nist -纯碳纳米管薄膜中声子发射与吸收的拉曼散射共振结构

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TheResonanceStructureofRamanScatteringforEmittedandAbsorbedPhononsinChirality-PureCarbonNanotubeFilms

PaulFinnie*,†,AdamWind†,¥,JianyingOuyang†,PavelShapturenka§,JeffreyA.Fagan§

†QuantumandNanotechnologiesResearchCentre,NationalResearchCouncilCanada,1200MontrealRoad,Ottawa,Ontario,K1A0R6,Canada

¥UniversityofWaterloo,200UniversityAvenueWest,Waterloo,ON,N2L3G1,Canada

§MaterialsScienceandEngineeringDivision,NationalInstituteofStandardsandTechnology(NIST),Gaithersburg,MD20899,USA

Certainequipment,instruments,software,ormaterials,commercialornon-commercial,areidentifiedinthispaperinordertospecifytheexperimentalprocedureadequately.SuchidentificationisnotintendedtoimplyrecommendationorendorsementofanyproductorservicebytheNationalInstituteofStandardsandTechnology(NIST)ortheNationalResearchCouncilCanada,norisitintendedtoimplythatthematerialsorequipmentidentifiedarenecessarilythebestavailableforthepurpose.

KEYWORDSSingle-wallcarbonnanotubes,Ramanspectroscopy,Ramanexcitationmapping,Stokes,Anti-Stokes

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INTRODUCTION

Ramanscattering(RS)revealsvibrationalmodesofmoleculesandmaterials.Forsingle-wallcarbonnanotubes(SWCNTs),Jorio,A.,Saito,R.RamanSpectroscopyForCarbonNanotubeApplications.J.Appl.Phys.2021,129,021102.

amongmanyothermaterials,RSisanessentialcharacterizationtool.Whenphotons-eitherincidentorscattered-areatenergieswhichmatchelectronicresonancesofthesample,scatteringprobabilitiesareenhanced,andtheprocessiscalledresonantRS(RRS).Differentmolecularstructures(species,orcolloquially“chiralities”)ofSWCNTs,whichareclassifiedbytwowholenumberindices(n,m)identifyingthe“rollup”latticevectoronagraphenesheetthatwouldformaseamlesscylinder,havedifferentvibrationalmodesanddifferentelectronic(/excitonic)resonances.Weisman,R.B.;Bachilo,S.M.DependenceofOpticalTransitionEnergiesonStructureforSingle-WalledCarbonNanotubesinAqueousSuspension:AnEmpiricalKatauraplot.NanoLett.2003,3,1235–1238.

Thewidely-usedKatauraplot,1,2,Kataura,H.;Kumazawa,Y.;Maniwa,Y.;Umezu,I.;Suzuki,S,;Ohtsuka,Y;Achiba,Y.Opticalpropertiesofsingle-wallcarbonnanotubes.SyntheticMetals1999,103,2555-2558.

,Saito,R.;Dresselhaus,G.;Dresselhaus,M.S.Trigonalwarpingeffectofcarbonnanotubes,PhysicalReviewB2000,61,2981-2990.

whichmapsthepeakresonantwavelength(usuallyinthevisiblefortheE22resonance)versustheradialbreathing(RB)vibrationalmodefrequency(≈150cm-1to350cm-1),allowsonetoinferthediameterandeventheexact(n,m)ofindividualSWCNTs.Thismodeismostcommonlycalledtheradialbreathingmode(RBM),howeverhereweuseRBforbrevity.(Itisapeculiaritythatthisbandaloneincludes“M”for“mode”initsmostcommonlyusednotation,whilealltheotherbandsdonot,soRBisarguablyamoreconsistentchoicefornomenclature.)Theresonanceexcitationprofile(REP),i.e.,theRRSintensityversuswavelength,ofsingleRamanmodessuchastheGband(≈1592cm-1)hasalsobeenexploredforanumberof(n,m)species,revealingquantuminterferenceeffectsandstimulatinginterestinRRSphotophysics.Duque,J.G.;Chen,H.;Swan,A.K.;Shreve,A.P.;Kilina,S.;Tretiak,S.;Tu,X.;Zheng,M.;Doorn,S.K.ViolationoftheCondonApproximationinSemiconductingCarbonNanotubes.ACSNano2011,5,5233–5241.

