Spatial ordering of charge and spin in quasi one-dimensional Wigner molecules

ew-electron systems confined in quasi one-dimensional quantum dots are studied by the configuration interaction approach. We consider the parity symmetry of states forming Wigner molecules in large quantum dots and find that for the spin-polarized Wigner m

Spatialorderingofchargeandspininquasione-dimensionalWignermolecules

B.Szafran,1,2F.M.Peeters,1S.Bednarek,2T.Chwiej,1,2andJ.Adamowski2

1

May 2004

DepartementNatuurkunde,UniversiteitAntwerpen(CampusDrieEiken),B-2610Antwerpen,Belgium

2

FacultyofPhysicsandNuclearTechniques,AGHUniversityofScienceandTechnology,al.Mickiewicza30,30-059Krak´ow,PolandFew-electronsystemscon nedinquasione-dimensionalquantumdotsarestudiedbythecon g-urationinteractionapproach.WeconsidertheparitysymmetryofstatesformingWignermolecules

inlargequantumdotsand ndthatforthespin-polarizedWignermoleculesitstrictlydependsonthenumberofelectrons.Weinvestigatethespatialspin-orderingintheinnercoordinatesofthequantumsystemandconcludethatforsmalldotsithasashort-rangecharacterandresultsmainlyfromthePauliexclusionprinciplewhiletheWignercrystallizationinlargedotsisaccompaniedby 6spinorderingovertheentirelengthofthedot.

1 ]I.

INTRODUCTION

llahStrongcon nementofchargecarriersintwodirections-sresultsinreductionoftheirdegreesoffreedomtoasin-egleone,i.e.,inquasione-dimensionalmotion.Suchone-mdimensionalsystemsarerealizedtypicallyinsplit-gate1,2.tandcleaved-edgeovergrowth3semiconductora4quantumwires,aswellasincarbonnanotubes,butcanalsobemrealizedin-5 nite-sizesystems,i.e.,inanisotropicquan-tumdotsorquantumrings.6Thereisarenewedin-dnterestintheone-dimensionalsystemsrelatedtothere-ocentprogressofvapour-liquid-solidfabricationofquan-ctumwiresofveryhighquality.7,8,9

[ Thepresentpaperisdevotedtoelectronsystemscon- 1 nedinone-dimensionalquantumdotsandinparticu-vlartotheirWignercrystallization10appearingwhenthe0electron-electroninteractiondominatesoverthekinetic5energy.Wignerelectronsolids(Wignermolecules)are3predictedtoappearinlargedots11orinstrongmag-50netic elds.12IntheWignermoleculesthechargeden-4sityseparatesintodistinctchargemaximaeachcorre-0spondingtooneofcon nedelectrons.Formationof/Wignermoleculesintheground-statechargedensityintaone-dimensionalquantumdotswaspreviouslyobtainedminexact13,14,15anddensityfunctional-approach.16Inone-dimensionaldotstheWignerlocal-dizationappearsinthelaboratoryframe,incontrasttontheinner-coordinatecrystallizationappearingincircu-colarquantumincludingquantumrings.Trans-:portpropertiesofWignercrystalsformedinopenin -vniteone-dimensionalsystemshavealsobeenstudied.17,18iXTheLuttingerliquidformalismhasbeenapplied19torquantumwireswithbox-likeboundaryconditions,i.e.,atotheone-dimensionalquantumdots.Meltingofclas-sicalone-dimensional20

WignercrystalshasrecentlybeenWestudythequasione-dimensionalquantumdotsus-ingacon gurationinteractionapproachwiththee ec-tiveelectron-electroninteractionpotentialwhichwede-rivedrecently.21Thisworkisageneralizationofourex-acttwo-electronstudy15tolargernumberofelectrons.Intheweakcon nementlimittheground-statebecomesnearlydegeneratewithrespecttothespincon guration

oftheelectronsystem.14,15Similarapproximatedegen-eracyhasbeenfoundinquantumringsoflargeradius.22InthispaperwestudytheparitysymmetryofthenearlydegeneratestatesformingWignermoleculesinlargedots.Weshowthatforspin-polarizedelectronstheWignerlo-calizationisformedonlyforone(evenorodd)spatialparityofthestatestrictlydependentonthenumberofelectrons.Wepresentthisdependenceintheformofatheoremforwhichweprovidearigorousanalyticalproof.Thefounddependenceoftheparityofone-dimensionalWignermoleculestatesonthenumberofelectronsissim-ilartotheappearanceofthemagicangularmomentastatesforwhichWignercrystallizationispossibleincir-culardots.23,24Furthermore,wediscussaninhibitionofWignercrystallizationbyaperturbationofthecon ne-mentpotentialthroughacentralinversion-invariantpo-tentialwell.

Magneticspin-orderingofelectronsinone-dimensionalspacehasbeenextensivelystudied25inHubbardmod-elswhich,inonedimensionwithonlynearest-neighborhoppinginteractions,predicttheappearanceofalow-spingroundThisisaconsequence26oftheLieb-Mattistheorem27whichimpliesthatwith-outspin-dependentinteractionstheground-stateofone-dimensionalelectronsystemscorrespondstothelowestpossiblespinquantumnumber(S=0or1/2).Thisfea-turegenerallydoesnothavetoresultinanyspatialspinordering.Inthispaperweusetheexactnumericalsolu-tionoftheSchr¨odingerequationtoinvestigatethespatialdistributionofspinsintheone-dimensionalquantumdotandtherelationbetweenthechargeandspatialspinor-deringintheWignercrystallizationlimit.We ndthatWignercrystallizationisaccompaniedbyalong-rangespin-orderingintheinnercoordinatesofthesystemin-steadofaspin-symmetrybreakingpredictedbydensityfunctional16,28Intheground-statethisorderinghasaclearantiferromagneticcharacter.

Thispaperisorganizedasfollows.InSectionIIwepresentthetheoreticalmethod.SectionIIIcontainstheresultsfortheWignerlocalizationandgroundstatede-generacyofthefewelectronsystems.InSectionIVwepresentproofforthedependenceoftheparityofspin-polarizedWignermoleculesonthenumberofelectrons.

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