ELECTRONICSTRUCTUREANDDYNAMICSINCOLLOIDALGRAPHENE QUANTUMDOTS By ChengSun ADISSERTATION Submittedto MichiganStateUniversity inpartialtoftherequirements forthedegreeof Physics-DoctorofPhilosophy 2016 ABSTRACT ELECTRONICSTRUCTUREANDDYNAMICSINCOLLOIDAL GRAPHENEQUANTUMDOTS By ChengSun Ipresentstudiesofexcitonsingraphenequantumdots(GQDs),aclassofelectronical- lyquasi-zero-dimensionalmaterialswithatwo-dimensional sp 2 -hybridizedcarbonlattice. Theweakscreeningassociatedwithsuchatwo-dimensionallatticeoflightatomsresultsin strongcarrierinteractions.Semiconductorquantumdotsandlow-dimensionalcarbonbased materialshavebeendevelopedandstudiedfordecadesandgraduallyappliedinareassuch asphotovoltaics.Oneoftheoriginalmotivationsforourcollaborators'synthesisofthese particularGQDsistheirpotentialassensitizersforsolarcells,andresearchontheelectronic structureandtheexcitonbehaviorofGQDswillhelptorevealthepotentialofthiscandidate material. ThisthesisdescribesexperimentalinvestigationsofbiexcitonsinGQDs.Weusetransient absorptionspectroscopytodeterminethebiexcitonbindingenergy.Weavalueof ˘ 140meVforacertaintypeofbiexciton,whichisinroughagreementwiththetheoretically calculatedvalue.Comparedwithsemiconductorquantumdots,GQDsdisplaystronger biexcitonbinding,whichhighlightstheimportanceofexcitonicinexplainingthe opticalandelectronicpropertiesofthesesystems.Whileweobserveclearsignaturesof biexcitons,thesestatesareshort-lived.WeobservebiexcitonAugerrecombinationtimes of ˘ 0.3ps,whichiscomparabletothetimescaleofbiexcitonAugerrecombinationin single-wallcarbonnanotubeswithcircumferencecomparabletothelongestedgelengthof theGQDsstudiedhere.Slowerrelaxation(afewpsandtensofps)ofexcitonsisbelieved toberelatedtocoolingofthelattice.Thestronginteractionbetweencarriersandrapid biexcitonAugerrecombinationsuggestthatGQDscouldbeusedforcarriermultiplication andthusincreasetheofGQD-basedsolarcells. TABLEOFCONTENTS LISTOFTABLES .................................... vi LISTOFFIGURES ................................... vii Chapter1Introductionandbackground ..................... 1 1.1Dissertationoutline................................2 1.2Carbon,graphiteandgraphene.........................2 1.3Electronicstructureofthegraphenelattice...................6 1.4Graphenequantumdots(GQD).........................9 1.5C132andC168GQDs..............................14 1.6Excitont...............................23 1.7BiexcitonbindingandAugerrecombination..................25 1.7.1Introductiontobiexcitonbinding....................25 1.7.2IntroductiontoAugerrecombination..................26 Chapter2EquipmentandmethodsforthestudyofGQDs ......... 28 2.1Comparisonofmeasurementtechniques.....................28 2.2Transientabsorptionmeasurement.......................29 2.2.1OPA-derivedpump............................30 2.2.1.1OptimizationoftheOPA...................31 2.2.2DatacollectionandLabviewVIs....................34 2.2.2.1BroadbandmeasurementbyCCD...............34 2.2.2.2Single-wavelengthmeasurementbyphotodiodes.......40 2.2.3TAsetupoptimization..........................41 2.2.3.1TAsetup............................41 2.2.3.2AlignmentprocedureofourTAsetup............44 2.3Upconversionofphotoluminescence.......................47 2.4Samplesstudiedinthisthesis..........................50 Chapter3BiexcitonBinding ............................ 