,Moura,L.;Moutinho,M.;Venezuela,P.;Fantini,C.;Righi,A.;Strano,M.;Pimenta,M.RamanexcitationprofileoftheGbandinsingle-chiralitycarbonnanotubes.Phys.Rev.B2014,89,035402.

,Gordeev,G.;Flavel,B.;Krupke,R.;Kusch,P.;Reich,S.AsymmetryofresonanceRamanprofilesinsemiconductingsingle-walledcarbonnanotubesatthefirstexcitonictransition.Phys.Rev.B2019,99,045404.

,Tran,H.;Blancon,J.;Huntzinger,J.;Arenal,R.;Popov.V.;Zahab,A.;Ayari.A.;San-Migeul,A.;DelFatti,N.;Sauvajol,J.-L.;Paillet,M.ExcitonicopticaltransitionscharacterizedbyRamanexcitationprofilesinsingle-walledcarbonnanotubes.Phys.Rev.B2016,94,075430.

Moregenerally,SWCNTshavemanyRRSbands,witheachhavingitsownparticularREP.

UntilrecentlyithasbeenoneroustoobtainREPsoverabroadrangewithhighresolutionduetotheneedforahigh-qualitylightsource,suchasatunablelaser,whichhaslimitedrangesandnon-trivialtuning.Progressinoptoelectronics,suchasthehyperspectralmethodusedhere(videinfra),however,aremakingiteasiertobuildupmultiwavelength-essentiallyfullcolor-plotsofRSintensitiesmappedoutversusincidentandoutgoingphotonenergies.Finnie,P.;Ouyang,J.;Lefebvre,J.FullSpectrumRamanExcitationMappingSpectroscopy.Sci.Rep.2020,10,9172.

,Finnie,P.;Ouyang,J.;Fagan,J.A.BroadbandFull-SpectrumRamanExcitationMappingRevealsIntricateOptoelectronic–VibrationalResonanceStructureofChirality-PureSingle-WalledCarbonNanotubes.ACSNano2023,17,7285–7295.

,Finnie,P.;Ouyang,J.;Fagan,J.A.StaticandDynamicRamanExcitationMappingofChirality-PureCarbonNanotubeFilms.CommunicationMaterials2025,6,10.

ThesearetermedRamanexcitationmaps(REMs)andincludebothRSspectraandREPsassubsets.TheKatauraplotcanbeviewedastheREMfortheRBmodealone,andbyitselfisanimportantfingerprintingtoolforSWCNTs.Morebroadly,everymodenecessarilyhasarelated,butdifferent,REP,andSWCNTshavemultiplevibrationalmodes,manyofwhichproducestrongRS.Itisinterestingtoinvestigateallthesemodestogetherandobservehowtheychangewithingoingphotonenergy(i.e.,laserwavelength),bothintermsofphotophysics,becausetheyreflectintrinsicmaterialpropertiesandcouplingtotheirenvironment,andasapotentiallyveryselectiveandsensitivecharacterizationtoolforgeneralchemicalanalysis.

InRamanscattering,bothphononemission,knownasStokes(St)scattering,andphononabsorption,knownasanti-Stokes(ASt)scatteringarepossible.EarlystudiesonSWCNTshadtocontendwithmixedchiralityensemblesanditwasquicklyapparentthatfixedwavelengthRSsampleddifferentpopulationsofSWCNTsinStandinASt.S.D.M.Brown,P.Corio,A.Marucci,andM.S.Dresselhaus;Pimenta,M.;Kneipp,K.Anti-StokesRamanspectraofsingle-walledcarbonnanotubes.Phys.Rev.B2000,61,R5137-R5140.

SingleSWCNTsRSstudiesdemonstratedwavelengthdependentSt/AStpeakratiosfortheRBmodeandrationalizedthisintermsofthedifferentresonanceenhancementstructureforStvs.ASt.SouzaFilho,A.;Chou,S.;Samsonidze,G.;Dresselhaus,G.;Dresselhaus.,M.S.;An,L.Liu,J.;Swan,A.;Ünlü,M.;Goldberg,G.G.Stokesandanti-StokesRamanspectraofsmall-diameterisolatedcarbonnanotubes.Phys.Rev.B.2004,69,115428-1-115428-8.

