Statistical Properties of Level Widths and Conduct

arXiv:cond-mat/9406048v110Jun1994submittedtoPhys.Rev.BStatisticalpropertiesoflevelwidthsandconductancepeaksinaquantumdotE.R.Mucciolo†,V.N.Prigodin‡,andB.L.Altshuler††DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139and‡Max-Planck-Institutf¨urFestk¨orperforschung,Postfach800665,D-70506Stuttgart,Germany(June10,1994)AbstractWestudythestatisticsoflevelwidthsofaquantumdotwithextendedcon-tactsintheabsenceoftime-reversalsymmetry.Thewidthsaredeterminedbytheamplitudeofthewavefunctionaveragedoverthecontactarea.Thedis-tributionfunctionoflevelwidthsforatwo-pointcontactisevaluatedexactly.Thedistributionresemblescloselytheresultobtainedwhenthewavefunctionfluctuatesindependentlyateachpoint,butdiffersfromtheone-pointcase.Analyticalcalculationsandnumericalsimulationsshowthatthedistributionformany-pointcontactshasapower-lawbehavioratsmalllevelwidths.Theexponentisgivenbythenumberofpointsintheleadanddivergesinthecontinuouslimit.Thedistributionoflevelwidthsisusedtodeterminethedistributionofconductancepeaksintheresonanceregime.Atintermediatetemperatures,wefindthatthedistributiontendstonormalandfluctuationsintheheightofthepeaksaresuppressedastheleadsizeisincreased.73.40.Gk,73.20.Dx,05.45.+bTypesetusingREVTEX1I.INTRODUCTIONUsuallyanymeasurementofconductanceinametallicsystemassumesatleasttwoattachedleads.Theexistenceoftheseleadsconnectingthesystemtoreservoirsbroadenstheelectronicenergylevels.Foramacroscopicsample,thewidthofanenergylevelistypicallymuchlargerthanthedistancebetweenneighboringlevels.Asaresult,oneobservesasmoothdependenceoftheconductanceontheFermienergy.Recently,theadvancesinnanometertechnologyhavemadepossiblethefabricationofverysmallsemiconductordevices,knowngenericallyasquantumdots,1–3whereonehasafixednumberofconductionelectronsconfinedintoanisland.Inthesesystems,onecannarrowthelevelwidthbyreducingeitherthesizeoftheleadsortheirtransmittance.Inthefirstcaseitiscustomarytospeakaboutchannels:Ifweconsideranidealleadasawaveguidewithagivencrosssection,wecanassociateachanneltoeachstateduetotransversequantization.Thenumberofchannelsisapproximatelythearea,inunitsofelectronwavelength,ofthecontactbetweenleadanddotandeachchannelischaracterizedbyitstransmittance.Theconductanceofasystemcanbecalculatedthroughthewell-knownLandauer-B¨uttikerformula.4,5Followingthisapproach,thedistributionofconductancesofaquantumdothasbeenevaluatedfortwodistinctcases:forweaklycoupled,pointlike,leads6ofaclosedsystem,andforleadswithanynumberofchannelsattachedtoaballisticcavity.7,8Herewewillconsiderthesituationatlowtransmittance,whenleadsbehaveastunnelcontacts.Inthiscase,eachelectroneigenstatecorrespondstoapeakintheFermienergy(gatevoltage)dependenceoftheconductance.2,3Thisallowsonetomakearealspectroscopyofelectronsinquantumdots.Theheightofeachpeakisdeterminedbytheprobabilitiesoftunnelingthroughtheleads,aswellasbytheamplitudeofthewavefunctionnearthecontacts.Thelatermeansthattheseheightsarerandomlydistributed.ThedistributionfunctionofthepeakshasbeendeterminedforthecaseofpointlikecontactsinRefs.6,9,and10.Inthispaperwewilldeducethedistributionconsideringleadsofarbitrarysize.Webeginbystudyingthestatisticsoflevelwidths,whichisconnectedtothefluctu-ationsofconductancepeaksthroughtheLandauer-B¨uttikerformula.Thesupersymmetrytechniqueandthenonlinearσmodel11,12providetheframeworkfortheexactcalculationofthedistributionoflevelwidthsforleadswithtwo-pointcontact.Thedistributiondif-fersdrasticallyfromthesingle-pointcase,butwefindthatinclusionofcorrelationbetweenthewavefunctionfluctuationsatthetwopointsintheleaddoesnotsignificantlyaltertheoverallformofthedistributionoflevelwidths.Thisanalyticalresultisconfirmedthroughnumericalsimulationsofaquantumdot.Theexactdistributionofconductancepeaksisalsoevaluatedfortwo-pointleadsandwefindthatitdeviatesfromthesingle-pointdistributionmainlyforsmallpeakheights,whereitshowsalineardependence.Forleadswithmany-pointcontactswedonotknowofanymethodwhichenablesageneralevaluationofthedistributionfunctionoflevelwidths.Thusweassumethatthewavefunctionateachpointfluctuatesindependentlyandproceedtocalculatethetotaldis-tribution.Bycomparingnumericalwithanalyticalresults,weshowthatthisapproximationyieldsagoodqualitativeunderstandingofthelarge-leadlimit,sincecorrelationsdonoteffectthebehaviorofthedistributionverystrongly.Animportantuseofquantumdotsisfoundinthestudyofchaos.13–18Becauseonecan2obtainextremelycleansamplesandshapethemintodifferentforms,itisnowpossibletofabricatetheso-calledquantumbilliards,whereonehasauniquechancetostudyquantumandsemiclassicalphysics.13,19Althoughthesupersymmetrymethodassumesaveragingoverdisorder,onecanconjectureonveryfirmgrounds(theergodichypothesis)thatitappliesquitegenerallytoquantumchaosproblems.Therefore,weexpectthatourresultsshouldbeabletodescribeanysystemwheretheunderlyingdynamicsischaotic,regardlessastowhetheritisdiffusiveorballisticTheorganizationofthispapergoesasfollows.InSec.IIwedescribethemodelusedandthebasicconceptsrelatedtoconductancepeaksandlevelwidths.Thewaythenumericalsimulationsweredoneisex