On the Line Widths of Vibrational Features in Inelastic Electron Tunneling Spectroscopy - PDF document

OntheLineWidthsofVibrationalFeaturesinInelasticElectronTunnelingMichaelGalperinandMarkA.RatnerDepartmentofChemistry,NorthwesternUniersity,Eanston,Illinois60208AbrahamNitzan*SchoolofChemistry,TheSacklerFacultyofScience,TelATelA,69978,IsraelReceivedMay6,2004;RevisedManuscriptReceivedJuly5,2004Weaddressthelineshapeandlinewidthsobservedinrecentinelasticelectrontunnelingspectroscopy(IETS)experiments.ThenonequilibriumGreenfunction(NEGF)formalismisusedtoanalyzetheeffectoftheelectronphononinteractiononthetunnelingspectra.WefindthatIETSlineshapesaresensitivetojunctionparameters,inparticularthepositionofthebridgeelectronicresonanceandthemoleculeleadcouplingthatmaybecontrolledexperimentally.IntrinsicIETSlinewidthsarefoundtobedominatedbythecouplingofmolecularvibrationstoelectronholepairsexcitationsinthelead(s)towhichthemoleculeisbondedchemically.WhileestimatedwidthsareofsimilarorderofmagnitudeasobservedintherecentexperimentofWangetal.(NanoLett.,643),onecannotruleoutinhomogeneouscontributiontothelinewidthinthismonolayerexperiment.Inelasticelectrontunnelingspectroscopy(IETS)hasbeenanimportanttoolforidentifyingmolecularspeciesintunneljunctionsforalongtime.Withthedevelopmentandadvancesinscanningtunnelingmicroscopy(STM)andspectroscopy(STS)ithasproveninvaluableasatoolforidentifyingandcharacterizingmolecularspecieswithintheconductionregion.Indeed,thisistheonlydirectwaytoascertainthatamolecularspeciesparticipatesintheconduc-tionprocessandatthesametimetoprovideimportantspectroscopicandstructuraldataontheconductingmolecule,inparticular,informationonthestrengthofthevibroniccouplingitself.MostpasttheoreticaldiscussionsofIETSarebasedonloworderperturbativetreatmentswherethetunnelingcurrentiscomputedinthelowestorderintheelectronphononcoupling.Suchanapproachisveryusefulforroughestimatesusingrealisticmolecularmodels,and,whencarriedoutcarefully,canqualitativelyexplainsomesubtleissues,e.g.,thesometimesnegativeresponseindthecurrentandistheimposedpotentialbias),butitisnotfullyconsistentwiththenonequilibriumconditionsunderwhichsuchmeasurementsaredoneaswellaswiththeboundaryrestrictionsimposedbythePauliprinciple.OurpresentdiscussionisbasedonthenonequilibriumGreen'sfunction(NEGF)formulation,whichprovidesasys-tematicframeworkfordescribingtransportphenomenaininteractingparticlesystems.SuchanapproachwasrecentlytakenbyUebaandco-whohaveappliedtheNEGFformalismtotheresonantlevelmodelofphononassistedtunnelingwhereasinglebridgelevelrepresentsajunctionconnectingtwofreeelectronreservoirswhilebeingalsocoupledtoasingleharmonicmode.ThefreeparticleHamiltonianisarecreationandannihilationoperatorsforelectronsonthebridginglevelofenergyaresetsofelectronicstatesrepresentingtheleft(L)andtheright(R)electrodeswiththecorrespondingcreationandannihilationoperators,andÃarecreation *Correspondingauthor.Tel:972-3-6408904.Fax:E-mail:Nitzan@post.tau.ac.il.