51 3.1Introduction....................................51 3.1.1Biexcitonbindinginquantumsystems............51 3.1.2Experimentaltechniquesformeasuringbiexcitonbinding.......55 3.2Experimentandresults..............................59 3.2.1Experimentalsetup............................59 3.2.2Experimentalresultsanddiscussion...................59 3.3Conclusion.....................................70 Chapter4BiexcitonAugerrecombination ................... 71 4.1Carrierrelaxationinquantumsystem.................71 iv 4.2Experimentanddataanalysis..........................73 4.2.1Experimentalsetup............................73 4.2.2Dataanalysis...............................74 4.2.3DiscussiononAugerrecombinationofGQDs..............85 4.3Conclusion.....................................88 Chapter5Summary ................................. 89 5.1Generalsummary.................................89 5.2Futurework....................................90 BIBLIOGRAPHY .................................... 91 v LISTOFTABLES Table1.1:TheoreticalelectronandholestatesinC168 ac ............21 Table2.1:22-pinaccessoryconnectorpinoutdiagramwhenfacingthe22-pin AccessoryConnectoronthefrontoftheverticalwalloftheUSB2000+37 Table2.2:anddescriptionsofpin6and7,whichareusedtoconnect withtheUSB2000+triggersource[1].................38 vi LISTOFFIGURES Figure1.1:(a)Diamondlattice.(b)Graphitelattice.(c)Graphenelattice.(d) Singlewallcarbonnanotubes......................3 Figure1.2:(Left)Real-spacestructureofgraphene.Theyellowandbluedots representthetwosublatticesofgraphene. 1 ; 2 and 3 arethe nearest-neighborvectors.(Right)Brillouinzoneofthegraphenelat- tice(reproducedfromRef.[2])......................6 Figure1.3:Calculatedbandstructureingraphenebasedonthetight-binding approximation...............................8 Figure1.4:(Left)Bandgapof ˇ - ˇ transitionsonchemicallyderivedGOcalcu- latedbyDFTasafunctionofthenumberofaromaticrings.Re- producedfromRef.[3].(Right)Size-dependenceofthebandgapof GQDswithzigzagandarmchairedges( N isthenumberofhexagonal unitsalonganedge).ReproducedfromRef.[4]............11 Figure1.5:Illustrationsofthesublatticesymmetryofarmchair(left)andzigzag edges(right).Theup-anddown-arrowsrepresentspins.Reproduced fromRef.[5]................................12 Figure1.6:Tight-bindingspectraof(a)armchairhexagonal,(b)zigzaghexago- nal,and(c)zigzagtriangularGQDs.ReproducedfromRef.[6]...13 Figure1.7:Improvedpowerconversion(left)andexternalquantumef- (right)ofsolarcellwith/withoutGQDlayers.Reproduced fromRef.[7]................................15 Figure1.8:(Left)StructureofC132andC168GQDs[8].Theblueareaisthe two-dimensionalgraphenesheet,andtheblackstructuresareligands topromotesolubilityandpreventaggregation.(Right)Solidlines: Ground-stateabsorptionspectraofC132andC168GQDs.Dashed line:photoluminescencespectrumofC132excitedat3.1eV.....16 Figure1.9:SynthesisofC168GQD.ReproducedfromRef.[8]..........17 Figure1.10:Detailed-balancelimitoftheofasinglep-njunctionsolar cellcalculatedasafunctionofbandgapbyWilliamShockleyand HansQueisser[9].Theupperhorizontalaxisshowsthebandgapin eV.....................................18 vii Figure1.11:(Left)C168GQDsynthesizedbyYanandLi[8].(Right)C168 ac : GQDwitharmchairedges[10]......................19 Figure1.12:Theallowedvaluesof ~ k n;m ofatriangulararmchairGQDof60C atoms.Theallowed ~ k n;m occupyonesixthofthegrapheneBrillouin zone.Everycirclerepresentsoneorbitalstate(i.e,twostates includingspin),whileeachopencirclerepresentsahalfstate[10]..20 Figure1.13:Fluorescencespectraforvariousexcitationwavelengths ex ofC132 intoluene.(reproducedfromRef.[11])................22 Figure1.14:Summaryofthedensityofstatesforabulkcrystal(3D),aquantum well(2D),aquantumwire(1D),andaquantumdot(0D)(reproduced fromRef.