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Correlation effects in the transport through quantum dots

We study the charge and heat transport through the correlated quantum dot with a finite value of the charging energy U \neq \infty . The Kondo resonance appearing at temperatures below T_K is responsible for several qualitative changes of the electric and

Correlatione ectsinthetransportthroughquantumdots

A.Donabidowicz,T.Doma´nski,K.I.Wysoki´nski

InstituteofPhysicsandNanotechnologyCentre,M.CurieSk lodowskaUniversity,20-031Lublin,Poland

(Dated:February6,2008)

arXiv:cond-mat/0701217v1 [cond-mat.str-el] 10 Jan 2007

Westudythechargeandheattransportthroughthecorrelatedquantumdotwitha nitevalueofthechargingenergyU=∞.TheKondoresonanceappearingattemperaturesbelowTKisresponsibleforseveralqualitativechangesoftheelectricandthermaltransport.WeshowthatundersuchconditionsthesemiclassicalMottrelationbetweenthethermopowerandelectricconductivityisviolated.Wealsoanalyzetheothertransportpropertieswherea nitechargingenergyUhasasigni cantin uence.Theyareconsideredhereboth,inthelimitofsmallandforarbitrarilylargevaluesoftheexternalvoltageeV=µL µRand/ortemperaturedi erenceTL TR.Inparticular,wecheckvalidityoftheWiedemann-FranzlawandthesemiclassicalMottrelation.

Recentdevelopmentinnanoelectronicshasrenewedtheinterestsforstudyingthecorrelatione ectsinsys-temswherethelocalizedquantumstatescoexistandin-teractwithaseaofitinerantelectrons.PreviouslysomerelatedideashavebeenexploredinthesolidstatephysicswheninvestigatingthescreeningofmagneticimpuritieshybridizedwiththeconductionbandelectronsCor-relationsleadtheretotheKondoe ect[2]whichshowsupbyalogarithmicincreaseoftheelectricalresistanceattemperaturesbelowTK.Similarmechanismhasbeenlaterfoundtoberesponsibleforasuperconductivityoftheheavyfermioncompounds[3].

Inthenano-andmesoscopicsystems(suchasquan-tumdotsorcarbonnanotubesconnectedtoexternalleads)therecanalsoarisetheKondoe ectThishasbeenpredictedtheoretically6]andobservedex-perimentallyinmeasurementsofthedi erentialconduc-tanceG(V)=dI(V)

T|I(V)=0[8]provided

anadditionalevidenceinfavoroftheKondoe ectatlowtemperatures.

Varioustransportpropertiesthroughthecorrelatedquantumdotshavebeenstudiedinquiteadetail6,7,8,9,Authorsinvestigatedthequantumdotscoupledtothenormal,superconductingandferro-magneticleads.Inseveralcasestherehavebeenobservedsomequalitativelynewe ects,forinstanceasplittingofthezerobiasresonanceundermagnetic eld[7,12],ab-senceoftheeven-oddparitye ectsortheoutofequi-libriumKondoe ect[9,Kondoe ecthasbeenfoundinsuchnanosystemslikethesingleatom[15],sin-glemolecule[16]andcarbonnanotubes[17].Moreover,ithasbeenalsoobservedinthequantumdotsattachedtoferromagneticleadswhere,inprinciple,transportiscontrolledbythespindegreeoffreedom.

Quantumdotsareoftenthoughttobepromisingob-jectsforthenanotechnologye.g.invariousspintronicsapplicationsasthegatesofquantumcomputers,inelectronspinentanglersore cientenergyconver-siondevicesThereforeadetailedunderstandingofthechargeandheattransportthroughsys-temsseemstobeofprimaryimportance[22].Besidesthedi erentialconductanceitisimportanttoanalyzeon

equalfootingalsothethermoelectricpropertiesbecausetheyareverysensitivetodimensionalityaswellastheelectronicspectrumneartheFermilevel.IntheCoulombblockaderegimethethermoelectriccoe cientShasbeenshown[23]tooscillateasafunctionofthegatevoltage(acharacteristicsaw-toothshape).AttemperaturesoftheorderorsmallerthanTKthereformsanarrowresonanceslightlyabovetheFermienergy.Itsappearanceisre-sponsibleforincreasingthedi erentialconductance(thezerobiasanomaly)andsimultaneouslyleadstoachangeofsignofthethermopowerfrompositive(whenT>TK)tonegativevalues(forTsmallerthanTK).Boththesee ectsareusefulfordetectingtheKondoe ectinthequantumdots24].

Inmostoftheexperimentalmeasurementsithasbeenfoundthatthechargingenergy(theon-dotCoulombre-pulsion)Uislarge,yetbeing http://www.wenkuxiazai.comuallyitvariesbe-tween2and5inunitsofthee ectivecouplingΓwhosemeaningwillbeexplainedintheSectionII.

Itisapurposeofourworkheretofocusonstudy-ingthetransportpropertiesforthequantumdotwitha niteCoulombenergyU.Weanalyzethelinearre-sponsewithrespecttothebiasVandtemperaturedif-ference Tappliedacrossthejunctionanddiscusssomenonlineare ectsinthecaseofarbitraryVand Tval-ues.Transportcoe cientsaredeterminedwithuseofthenon-equilibriumKeldyshGreen’sfunctionsWetreatthecorrelationsusingtheequationofmotionapproach[26]whosejusti cationhasbeenrecentlythoroughlydis-cussedintheRef.[27].

Studyingtheequilibriumandthenon-equilibriumas-pectsoftheelectricandthermaltransportweshallcheckavalidityoftheWiedemann-FranzlawandtheMottre-lation.Thepaperisorganizedasfollows.InSectionIIwebrie yintroducethemodelandtheapproximationusedforcalculatingthetransportcoe cients.ResultsobtainedwithinthelinearresponsetheoryarepresentedinSectionIII.InthenextSectionIVwediscussanon-equilibriumcasecorrespondingtoarbitrarilylargevaluesofVand T.Weenclosethispaperbyasummaryandsomeoverviewofthefutureprospects.

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