Ã(1)Vol.4,No.910.1021/nl049319yCCC:$27.502004AmericanChemicalSocietyPublishedonWeb07/30/2004 andannihilationoperatorsforthephononmodeoffrequency.TheinteractionsaregivenbyWithinthismodelUebaetal.havereproducedandimprovedresultsobtainedearlierbyPerssonandBaratoff.Inbothtreatmentsinelastictunnelingspectraareanalyzedintheleadingorderoftheelectronphononinteraction.PerssonandBaratoffhaveobserved(followingDavis)thatinthisorderthereisanimportantcorrectiontotheelasticcomponentofthetunnelingcurrentattheonset(isthebiaspotential)oftheinelasticchannel.Thiscontributiontothetunnelingfluxstemsfromwhatmaybeseenasinterferencebetweenthepurelyelasticcurrentamplitudethatdoesnotinvolveelectronphononinteractionandtheelasticamplitudeassociatedwithtwoelectronphononinteractioneventsinvolvingvirtualphononemissionandabsorption.Dependingontheenergeticparametersofthesystem,theresultingcorrectiontotheelasticcurrentmaybenegativeand,furthermore,mayoutweighthepositivecontributionoftheinelasticcurrent,leadingtoanegativepeakinthesecondderivativeofthecurrent/voltagerelation-ship.Suchnegativefeatureshaveindeedbeenobservedinsingle-moleculevibrationalspectroscopyofmethylisocya-nideadsorbedonaluminia-supportedrhodiumparticlesofoxygenmoleculeschemisorbedonAg(100).TherecentresultsbyReedandco-workersthatshowrelativelystrongderivative-likefeaturesinthelow-temperatureIETSspectrumofC8alkanethiolsmaybeanothermanifestationofthesameSpectroscopiclinewidthsareoftendifficulttointerpretsincetheiroriginsmaylieindiversephysicalfactors.IntherecentIETSexperimentbyWangetal.itwasfoundpossibletoeliminateortoestimatesomeoftheimportantcontributionsofthethermalFermidistributioninthesubstrateandofthedistributionofthelocalelectrostaticfieldandtocomeupwithwhattheauthorscallanªintrinsiclinewidthºof3.730.98mVat4KinananojunctioncontainingalayerofalkaneC8moleculesbetweengoldelectrodes.Asalreadymentioned,therateofvibrationalrelaxationduetonuclearcouplingwiththethermalenviron-mentisexpectednottoexceedafewwavenumbersatnearzerotemperaturesandcannotaccountforthisobservation.Ontheotherhand,inhomogeneousbroadeningisobviouslyapossiblecontributiontotheobservedlinewidthinameasurementsuchasofref13thatinvolvesasampleofafewthousandsmolecules.Foramoleculeadsorbedonametallicsubstrate,anotherchannelofrelaxationinvolvesthevibroniccouplingtothecontinuumofelectronholepairsinthemetal.Indeed,suchcouplinghasbeenshowntobeanimportantandsometimesdominatingsourceofbroadeningintheinfraredspectraofmoleculesadsorbedonmetalInthispaperweapplytheNEGFapproachtotheanalysisoflineshapesandinparticularlinewidthsofIETSfeatures.Themodelusedinrefs2527isgeneralizedtoincludecouplingofthemolecularbridgetoitsthermalenvironment.Furthermore,previouscalculationsaregeneralizedbycom-putingtheinelastictunnelingfluxtoallordersinthevibronic,usingtheself-consistentBornapproximationWeshowthatwhilethesecond-orderap-proximationcapturesmuchoftheessentialphysicsoftheIETprocess,infiniteordercorrectionscanleadtoquantitativedifferenceswithqualitativeimplications,e.g.,overtonesintheIETSspectrathatareabsentinthelowordertheoryareobtainedintheinfiniteordertreatment,andpeaks(dips)inspectrum,predictedbythelowordertheory,canappearasdips(peaks)intheinfiniteordercalculation.Atthesametime,experimentallyverifiablepredictionscanbemadewithrespecttothedependenceoftheshapesandwidthsoftheIETSvibrationalfeaturesontheparametersthatcharacterizethejunction.Inparticular,usingmodelparametersinferredfromexperimentalinformation,wearguethatthemainintrinsiccontributiontoIETSlinewidthsresultsfromthecouplingofmolecularvibrationstoelectronexcitationsinthelead(s)towhichthemoleculeischemicallyOurmodelisdefinedbytheHamiltonianwherethefourtermsontheright-handsiderepresent,respectively,electronsonthemolecules,electronsonthe(left,Landright,R)leads,aprimarymolecularharmonicmodeoffrequency,andasecondarysubsetofharmonicmodesoffrequenciesthatrepresentthethermalenviron-ment.Thefirstthreetermsarethesameasineq1.Inthelastterm,)representstheannihilation(creation)operatorsofthephononbathmodes.ThiszeroorderdescriptionissupplementedbytheinteractionHamiltonianÃand.Thethreetermsineq4correspondrespectivelytocouplingbetweenthebridgeelectronicsystemandtheleads,on-bridgecouplingoftheprimaryphonontotheelectronicsystem(thepolaronicformusedherecorrespondstotheassumptionthattheequilibriumpositionofthemolecularvibrationdependsonwhethertheelectronisonthemoleculeoronthemetals),andinteractionofthelocalphononmodewithitsthermalenvironment.Thephononbathandtheelectronicreservoirsthatrepresenttheleftandrightelectrodesareassumedtobeatthermalequilibriumwithtemperaturehowever,theelectronicelectrochemicalpotentialsoftheelectrodesaredifferentsuchthatistheelectronchargeandistheimposedbiaspotential.ThewidebandapproximationisinvokedforbridgeleadcouplingandforthecouplingbetweenthebridgephononandthethermalNanoLett.,Vol.4,No.9, environment,sothatthesecouplingsmaybecharacterizedbyconstantwidthparametersInaddition,thecouplingbetweentheelectronicandnuclearmotionsonthebridgeischaracterizedbytheparameterineq4.Ordersofmagnitudesoftheseparameterscanbeinferredfromdifferentexperimentaldata.Thewidthparametersarerelatedtoobservedlifetimesofexcesselectronsonmoleculesadsorbedonmetalsurfacesandcanbeestimatedtheoreticallyandfromtime-resolved2-photonphotoemissionexperiments(see,e.g.,refs34,35)tobeintherange0.11eVforchemisorbedspecies.canbeestimatedfromstudiesofvibrationalrelaxation(VR)ofmoleculesembeddedincoldmatrices.VRratesdependstronglyontheoscillatorfrequency,thebathspectrum,andthetemperature;however,forourpurposeitissufficienttoassertthatforlargerthanthebathDebyefrequencythecorrespondingwidthatlowisgenerallylessthan10eV.Finally,theelectronphononcouplingcanbeestimatedinmolecularsystemsfromreorganizationenergies,,inferredfromelectron-transferratestudiesinsimilarenvironments.Observedvaluesforare0.1eV,andtaking0.1eVplacesthemagnitudeoftherangeofafewtenthsofeV.Nextwebrieflyoutlineourtheoreticalapproach.WeusetheNEGF(Keldysh)formalism.ThecentralobjectsthatneedtobeevaluatedaretheelectronandphononGreen'sfunctionsrespectively,whoseprojectionsontherealtimeaxisareAtsteady-stateonecantransformtotheenergydomainandsolveself-consistentlytheDysonequationsfortheretardedandadvancedGFs(andsamewith)andtheKeldyshequationsforthelesserandgreaterprojections(andsamewithistheelectronself-energythat,forthepresentmodelHamiltonian,expressestheconse-quenceoftheinteractionoftheelectrononthebridgewiththeexternalelectronicreservoirsontheelectrodesandwithitsphononenvironment.istheself-energyoftheprimaryphonon,expressingtheconsequenceofitsinteractionwiththephononicthermalenvironmentaswellaswiththeelectronicsubsystem.Intheso-calledªnon-crossingºap-thecorrespondingcontributionsareassumedadditive,i.e.,usingobviousnotation,thecomponentsoftheelectronicself-energyassociatedwiththecouplingtotheleftandrightleads,aregiveninthewidebandapproximationbydenotetheleftandrightleads,)aretheDiracdistributionsfortheleftandrightelectrodes,characterizedbythecorrespondingchemicalpotentialspotentials(E-íK)/kBT]+1]-1,andisdefinedbyeq5.Inthesamewidebandapproximation,thephonon-bathcontributiontotheself-energiesoftheprimaryphonontakestheformsThephononcontributiontotheelectronicself-energyandtheelectroniccontributiontotheself-energyoftheprimary)(6)(6)Aö(t),Aö+(t¢)]ñ;Da(t,t¢))i£(t¢-t)á[Aö(t),Aö+(t¢)]ñ(8a)D(t,t¢))-iáAö+(t¢)Aö(t)ñ;D&#xTj /;༙ ;
Tf;&#x 9.9;x 0;&#x 0 9;&#x.978;&#x 85.;न ;Ɛ.;& ;&#xTm 0;(t,t¢))-iáAö(t)Aö+(t¢)ñ(8b)Gr(E))([G0r(E)]-1-ªr(E))-1(9)Dr(ö))([D0r(ö)]-1-¦r(ö))-1(10)G(E))Gr(E)ª(E)Ga(E)(11))(12))(13))(14))(15b)(15b)-fK(E)](15c))(16b))(16c)(16c)(ö/kBT)-1]-1(17)NanoLett.,Vol.4,No.9, phononmayberepresented,withintheself-consistentBornapproximation(SCBA),intermsoftheelectronandphononGFs.ThecorrespondingexpressionsareEquations914,18,19maybesolvedself-consistently.TheprocedurestartswiththeexpressionsfortheGreen'sfunc-tionsoftheelectronicsystemandtheprimaryphononsthatarezeroorderintheelectronphononinteractionand,ateachiterationstepuntilconvergence,updatestheelectronandphononself-energiesusingtheGFsobtainedinthepreviousiterationstep.ThenumericalcalculationoftheseGreenfunctionsandself-energiesinvolvesrepeatedintegra-tionsovertheelectronicenergyandthefrequencyvariable.Thesearedoneusingnumericalgridsthatarechosenlargeenoughtospantheessentialenergyandfrequencyregionsofthecorrespondingspectra,anddenseenoughrelativetothespectralwidthstoyieldreliablequadratures.Formorediscussionofthetheoryandfordetailsofthenumericalprocedure,seeref30.Afterconvergenceisachieved,theresultingGreenfunc-tionsandself-energiescanbeusedtocalculatemanyimportantcharacteristicsofthejunction.Inparticular,thetotalcurrentthroughthejunctionisgivenbyisthecurrentattheleft(right)molecule-leadcontact.ItcanbeshownthatinaccordancewithKirchoff'slaw.Usingeq11andtheassumedadditiveform(eq13)oftheelectronicself-energy,thetotalcurrenteq20canberecastasasumofelasticandinelasticcontributionswrittenbelowattheleftcontact:Acommonapproximationtotheseresultsisobtainedbyconsideringonlytermsuptosecondorderintheelectronphononinteraction.InthiscasetheGFsineq22arereplacedbytheirzero-ordercounterparts,whileistakenfromthelowestorderequivalentofeq18inwhichtheGFsarerepresentedbytheirzeroordercounterparts:Toobtaineq21inthesameorder,theGFsareexpressedbythelowestorderDysonformstogetEquations23and25wererecentlyusedbyMiietal.rederivetheresultsofPerssonandBaratofffortheIETSspectraforamodelofasingleelectroniclevelconnectingbetweentheleads(inthiscaseallGFsandself-energiesarescalarsandthetraceoperationineqs23and25isunneeded).Intheexamplesdisplayedbelow,wecomparetheresultsofthislowestorderperturbationtheory(LOPT)approximationtothefullSCBAcalculation.NextweusethetheoreticaltoolspresentedabovetoestimatetheeffectofvibroniccouplingonthelinewidthofvibrationalfeaturesinIETS.Sinceatrulyintrinsiclinewidthcanbeobservedonlyinasinglemoleculemeasurement,therelevantenergyparametersarethosesuitabletoanSTMexperiment,i.e.,essentiallypinnedtotheFermienergyoftherightelectrode.However,inwhatfollowsweconsiderthegeneralcaserepresentedbykeepingpinnedtotheunbiasedFermienergyandmovingthe 2ð[D(ö)Gr(E-ö)+Dr(ö)G(E-ö)+Dr(ö)Gr(E-ö)]-ijMj2Dr(ö)0)sdE )(18a) )(18b) )(18c) G(E)Ga(E-ö)+Gr(E)G(E-ö)](19a) )(19b) )(19c) psdE ªL(R)(E)G&#xTj /;༙ ;
Tf;&#x 9.9;x 0;&#x 0 9;&#x.978;&#x 158;&#x.232;&#x 141;&#x.171;&#x Tm ;(E)-ªL(R)&#xTj /;༙ ;
Tf;&#x 9.9;x 0;&#x 0 9;&#x.978;&#x 158;&#x.232;&#x 141;&#x.171;&#x Tm ;(E)G(E)](20) psdE 2ðTr[ªL(E)Gr(E)[ªL�(E)+ªR�(E)]Ga(E)-ªL�(E)Gr(E)[ªL(E)+ªR(E)]Ga(E)])2e ps-¥¥dE 2ð(fL(E)-fR(E))Tr[¡L(E)Gr(E)¡R(E)Ga(E)](21)Iinel)2e psdE ªL(E)Gr(E)ªph&#xTj /;༙ ;
Tf;&#x 9.9;x 0;&#x 0 9;&#x.978;&#x 399;&#x.931;&#x 631;&#x.182;&#x Tm ;(E)Ga(E)-ªL&#xTj /;༙ ;
Tf;&#x 9.9;x 0;&#x 0 9;&#x.978;&#x 399;&#x.931;&#x 631;&#x.182;&#x Tm ;(E)Gr(E)ªph(E)Ga(E)](22) psdE ªL(E)G0r(E)ªph,0&#xTj /;༙ ;
Tf;&#x 9.9;x 0;&#x 0 9;&#x.978;&#x 399;&#x.931;&#x 485;&#x.674;&#x Tm ;(E)G0a(E)-ªL&#xTj /;༙ ;
Tf;&#x 9.9;x 0;&#x 0 9;&#x.978;&#x 399;&#x.931;&#x 485;&#x.674;&#x Tm ;(E)G0r(E)ªph,0(E)G0a(E)](23))(andsamefor ps-¥¥dE 2ð(fL(E)-fR(E))Tr[¡L(E)G0r(E)¡R(E)G0a(E)]+2e ps-¥+¥dE ¡L(E)G0r(E)ªph,0r(E)G0r(E)¡R(E)G0a(E)+hc](25)NanoLett.,Vol.4,No.9, chemicalpotentialsoftheleftandrightelectrodeswithaªvoltagedivisionfactorºaccordingtoTheSTMlimit,withtheleftelectroderepresentingthetip,isgivenby1.Inthecalculationsdisplayedbelowwehavetakenastheenergyorigin,i.e.,0,andhaveusedthemodelthatimpliesstrongerpinningofthemolecularleveltotheelectrodethatprovideslargerelectroniccoupling.IETSspectraareusuallydisplayedasthesecondderivativeofthecurrentwithrespecttothebiasvoltageplottedagainstthisvoltage.Withinthepresentformalismthelow-temper-aturestructureofthisspectrummaybeinvestigatedbystartingfromandusingthe0limitsofeqs21and22.Itcanbeshownthat,providedthat)and)with),discharacterizedbyfundamentalandovertonesofathreshold(featurewhosewidthisoftheorder)andwhoseshape(peak,dip,peak-derivative-like)dependsonthejunctionparameters.AdemonstrationofthelatterstatementisshowninFigure1(seealsoFigure1ofref25)inwhichthisfundamentalfeatureisdisplayedfordifferentchoicesoftheresonanceenergy.Wenotethat,