[12])..............................24 Figure1.15:Illustrationofexcitonandbiexcitiondispersionforpositivebiexciton bindingenergy..............................26 Figure1.16:Energy-leveldiagramillustratingcarriermultiplicationandAugerre- combination(reproducedfromRef.[13])................27 Figure2.1:SetupoftheopticalparametricThepoweroftheincoming 800nmbeambeforetheOPAentranceis ˘ 0.5to1W. 2represents thehalfwaveplate.BSrepresentsa20/80beamsplitterwith80% ofthepowersenttothesecondpass.TFPrepresentsathin polarizer.DMrepresentsdichroicmirrors.Theboxeswithdashed linesrepresenttranslationstagesintheOPAsetup.Allthebeams thatdonotpasstheTFPhavelinearpolarizationparalleltothetable surfaceandthebeamaftertheTFPhaslinearpolarizationvertical tothetablesurface.TypicallywecangetafewhundredmWoutof theOPAwiththepowerdependingonoutputwavelength......32 Figure2.2:USB2000+hardwareedgetriggermodetimetable..........35 Figure2.3:Timingofspectrometertrigger.....................36 Figure2.4:Layoutofopticalpathsandelectricalconnections..........37 Figure2.5:LocationofUSB2000+Accessoryconnector..............38 Figure2.6:Timingdiagramforchopperandspectrometertrigger........38 Figure2.7:FlowchartofLabviewVI(dynamicmeasurement)..........42 viii Figure2.8:ofTAmeasurement...................43 Figure2.9:SetupofuPLmeasurement.......................48 Figure3.1:(A)Experimental(blue)andtheoretical(black)groundstateabsorp- tionofC168.(B)Calculatedband-edgesingletexciton(X)andbiex- citon(XX)states(blacklines)derivedfromthedegenerateHOMO andLUMOstates.Greylinesshowexcitedexcitonstatesaccessible fromX 1 ; 2 .Dipole-allowedelectronictransitions,whichcorrespondto achangein m of 1,fromthegroundstateandfromthelowest singletexcitonstatesareshownrespectivelybysolidredandblue arrows.Dashedarrowsindicateopticallydarkelectronictransitions. [14]....................................52 Figure3.2:Single-pairexcitationwithtotalangularmomentum 1(opti- callybrightexciton)and(opticallydarkexciton)(reproduced fromRef.[15])..............................53 Figure3.3:Predictedexciton(X)andbiexcitonstates(XX)inGQDs.Dipole- allowedelectronictransitions,correspondingto= 1,arelabeled witharrows.Red(blue)arrowrepresents=+1(-1)corresponding to ˙ +( ˙ )photonpolarization[14]...................56 Figure3.4:Excitation,cooling,andopticaltransitionsinvolvedinphotolumines- cenceinGQDs..............................57 Figure3.5:NormalizeddynamicdataofC132pumpedat3.1eVandprobedat 1.68eVand1.84eV...........................58 Figure3.6: L asafunctionofwavelengthanddelayforC168excitedat 3.1eVatanintensitycorrespondingto ˘ 1.2.Thescale correspondstothedataatdelayst 5.0ps.Thedataintheright panel(t > 8ps)aremultipliedby3.Theblackcurvesindicatethe L =0contours[14]..........................60 Figure3.7: L atdelayt=100psforC168at =1.2and ~ ! pump =3.1eVwithTAspectrumofintrabandtransitions..........62 Figure3.8:PossibletransitionsofgivingrisetoX 3 andX 4 (left)andX 1 ; 2 ! X 1 ; 2 +X 3 ; 4 i.e.,X 1 ; 2 ! XX 4 7 (right)showingtheofstate onthetransitionstothelowestbrightsingletexcitons....63 Figure3.9:Stepbystepanalysisoftcontributionson L inTAmea- surement,rightsideofeachspectrumillustratethetcontri- butiontotheabsorptionspectrum...................64 ix Figure3.10:Experimentalandtheoreticalabsorptionspectrafromthelowests- ingletexcitonstate.Bluecirclesindicatemeasured L (t=100ps) ofC168.Theredbarsindicatecalculatedtransitionsfromthestates X 1 ; 2 