Morerecently,theGbandREPhasbeeninvestigatedinStandAStforchiralitysortedSWCNTs,showingasymmetricREPswithdifferentpeaksforStandASt.Gordeev,G.;Jorio,A.;Kusch,P.,Vieira.B.;Flavel,B.;Krupke,R.;Barros,E.;Reich,S.Resonantanti-StokesRamanscatteringinsingle-walledcarbonnanotubes.Phys.Rev.B2017,96,245415

Beyondthis,broadbandREMshavenotbeenappliedtomeasuringStversusAStphenomenononthesameSWCNTsample;itisinterestingtolookatbothStandASttogether,toseehowtheycompare,andexperimentallyevaluatewhatmorecanbelearnedfromthecombination.

Iftheyarereasonablyeasytoobtain,thereisgreatpotentialformorespecificandsensitivechemicalsensingusingREMs.Thereisalsogreatpotentialtotestphotophysicalmodelsofmaterialsandlightscattering.AsananalyticaltooltheREMisexquisitelyspecific,containingallRSinformation,allREPinformationandthecouplingbetweenthem.JustastheRBmodeKatauraplot“unfolds”theRBregionandthephotoluminescenceexcitation(PLE)mapuncluttersfluorescencespectra,allRScanbesectionedinthisway–andfromscatteringtheory,everybandbehavesdifferentlyandhasitsownprofile.REMalsohaspotentialtoseparateoutphysicalphenomenaarisingfromdensityofstates,temperatureandothereffects.EachRSbandhasitsowncouplingtoelectronicstatesandtheirvariationinmagnitudeismanifestintheREM.ItshouldbecomepossibletorigorouslytestdetailedmodelsofRSsincetheymakedifferentpredictionsforREPsandREMs.IfREMsareslowtoacquire,suchinvestigationsbecometimeconsumingandlesspractical.

InthisworkweshowexperimentalbroadbandREMsforspecies-pureSWCNTs.WeshowhowallRSmodesareaffectedbyresonanceeffectsonbothsidesofthelaserRayleigh(Ry,elasticscattering)line.EachREMisinformationdense,reflectingdetailsofmaterialpropertiesandquantummechanicalinteractions,andsocanbeusedasaspecific,sensitivefingerprint-morespecificthanRSaloneoropticalabsorptionalonewouldbe–orasavehicletoexploredetailedphotophysics.TheSt/AStmapisalsosensitivetotheelectronicdensity-of-stateoccupancy,andhencethetemperature.

Mostcommonly,RSisperformedwithfixedwavelengthlaserillumination.Thislaserlightistypicallyfocusedtoaspotontoasamplebyamicroscopeobjective,andthescatteredlightiscollected,oftenbythesameobjective,withtheelasticallyscatteredlaserlightblockedbyfiltersandtheremainingscatterdispersedbyagratingforspectralanalysis.McCreery,R.LM.RamanSpectroscopyforChemicalAnalysis;Wiley-Interscience:2000

SinceSWCNTshowhighlyresonantRS,oftentwoormorelaserwavelengthsareusedinsequencetocharacterizesamples.Takingadvantageofprogressinopticalequipment,wehavebeenusingageneralizationofthismethodthatwecallfullspectrumREM(FS-REM).

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Weuseabroadbandwhitelightsourcedispersedspectrallyacrossthesampleandfocusedlaterally,makingakindof“rainbow”line.Lefebvre,J.RealTimeHyperspectroscopyforDynamicalStudyofCarbonNanotubes.ACSNano2016,10,9602-9607.

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Thislightiscollectedsimultaneouslyfromeachspatial(i.e.,ingoingphotonenergy)position,filtered,anddispersedspectrallyacrossa2Ddetector.ThismakesitpossibletosimultaneouslycollecttheRSforabroadwavelengthrangeofingoingphotonenergiesinasinglecapturesequence,suchthatwecancollecttheentireREMveryquicklywithacquisitiontimesintherangefrommilliseconds(inthecaseofsinglecapturewindowswherewedonotchangeanyoftheRylineblockingfilters)tominutes(acrossthefullrangeofingoingphotonenergiesandoutgoingphotonenergiesinthecaseofaweakerscatterer).