inprinciple,canbecontrolledbyagateelectrode.Inwhatfollows,considerthelow-temperaturewidthofthisvibrationalfeature.Focusingontheelectroniccontribu-),weusetheformalismdescribedabovetocomputetheelectronicself-energyofthebridgephononthatyieldsthiscontributionviaitsimaginarypart.Unlessotherwisestated,weuseasarepresentativesetofbridgeparametersthevalues0.05eV,0.5eV,0.0001eV,0.3eV,1eV,and0.13eV.Thevoltagedistributionistakenaccordingtoeq26with.Thetemperatureistaken10K;however,wefindthattheresultforispracticallyindependentofuptoroomtemperature.Figures25showtheresultsobtainedforthedependenceontheparametersthatcharacterizethejunction.(ThetotalIETSwidthisgivenbytowhichmakesasmallfixedcontribution.)Figure2showsthevoltagedependence.Physicallymeaningfulvaluesoftothethresholdvoltage,,atwhichthevibrationalfeatureisobserved.Atthisrangeandforourchoiceofmolecularparametersisnotstronglysensitivetotheappliedvoltage.Thissuggeststhattheshapeofthecorre-spondingIETSfeatureisnotstronglyaffectedbythisdependence.Theotherfiguresdisplaythedependenceoftheelectroniccontributiontothetotallinewidth,,ontheparametersthatcharacterizethemolecularjunction:thepositionoftheresonanceenergy(Figure3),thecouplingtothetip(left)electrode(Figure4), Figure1.TheIETSthresholdfeatureindforthemodeldefinedbyeqs3and4using0.5eV,0.0001eV,0.3eVand10K.Fullline(red)0.70eV,dashedline(green)0.60eV,dottedline(blue)0.55eV. Figure2.Theelectroniccontribution)totheimaginarypartofthevibrationalself-energyplottedagainstthebiaspotentialFullline:SCBAresults.Dashedline:Lowest(second)orderperturbationtheory.Seetextforparameters. Figure3.Theelectroniccontribution)tothewidthoftheIETSsignalplottedagainstthepositionofthebridgelevelrelativetotheFermienergyoftheunbiasedjunction.Fullline:SCBAresults(multipliedbyafactor10).Dashedline:Lowest(second)orderperturbationtheory.Seetextforparameters.NanoLett.,Vol.4,No.9, andthestrengthoftheelectronphononcoupling(orthecorrespondingreorganizationenergy,FigureThefollowingpointsarenoteworthy.(1)decreaseswithincreasingspacingbetweentheFermienergyoftheunbiasedjunctionandtheelectronicresonanceenergyofthebridge.Asalreadynoted,maybevaried,atleastinprinciple,byagateelectrode.Thisdependence,mostpronouncedathighermayprovideawaytoidentifycaseswheretheobservedwidthisindeeddominatedbythiscontribution.(2)dependsmildlyon,aparameterthatmaybevariedbychangingthetipmoleculedistance.For0.5eV,valuestypicaltoexperimentalsituationssuchasthatofref13,theelectroniccontributiontotheIETSwidthexceeds1meV,closetotheorderofmagnitudededucedfromthisexperiment.(3)ThevariationoftheelectronphononcouplingchangesfromarelativelymilddependenceatsmalltoarapidincreasewhenMincreasesbeyond0.4eV,theorderofmagnitudeof.Weshould,however,keepinmindthatforlargeelectronphononcouplingthevalidityofeventheSCBAlevelofcomputationisuncertain.(4)Forourchoiceofelectronphononcoupling,0.3eV,substantialdifferencesareobservedbetweentheSCBAresultsandthoseobtainedinsecond-orderperturbationtheory,whichisusedinstandardtreatmentsofIETS.TheobservationsmadeabovearebasedontheSCBAresults.