accountingforintra+interbandtransitions.Theblacklineisthe theoreticallycalculated,Gaussianbroadened = ( 1 ; 2 0 ) assumingequallypopulatedX 1 ; 2 states.Toppanelshowssingletex- citons(lightgrey),band-edgeexcitons(colorcorrespondingtoFigure 3.3)andhigherXXs(darkgrey)accessiblefromX 1 ; 2 [14].......65 Figure3.11:Groundstateabsorptionisplottedinblackdotsandthemodel ispresentedinredline.Thepeaksof [( d 2 0 ) = ( d ( ) 2 )]indicates availabletransitions,whichareshownbydashedlineinthe.66 Figure3.12: L atdelayt=100psforC132at =1.2and ~ ! pump =3.1eVwithTAspectrumofintrabandtransitions.........69 Figure4.1:Aschematicoftheprocessesbywhichopticallyexcitedelectronand holedistributionsapproachequilibriuminepitaxialgraphene.Dis- tributionatthetimeofexcitationneartheDiracpointshowsanin- trinsicthermalpopulationofelectronsandholes.(Reproducedfrom Ref.[16]).................................72 Figure4.2: L asafunctionofprobeenergyanddelayforC132pumpedat 1.94eVandof(a)1 : 3 10 16 and(b)1 : 4 10 14 photons cm 2 perpulse.Thecolorscalecorrespondstothedatafort 5.0 psin(a).Thedataintheotherthreequadrantsaremultipliedby thefactorsshowntomatchthescales..................75 Figure4.3: ;t ) L versusprobedelayforC132pumpedat1.94eVand probedat2.34eV.Solidcurvesarethedescribedinthetext. Thedashedcurverepresentstheinstrumentresponsefunction....76 Figure4.4: ( t ) = ( t long )versusprobedelayforC132intolueneexcitedat ~ ! pump =1 : 94eVandprobedat ~ ! probe =2 : 34eV : .........77 Figure4.5:Amplitudes A i fromEq.4.1describingtheof ;t ) L of C132inFigure4.3.Theinsetshowsthequantity A f 0 inthe textandassociatedwithmultiexcitonsandusesthehorizontalscale ofthemainCurvesshowthedescribedinthetext.....79 x Figure4.6: L asafunctionofprobeenergyanddelayforC168excitedat 1.70eVatof2 : 0 10 15 photonscm 2 perpulse(a)and 2 : 4 10 13 photonscm 2 perpulse(b).Thecolorscalecorresponds tothedatafort 5.0psinpanel(a).Thedataintheotherthree quadrantsaremultipliedbythefactorsshowntomatchthescales..81 Figure4.7:(a) L forC168intolueneexcitedat ~ ! pump =1.70eVand probedat ~ ! probe =2.21eVataseriesofexcitationfrom 2 : 4 10 13 to2 : 0 10 15 photonscm 2 perpulse.Solidcurvesare triexponentialdescribedbyEq.4.1,andthedashedcurveisthe instrumentresponsefunction.......................82 Figure4.8: ( t ) = ( t long )versusprobedelayforC168intolueneexcitedat ~ ! pump =1 : 70eVandprobedat ~ ! probe =2 : 21eV : .........83 Figure4.9:Amplitudes A i fromEq.4.1describingtheof ;t ) L of C168inFigure4.7.Theinsetshowsthequantity A f 0 inthe textandassociatedwithmultiexcitonsandusesthehorizontalscale ofthemainCurvesshowthedescribedinthetext.....84 Figure4.10: ( t ) = ( t long )versusprobedelayforC168intolueneexcitedat ~ ! pump =3 : 10eVandprobedat ~ ! probe =0 : 76eV : .........86 xi Chapter1 Introductionandbackground Carbonisoneofthekeyelementsintheuniverse.Becauseoftheyofitsbonding, carbon-basedsystemspresentavarietyofchemicalstructureswithtphysicalproper- ties.Amongtherichestcarbon-basedsystems,grapheneconsistsofasingleplanarsheetof sp 2 -bondedcarbon.Graphenehasplayedanimportantroleinhelpingusbetterunderstand theelectronicpropertiesofothercarbon-basedmaterials.Thesematerialsincludegraphite (athreedimensionalallotropeofcarbonconsistingofgraphenelayers),carbonnanotubes (structuresformedbyrollinggrapheneintoacylinder),sphericalfullerenes(introductionof pentagonsintothegraphenelattice),andgraphenequantumdots(single-atom-thicksheets ofgraphenewithalldimensionstypicallylessthan100nm,whereelectronictransportis