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ThoughAStscatterismuchweakerthanStscatter,thesignal-to-noiseonourFS-REMinstrumentisadequatetoreadilyobtainexperimentalAStREMsfromSWCNTswithexposuretimesasshortasseconds.Forafixedspectralwindow,hereweacquireStandAStinseparateexposuresusingdifferentfilters,andtocoveralargebandwidthwecombineseveralsmallerspectralwindowsusingseveraldifferentfilterpairs.ThisapproachtoblockingunwantedRylightscatteringandpassingwantedRSisextremelyefficient,however,notethatwhiletheresultingstepwisemapscoverlargeRamanshiftswell,lowerenergybandssuchastheRBmodeendupgettingblockedintermittently.MoreexperimentaldetailsareintheMethodssection.

Atthesametimeastherehasbeenprogressinopticalinstrumentation,therehasbeenprogressinseparationscience,suchthathighly(n,m)pureSWCNTsamplesareavailableinpracticallyusefulquantitiesforexperimentandapplication.Sims,C.M.;Zheng,M.;Fagan,J.A.Single-wallcarbonnanotubeseparationsviaaqueoustwo-phaseextraction:newprospectsenabledbyhigh-throughputmethods.Chem.Commun.2025,61,2026-2039.

,18Here,weinvestigatethreedifferentseparated(n,m)speciesofsemiconductingSWNCTs,preparedindifferentways,andreportexperimentalbroadbandSt/AStREMsforeachofthem.AdetaileddescriptionofsamplepreparationisintheMethodssection.Briefly,allthreespecies,(7,6),(7,5)and(6,5)werepreparedbyaqueoustwo-phaseextraction(ATPE).Samplesofthe(7,5)and(6,5)specieswerealsoseparatelypreparedbyapolymerwrappingprocedure,which,forthepolymersusedhere,oftengoesbyamorespecifictechnicalname:conjugatedpolymerextraction(CPE).Thisacronymisusedbelow.Todistinguishthetwosetsofsamples,weappendan“s”afterthe(n,m)indicestothosepreparedbyATPE,anda“p”tothosepreparedbyCPE.

Homogenous,flatopaqueorsemi-opaquefilmsareidealforrapidREMmeasurementsandseveraldepositionmethodswereused.Liquidsamplescanalsobemeasuredthisway,butunlesstheliquidisopaquethefocaldepthtendstobelarger,whichcancausesomelossofspectralresolution,sincetherainbowlineistheonly“slit”.Also,forliquids,lightmayscatterfromthecontainer,whichcanrequireadditionalcare.Opaquefilmsscatterlightveryeffectively,withnocontainerrequired.Scatteringintensitiesareconcentrationdependent,soliquidsamples,iftheyaredilute,tendtohavecorrespondinglylowerintensities.Ofcourse,justlikeconventionalRS,inhomogeneousfilmscanbescannedand/oraveraged,buthomogeneityisveryconvenienttoenable“one-shot”measurementsthatarefullyrepresentativeofthesampleunderinvestigation.

TheATPEsampleswerealldropcastanddriedonCaFsubstrates,exceptforoneofthe(7,6)samples,denoted“(7,6)s,Si”,whichwasproducedbymembranefiltrationandtransferredasasemi-opaquethinfilmontoasiliconsubstrateremovingthesurfactantduringtheprocess.CPEsamplesweredepositedbyfiltrationintoafilmonapolytetrafluoroethylene(PTFE)filtermembrane.Exceptforthe“Si”appendedsample,itisworthrememberingthat“s”samplesremaininasurfactantmatrix,and“p”samplescoatedbytheirconjugatedpolymer.