(5)Aswasalreadymentionedonlyveryweaklysensitivetothetemperature.Obviously,however,athighertemperatures,othercontributiontotheoverallwidthwilldominatetheIETSlinewidth.Inconclusion,basedonthesimplemodeldescribedaboveandusingareasonablechoiceofparameters,wehavefoundthatthecouplingofmolecularvibrationalmodestotheelectroniccontinuaoftheleads(viathemolecularvibroniccoupling)makesasubstantialcontributionoftheorderof1meVtothewidthofIETSspectra.ThisisexpectedtobethelargestsourceofbroadeninginlowtemperaturessinglemoleculeIETSspectra;however,inexperimentssuchasthatofref13thatinvolveafewthousandmolecules,onecannotruleoutinhomogeneouscontributionsasthedominatingsourceofbroadening.WehavealsoshownthatIETSlineshapesandlinewidthsdependonjunctionparameters,inparticularthepositionofthebridgeelectronicresonancethatcanbeaffectedbyagatevoltageandthemoleculeleadcouplingthatcanbecontrolledbythetip-moleculedistanceinascanningtunnelingspectroscopysetup.Anotherobservation,oftechnicalnature,concernstheuseofsecond-orderperturbationtheoryintheseapplications.ComparingcalculationsdoneonthislevelofapproximationtoresultsobtainedfromtheSCBAtheory,wemayconcludethatsecondorderperturbationtheorycanaccountonlyqualitativelyfortheeffectsdiscussedabove.Withtypicalmolecularcouplingparameters,theSCBAcalculationseemstoyieldreliableresultsaslongastheelectronphononinteractionisnottoolarge.M.R.thankstheDoD/MURIinitiative,theNASAURETIprogramandtheNSFNCNprogramforsupport.A.N.thankstheIsraelScienceFoundation,theIsraelBinationalScienceFoundationandtheVolk-swagenFoundationforsupport.A.N.thanksProf.JanvanRuitenbeek(LeidenUniversity)forilluminatingdiscussions.(1)Wolf,E.L.Principlesofelectrontunnelingspectroscopy;OxfordUniversityPress:NewYork,1985;Vol.71.(2)Hipps,K.W.;Mazur,U.J.Phys.Chem.,7803.(3)Lee,H.J.;Ho,W.,1719.(4)Lorente,N.;Persson,M.;Lauhon,L.J.;Ho,W.Phys.Re.Lett.,2593(5)Hahn,J.R.;Ho,W.Phys.Re.Lett.,196102.(6)Lauhon,L.J.;Ho,W.Phys.Re.Lett.,4566.(7)Hahn,J.R.;Lee,H.J.;Ho,W.Phys.Re.Lett.,1914.(8)Gaudioso,J.;Laudon,J.L.;Ho,W.Phys.Re.Lett.,1918.(9)Stipe,B.C.;Rezaei,M.A.;Ho,W.Phys.Re.Lett.,1724.(10)Lauhon,L.J.;Ho,W.Phys.Re,R8525.(11)Zhitenev,N.B.;Meng,H.;Bao,Z.Phys.Re.Lett.,226801.(12)Park,H.;Park,J.;Lim,A.K.L.;Anderson,E.H.;Alivisatos,A.P.;McEuen,P.L.,57.(13)Wang,W.;Lee,T.;Kretzschmar,I.;Reed,M.A.NanoLett.,643. Figure4.SameasFigure3,whereevaluatedatplottedagainstthewidthofthebridgelevelthatresultsitscouplingtothetip(leftelectrode). Figure5.)plottedagainsttheelectroncouplingstrengthusingparametersgiveninthetext.FullanddottedlinesareSCBAresultsobtainedfor0.05eVand0.5eV,respectively.Dashedanddashdottedlinesare2ndorderperturbationtheoryresultsfor0.05eVand0.5eV,NanoLett.,Vol.4,No.9, (14)Kushmerick,J.G.;Lazorcik,J.;Patterson,C.H.;Shashidhar,R.;Seferos,D.S.;Bazan,G.C.NanoLett.,639.(15)Lee,H.J.;Ho,W.Phys.Re,R16347.(16)Smit,R.H.M.;Noat,Y.;Untiedt,C.;Lang,N.D.;Hemert,M.C.V.;Ruitenbeek,J.M.V.,906.(17)Persson,B.N.J.Phys.Scripta,282.