inallthreespatialdimensions[17]).P.R.Wallacetheoreticallyanalyzedgraphene asearlyas1947[18].Overthelast50years,therehavebeenmanystudiesrelatedto single-layergraphite[19,20,21],butnoonecouldaneasyandtwaytoproduce grapheneuntilAndreGeimandKostyaNovoselovdiscoveredin2004thattheycouldexfoli- ateitfrombulkgraphene[22,23].Thisenabledthepioneeringexperimentsforwhichthey wereawardedtheNobelPrizeinPhysicsin2010. Graphenehasattractedvastresearchinterestduetoitsremarkablepropertiessuchas largespsurfacearea,highelectronmobility,goodmechanicalstrength,andhighthermal conductivity.Thelineardispersionofitsconductionandvalencebandsatlow-energyhave alsodrawnmuchattention.Moreover,carbonisabundantandnontoxiccomparedtosome 1 othersolar-cellmaterialssuchasheavy-metalbasednanocrystals.Thismakesgraphene materialsattractivecandidatesforapplicationsinphotovoltaicas,forexample,transparent electrodes[24,25,26]. 1.1Dissertationoutline Thisthesispresentsexperimentsperformedonopticallyexcitedmultiple-electron-hole- pairstatesingraphenequantumdots(GQDs).GQDsaresynthesizedbyourcollaborators Liang-shiLi'sgroupatIndianaUniversity[8].Allexperimentalresultpresentedinthisthesis isperformedinourlab. Theoutlineisasfollows: Chapter1:Anoverviewofthebasicconceptsandbackgroundforunderstandingthe experimentalresultspresentedinChapters3-5. Chapter2:Equipmentandmethodsemployedintheexperimentsanddataanalysis. Chapter3:AnexperimentalstudyofbiexcitonbindinginGQDs. Chapter4:Transientabsorption(TA)measurementsoftherelaxationofbiexcitonsin GQDs. Chapter5:Conclusionsanddiscussionofopenproblems. 1.2Carbon,graphiteandgraphene Asalreadydescribed,carbonexistsintforms.Therearenumerousallotropes ofcarbonincludingdiamond,graphite,carbonnanotubesandgraphene.Thephysicaland electronicpropertiesvarygreatlyamongtformsofcarbonbecauseofthet waysinwhichcarbonatomsbondwitheachother.Theorbitaloccupationofthecarbon 2 atomis1 s 2 2 s 2 2 p 2 .The2 s and2 p orbitalscaneasilyhybridizewitheachotherint waysbecauseoftheirsimilarenergies.ThisyieldsthetstructuresshowninFigure 1.1.Amongitsmanyinterestingproperties,thehardnessofdiamondiswidelyknown.The electronsofasinglecarbonatomarearrangedinorbitalsformingthefourcornersofa tetrahedron(dueto sp 3 hybridization).Andallthebondshavethesamelengthwiththe samebondangle.Thesecovalentbondsgivediamonditsstrength. Figure1.1:(a)Diamondlattice.(b)Graphitelattice.(c)Graphenelattice.(d)Singlewall carbonnanotubes. Unlikediamond,graphiteis sp 2 -hybridizedandhasquitetphysicalandelectronic properties.GraphitewasnamedbyAbrahamGottlobWernerin1789foritsuseinpencils. 3 Graphiteisstructuredbystackingmultiplesingle-atom-thickplanarcarbonsheets,which areheldtogetherbyvanderWaalsforce.Itisanelectricalconductorandthemoststable formofcarbonunderstandardconditions. GrapheneisanothercarbonallotropediscussedtheoreticallybyP.R.Wallaceasearly as1947[18],butduetotheyofisolatingasinglelayerofgraphenefromgraphite,no transportmeasurementswereconductedongrapheneuntil2005byAndreGeimandKostya Novoselov[23].Theyusedadhesivetapetopeelthemonolayersaway,andthisprocessmade theproductionofgrapheneforlaboratorystudiesmucheasier.Graphenecanbeproduced throughmanymethodsincludingmechanicalexfoliationofgraphite[22],chemicalreduction fromgrapheneoxide[27],andchemicalvapordepositiononmetallicthin[28,29].The valenceandconductionbandsofgraphenemeetattheDiracpoints,leadingtozerobandgap. Moreover,thedispersionofgrapheneislinearsothatlow-energycarrierscanbedescribedin termsofmasslessDiracelectronsandholes,whichtravelwithaFermivelocityof10 