RESULTSANDDISCUSSION

Figure1showsanREMforthespecificsamplelabelled(7,6)s,Si,whichsignifiesthe(7,6)speciespreparedinsurfactantanddepositedonSi.Theingoingphotonenergy,fromthesupercontinuumlaser,isalongthey-axisrangingfrom1.55eV(800nm)to2.55eV(486nm).Thedetectedoutgoingphotonenergy,alongthex-axis,rangesfrom1.45eV(855nm)to2.45eV(506nm).Theintensity,incounts/s,ateachpixelonthecameraisrepresentedbythecolorscale.Theelastic(Rayleigh)scatteringline,alongwhichtheingoingphotonandoutgoingphotonenergiesareequal,isrepresentedbythediagonaldottedwhiteline,labelledRy,followingterminologyforelasticlightscatteringbymolecules.Here,themultiplebluestep-likeoverlappingsquaresadjacenttotheRylineareinstrumentalandcausedbythefiltersusedtoblocktheelasticallyscatteredlaserlight.RS,bydefinitioninelastic,isobservedasdiagonallinesofthesameslopeastheRyline(slightdeviationsarepossible),butofvaryingintensitybyband,(i.e.,theintensityoftheparticularphononmode)andalonganygivenband(i.e.,theintensityvariationalongthelengthforagivenphononmode)duetointeractionswithelectronicstates.

ConventionalStRS,inwhichphononsareemittedinthescatteringprocess,areaboveandtotheleftoftheRayleighline(lowerenergythantheingoingphoton.AStRS,whichinvolvesphononabsorption,isbelowandtotherightoftheRyline(greaterenergythantheingoingphoton).AStRSistypicallymuchweakerthanconventionalStRS.Here,toplotbothonthesamescale,theAStregionhasbeenmultipliedbyaconstantfactortobringtheAStGbanduptothesameorderofmagnitudeinintensityastheGbandintheStregion.Thisfactorisdisplayedonthemapasthenumberfollowing“ASt×”.

InFigure1,themapisadditionallylabelledwiththesecond-orderopticaltransition(E22)excitonenergyforthe(7,6)SWCNTonbothaxes.Theenergyofthisresonancewasdeterminedbyopticalabsorption(OA)ofthepurifiednanotubeliquidsuspensionsbeforesubstratedeposition.Thediagonalbandsareauto-labelledwithconventionalRamanscatteringphononmodenamesusingnamesfromtables19(TA,LA,oTo,I-[shortforIFM-],oTo,I+[shortforIFM+],D,G,M[combinedM+/M-],iT[shortforiTOLA],2D[alsosometimescalledG’],G+D,and2G).EachStmodehasacorrespondingAStmodeofthesamelabelbutwithaminussigninfront.InSt,somehigherordermodesbeyondthosetypicallytabulatedarealsovisibleandcanbeassignedasdescribedinearliercontributions(e.g.,G+M,G+iT,2iT,3D,2D+G,3G).

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Notably,thevariationoftheincidentlaserpowerwithingoingphotonenergy,andthevariationofresponseoftheopticalsystemisnotcorrectedforinFigure1ormapsinsubsequentfigures.However,itwillbeshownbelowthatoverall,forthisinstrument,theycouldbecorrectedwithalinearlyscalingcorrectionfactorovertheingoingphotonenergyrangefrom1.6eVto2.35eV.Abovethisenergythecorrectionisnonlinearbecausetheintensityofthelightsourcedrops.Wedonotapplythiscorrectiontothemaps,butwillapplyittoREPs,below.

Figure2showsthesametypeofREMplot,butformultiplespecies,includingthe(7,6),(7,5)and(6,5),andseveraldifferentsamplepreparations.Figure2(a),isthesame(7,6)mapasFigure1,shownagainhereforcomparisonpurposes.Figure2(b)andFigure2(c)are(7,5)and(6,5)SWCNTfilmsrespectively,bothonPTFEsubstrates,andhavingbeenpreparedbyCPE.Figures2(d),2(e)and2(f)arethesamesequenceofspeciesasthetoprow,(7,6),(7,5)and(6,5)respectively,butsortedbyATPEanddropcastonCaFsubstrates.Havingdifferentspeciesanddifferentpreparationsenablestheevaluationofsystematicchanges.