(18)Baratoff,A.;Persson,B.N.J.J.Vac.Sci.Technol.A,331.(19)Persson,B.N.J.;Baratoff,A.Phys.Re.Lett.,339.(20)Troisi,A.;Ratner,M.A.;Nitzan,A.J.Chem.Phys.,6072.(21)Keldysh,L.V..Phys.JETP,1018.(22)Kadanoff,L.P.;Baym,G.QuantumStatisticalMechanics.Green'sFunctionMethodsinEquilibriumandNonequilibriumProblemsBenjamin:Reading,MA,1962.(23)Wagner,M.Phys.Re,6104.(24)Datta,S.ElectrictransportinMesoscopicSystems;CambridgeUniversityPress:Cambridge,1995.(25)Mii,T.;Tikhodeev,S.G.;Ueba,H.Phys.Re,205406.(26)Mii,T.;Tikhodeev,S.;Ueba,H.Surf.Sci.,26.(27)Tikhodeev,S.;Natario,M.;Makoshi,K.;Mii,T.;Ueba,H.,63.(28)Davis,L.C.Phys.Re,1714.(29)Bayman,A.;Hansma,P.;Kaska,W.C.Phys.Re,2449.(30)Galperin,M.;Ratner,M.A.;Nitzan,A.,tobepublished.(31)Mahan,G.D.Many-particlephysics,3rded.;PlenumPress:NewYork,2000.(32)Haug,H.;Jauho,A.-P.QuantumKineticsinTransportandOpticsofSemiconductors;Springer:Berlin,1996;Vol.123.(33)Ouranalysis,aswelltheearlierworkofPersonandBaratoffanditsreassessmentbyUebaandco-workers,correspondtothelimitwhereelectrontransmissionacrossthejunctionisalowprobabilityeventthatdoesnotdisturbthethermalequilibriumintheleads.Intheoppositelimitwheretheleadbridgecouplingisstrongsothatthetransmissionprobabilityisnearly1(implyingasinglechannelconductionof),wemayencounterthesituationwhereinthenegativelybiasedbridgebackscatteredelectronsofenergiesintheconductionwindowbetweentheleftandrightFermienergiesarelocallydepletednearthejunction.Thiscausesincreasedreflectionattheonsetofinelasticscattering,whichispresumablythedominantmechanismforobservednegativepeaksinpointcontactspectroscopycharacterizedbylargetransmissionprobabilities,seee.g.,R.H.M.Smit,etal.,906;N.AgraõÈt,etal.Phys.Re.Lett.,216803.(34)Gauyacq,J.P.;Borisov,A.G.;Raseev,G.Surf.Sci.,99.(35)Kinoshita,I.;Misu,A.;Munakata,T.J.ChemPhys.,2970.(36)Wingreen,N.S.;Meir,Y.Phys.Re,11040.(37)Thisshouldprovideareasonablesimplificationprovidedthatthemolecularelectroniclevelisfarenough(relativetoitswidth)fromthemetalbandedge.Thesameassumptionusedforthephononbathineq16isgenerallyvalidbecauseofthesmallnessofthevibrational(38)Migdal,A.B..Phys.JETP,996.(39)OtherworkershaveusedSCBAforsimilarapplications,see,e.g.,Hyldgaard,P.;Hershfield,S.;Davies,J.H.;Wilkins,J.W.Ann.Phys.,1.Ourcalculationgoesbeyondthatdonebytheseauthorsinseveralways:first,wedonotlimitittozerotemperature,second,wecalculatethenonequilibriumbehaviorofbothelectronandprimaryphononsystemswhiletheseauthorsassumethatthephononremainsinequilibrium;andfinally,mostimportantinthepresentcontext,weaddresstheelectroniccontributiontothephononself-energyaspartoftheself-consistentcalculationastepthatwasnotdonebefore.(40)Datta,S.;Tian,W.D.;Hong,S.H.;Reifenberger,R.;Henderson,J.I.;Kubiak,C.P.Phys.Re.Lett.,2530.(41)Tian,W.D.;Datta,S.;Hong,S.H.;Reifenberger,R.;Henderson,J.I.;Kubiak,C.P.J.Chem.Phys.,2874.(42)Indirectdependenceontemperaturecouldcomefromthefactthatisoftenstronglytemperaturedependent,howeverwefindthatisalsoquiteinsensitivethemagnitudeofuptovaluesoftensofwavenumbersforthelatter.NanoLett.,Vol.4,No.9,