6 m/s. Thisleadstoanextremelyhighchargecarriermobilityof15000cm 2 V 1 s 1 [30]. Carbonnanotubes(CNT)andgraphenenanoribbons(GNRs)alsoattractintentionas graphene-relatedmaterials.Acarbonnanotubeisahollowcylinderconsistingof sp 2 - hybridizedcarbon.ThediscoveryofCNTscanbedatedto1991byIijima[31].CNTs canbepreparedbyttechniquessuchasarcdischarge[32],laserablation[33]and chemicalvapordeposition[34].AlthoughcurrentmethodsofpreparationofCNTsalways produceimpurities,chemicalvapordepositionhasbecomeastandardmethodfortheCNT productionbecauseofitsbettercontroloverthenanotubelength,diameter,orientationand density.Afterthat,single-wallcarbonnanotubes(SWCNT)wereobservedin1993[32,35]. ASWCNTcanbevisualizedasagraphenesheetrolledintoaseamlesscylindricaltube.A seamlesscylindercanonlybeachievedbyrollingincertaindirections.Thevectorconnecting 4 thetwocarbonatomsthatoverlapwitheachotherafterrollingiscalledthechiralvector. ThechiralvectordeterminesthefundamentalopticalandelectricalpropertiesofSWCNTs, forexample,whethertheindividualnanotubeshellisametalorsemiconductor. AGNRisastripofgraphenewithwidth.ItwasintroducedbyMitsutaka Fujita'sgroupasatheoreticalmodelfornanoscalesizeandedge-shapedependence ofGNRs[36].GNRskeeptheoutstandingtransportpropertiesofgraphene[37]buthave width-dependentbandgaps[38].Dai'sgroupproducedGNRswithwidthbelow10nmby sonicatingasolutionofexfoliatedgraphite[39].Johnson'sgroupproducedGNRsbyetching few-layergraphenewiththermallyactivatedmetallicnanoparticles[40].GNRswithwidth below10nmwerealsoetchedbySTMlithographyratherthanelectronbeamlithography [41].However,itistoobtainGNRswithsmoothedgesandcontrollablewithhigh yieldsbythesesonochemical,chemicalorlithographicmethods.Dai'sgrouphasintroduceda chemicalmethodby\unzipping"CNTswithnarrowwidthdistribution(10-20nm)andhigh yield[42].Morerecently,MullenandcollaboratorsrealizedGNRswithprecisewidthand edgestructuresbysurface-assistedorganicsynthesis[43].Armchairgraphenenanoribbons (AGNRs)havebeensynthesizedwithtwidthssuchas5-AGNRs(AGNRshave5 atomsacrosstheirwidth)[44],7-AGNRs[43]andeven13-AGNRs[45]bythisbottom- upfabrication.Besidessurface-assistedorganicsynthesis,Mullenandcollaboratorshave demonstratedsolution-phasesynthesisrecentlyofaclassofGNRswithlongsize( > 100nm), narrowwidthdispersions,wedges,andlowopticalbandgaps(1.2eV)[46,47]. 5 1.3Electronicstructureofthegraphenelattice Grapheneisasinglelayerofjoinedhexagonalringsofcarbonatoms.Figure1.2demon- stratesthatcarbonatomsAandBarenotequivalent.Therefore,grapheneistreatedasa triangularlatticewithabasisoftwoatoms. Figure1.2:(Left)Real-spacestructureofgraphene.Theyellowandbluedotsrepresentthe twosublatticesofgraphene. 1 ; 2 and 3 arethenearest-neighborvectors.(Right)Brillouin zoneofthegraphenelattice(reproducedfromRef.[2]). Thethreenearest-neighborvectorscanberepresentedby 1 = a 2 (1 ; p 3) ; 2 = a 2 (1 ; p 3) ; 3 = a 2 ( 1 ; 0) (1.1) where a ( ˇ 0.142nm)isthedistancebetweenthenearestneighbors.Thereciprocal-lattice vectorsaregivenbythevectors b 1 and b 2 inFigure1.2 b 1 = 2 ˇ 3 a (1 ; p 3) ;b 2 = 2 ˇ 3 a (1 ; p 3) (1.2) 6 TheDiracpointsareatthe K and K 0 pointsoftheBrillouinzone: K = 2 ˇ 3 a ( p 3 ; 1) ;K 0 = 2 ˇ 3 a ( p 3 ; 1) (1.3) FollowingP.R.Wallace[18]andusingatight-bindingapproach,inwhichweonlyconsid- erthenearest-neighboroverlap,atight-bindingHamiltonianwithonlynearest-neighbour interactionscanbewrittenas H = t X i;˙ ( a + ˙;i b ˙;i + a ˙;i b + ˙;i ) ; (1.4) where a + ˙;i and b + ˙;i arethecreationoperatorsand a ˙;i and b ˙;i aretheannihilationoperators