TheREMsinFigure1andFigure2arerichwithdetailandwehighlightsomeofthemajorfeatures.AmongthemostintensebandsaretheD,G,2DandtheASt-Gband,soweemphasizethem.Examiningthemaps,inallcasestheirintensitiesarehighestinthetop-leftandbottom-rightquadrantsofeachpanelasdelineatedbytheE22resonanceenergy,shownashorizontal(ingoingphotonenergy)andvertical(outgoingphotonenergy)dashedlines.IntensitiesofthestrongerRSbandsareoftheorderof10kcounts/sinthetopleftStquadrantbetweenthehorizontalandverticalE22lines,andsimilarlyintenseinthebottomrightAStquadrantbetweenE22lines,butonlyaftermultiplyingbytheAStfactorshownoneachplot.Crossingthesedottedlines,i.e.,movingoutsidethesequadrants,thebandintensitiesdropsignificantly,thoughtheyonlyweakenandarestilldetectable.InthecaseofFigure2(b)theyonlyweakenbyalimitedamountmovingupandtotheright.

Forcomparison,usingauniformintensityscale,manymoreRSbandsarevisibleontheStsidethanontheAStside.OntheAStside,the-2DispresentonmostAStplots,butbarelyvisibleonthescaleofFigure2.SupportingInformationFigureS1showsthesamedataonacolorscalewhichbringsoutweakerfeatureswhilesaturatingstrongerones.Thentheweaker-2Dbandcanbeseenfaintlyin(a)and(c)andmorestronglyin(b),(d)and(e).TheintensityofAStbandsdropsoffmovingdownandtotheright,awayfromtheRyline,inawaythatStlinesdonot.

AcleartrendemergesifwefollowanyoneSt/AStpairofbandsmovingfrombottomlefttotopright.WiththechoicesofcolorscalethisiseasiesttodofortheG/-Gbands.Itisapparentthattheintensitiesofthetwobandstrackeachotherclosely(asreflectedacrosstheRyline)movingupandtotheright.Infact,toagoodapproximation,thetrendoftheAStbandisthemirrorimageoftheStbandacrossthediagonalRyline.Below,thistrendwillberationalizedwithreferencetothemostbasicquantummechanicalpicturesofRRS.

Thereare,however,someothercomplicationstokeepinmind.Dropcastsamples(Figures2d,e,f)arenotasuniformasfilteredorfilteredandtransferredsamples(althoughhavingotheradvantages),andthisspatialnon-uniformitycancausesomevariationinthesesingle-shotREMs.Onallmapsthereareoccasionallybrighthorizontalstreakswhichareartifactsfromextralightscatteringduetoparticulatesonthesurface.Lessapparent,therearealsodarkhorizontalstreaksarisingfromplaceswherethescatteredlightiseitherblockedbyparticulates,orpossiblywheretherearenonanotubes.TheCPEsamples,inadditiontoStandAStRaman,alsoshowbroadlightemissionathighlaserenergieswhichisattributabletofluorescencefromthepolymersurroundingtheSWCNTs.

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Importantly,themultiplicationfactorrequiredtobringtheAStintensityuptotheStintensityintheshownREMsvarieswidely,fromaslowas20inFigure2(b)toashighas4000inFigure2(f).Thiswillbeexplainedbelow.

OtherfeaturesinFigure2arephysicallyinteresting.Inlinewithpreviousstudies,

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fortheGbandtheloweringoingphotonenergyresonance(wheretheingoingphotonisatenergiesclosetoE22)isconsistentlystrongerthanthehigherenergyresonance,exceptforthe(7,5)sampleforwhichthisisnotclear.However,notallbandsarelikethat.The2Dbandisgreaterneartheoutgoingphotonenergyresonance,withthepeakdownshiftedslightlyfromtheactualE22line.Theefficiencyoftheopticalsysteminingoingphotonenergyandoutgoingphotonenergywillbiasthesignalsomewhat,aswillbeseenbelow,butitdoesnotaccountforthislargedifference,especiallyincomparisontotheothersamples.

Next,weinterpretsomekeyaspectsofthemapsinthecontextofthequantumtheoryofRS.AlthoughthetheoryofRSisverywelldeveloped,thereareseveraldescriptionscorrespondingtovariouslevelsofapproximationinthescatteringprocess,usingdifferentordersofperturbationtheory,

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,assumptionsaboutwhetherrealorvirtualstatesareinvolved,20,21andassumptionsaboutthemotionofatomicnuclei.

