiscussions.AWacknowledgespartialsupportfromthePolishStateCom-mitteeforScientificResearch(KBN)grant2P03B11118.

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