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ThevariousmodelsmaypredictsimilarsinglelaserwavelengthRS,butshouldmakequantitativelydifferentpredictionsforREPsandREMs.Hopefully,withREMbecomingmoreaccessible,comparingexperimentalmapswithdetailedmodelscouldhelpconfirmwhatpicturebestrepresentstherealphysicalprocessandtheimportanceofspecificterms.

Withoutspecifyinganyparticularmodel,thereisaninterestingfundamentalfeatureoftheSt/AStREMs.Thatis,StRSisthetimereverseprocessofAStRS.Thescatteringprobabilityissymmetricundertimereversal,whichonthemapamountstoswappingingoingphotonandoutgoingphotonenergies–i.e.,reflectingthemapacrossthediagonalRyline.Infact,thissymmetryisalreadyapparentbyeyetoagoodapproximationifyoulookatanyoneband,e.g.,ifyoucomparetheshapeofGand-Gbandsforanyofthemaps.However,allAStbandsareattenuatedfromStbandsandtheyareattenuatedbydifferentratios,sothissymmetryisbrokenwhenlookingatmorethanoneband.

Brieflyignoringtheattenuation,therearedifferentmodelsusedtocalculatetheRRSprobability.OnemodelissecondorderKramers−Heisenberg−Dirac(KHD)theory

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,whichcanrepresentSWCNTREPswelliffourlevelsareused.

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AnotherpopulartreatmentofthequantumtheoryofRRSisthird-orderperturbationtheory,describedinRef.[22],andalsopresentedinananocarbonspecificcontextinRef.[

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].Wefollowthelatterhere.

Thebasicsofthistreatmentfollowfromconsideringthethreeinvolvedparticles:theincidentphoton,thescatteredphoton,andthephononthatisemittedinStRS(orabsorbedinAStRS).Allthreeinteractwiththesystemofelectronswhichgivesrisetosixpermutations,whichbecomeadditivetermsforthescatteringmatrixelement,allofwhichbothaddcoherentlyandinterfere.

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Thisissimplifiedmathematicallybyconsideringonlythemosthighlyresonantterms,thatis,thosethathavedenominatorstendingtozeroastheingoingphotonapproachessomeenergy.Inthelimitwhenthephononenergy(Q)issmall,oneofthesetermshasatwinresonanceenhancementandthematrixelementcanbesimplifiedto

P∝HEL−EX−iΓEL−Q−EX−iΓ+C2 . (1)

HereHisaneffectivescatteringmatrixelementintegratedovermanyvariables,EListhelaserenergy(i.e.,ingoingphotonenergy),EXistheexcitonenergy,Qisthephononenergy,Γisadampingtermwhichsoftenssingularities,andCisasumofthefiveotherscatteringtermswhichareallpresumedsmallbecausetheydonothavethetwinresonance.Equation1isforStbutisthesameforASt,onlysubstituting-QforQ.(QisalsopresentintheCterm.)Ingeneral,lightscattering,whetherRayleigh,StRSorAStRSalsohasanon-resonantscalingfactormultiplierforalltermswhichisnearlyfourthpowerinEL,butthisfactorisnotexplicitlywrittenoutinequation1.

Thefirsttermhastwopotentialzerosinthedenominator:onefromtheincidentphotonwhenEL=EXandonefromthescatteredphoton,EL=EX+Q(forAStRStheseareatEL=EX-QandEL=EX).IntheCterm,noneoftheadditivetermshasthetwinresonance,howevertherearetermswhicharesinglyresonantatEL-EXandalsoEL-Q-EX(EL+Q-EXandEL-EXforASt).Ifthephononislowenoughinenergy,thesetwoenergiesofvanishingdifferenceoverlap,andthetwinresonancemakesthefirsttermdominateoverC.Thisisagoodapproximationwhenthephononenergiesaresmall,e.g.,likelyfortheRBmode(≈20meV).Thisterm’sdominancebecomesweakerasQincreasesinenergy,becauseitisnotpossibleforbothincidentandou