onsublatticeAandB,respectively,withspin ˙ . t ( ˇ 2.8eV)isthehoppingenergybetween thenearestneighbors.TheenergybandsderivedfromthisHamiltonianare E ( k )= t s 1+4cos 2 ( p 3 2 k y a )+4cos( p 3 2 k y a )cos( 3 k x a 2 ) (1.5) wheretheplusandminussignreferrespectivelytotheelectronandholebranches. Figure1.3showsthebandstructureofgraphenebasedonthetight-bindingapproach. Undopedgrapheneissemimetallic,sincetheconductionandvalencebandstouchatthe Diracpoints.Thelineardispersionofgrapheneatlowenergiesleadstoadescriptionin termsofDiracfermion.Thisisindramaticcontrasttomostcrystallinematerials,which haveparabolicdispersionnearthebandextrema. 7 Figure1.3:Calculatedbandstructureingraphenebasedonthetight-bindingapproximation. 8 1.4Graphenequantumdots(GQD) Thetermquantumdotreferstothetofelectronsinallthreespatialdimen- sions.Thesizeofaquantumdotcanbeassmallas10to50atoms,sothattheydisplay propertiesofbothatoms/moleculesandbulkmaterials.Usually,graphenequantumdot referstoasingle-atom-thicksheetofgraphenewithdimensionstypicallylessthan100nm [17].Quantumtcanopenasize-dependentgapingraphene[8,10,17]. Todate,manyhavebeenmadetosynthesizeGQDs.Thesecanbeclas- intotwogroups:top-downandbottom-upmethods.Intop-down,GQDsarederived fromlargercarbonmaterialsandinclude,butarenotlimitedto,electronbeamlithography [17],acidicexfoliation[48,49],andelectrochemicaloxidation[50].Incontrast,bottom-up methodsstartfromsmallerorganicprecursors[51]. TheTop-downmethodsforthepreparationofGQDstakeadvantageofabundantraw materials,large-scaleproduction,andsimplicity.However,thismethoddoeshavesome disadvantages,suchaslowyield,uncontrollableedgetype,andbroaddistributionofGQD sizes[52,53].Incontrast,thebottom-upmethodsgreateropportunitiestocontrolthe GQDswithwmolecularsize,shape,andedges[51]. TherearetwaystocharacterizeGQDs.OneofthemisNMRspectroscopy, whichisoftenusedincharacterizingmolecules.Thesizeofthegraphenecoreanddynamic aggregationinsolutionontheNMRtimescalemakesittocharacterizeGQDsusing conventionalliquid-phaseNMRspectroscopy.Ourcollaboratorswerenotabletodetect anyaromaticprotonresonancepeakswiththismethod[51].Precisecharacterizationof GQDsremainsachallenge.Anothertechniquethathasbeenusedtodeterminethemass ofproteinsandpolymersisMALDI-TOF(matrix-assistedlaserdesorptionionizationtime- 9 t)massspectrometry[54].PeopleuseMALDI-TOFtocharacterizethesynthesized GQDs[51].However,MALDI-TOFcannotprovideinformationonthefractionoft speciesproducedbythesynthesis.PrecisecharacterizationofGQDsremainsachallenge. ThefactthatGQDshasbeensynthesizedwellrecentlymakesitpossibletostudyGQDs experimentally.Theresultofquantumtisasize-dependentbandgap,which resultsinsize-dependentopticalandspectroscopicproperties[55].Therehasbeenalotof workdoneonstructuressimilartoGQDs,e.g.,grapheneoxide(GO).Theenergygapbetween thehighestoccupiedmolecularorbital(HOMO)andthelowestunoccupiedmolecularorbital (LUMO)ofGOhasbeencalculatedbyEdaetal.byDFTcalculationasillustratedinthe leftpanelofFigure1.4[3].GOhasamixtureof sp 2 -and sp 3 -hybridizedcarbonwhileGQD consistsonlyof sp 2 -hybridizedcarbon.AccordingtoRobertsonetal.'swork,theoptical andelectronicpropertiesofmaterialswithsuchstructuresaredeterminedbythe ˇ states of sp 2 -hybridizedcarbon[56].TheDFTcalculationshownintheleftpanelofFigure1.4 illustratethatasthenumberoffusedaromaticringsgetssmaller,theenergygapofGOs risesfaster. Thebandgapofgraphenequantumdotswithtsizeshasalsobeenstudiedby tight-bindingcalculations[4].AsillustratedintherightpanelofFigure1.4,thecalculated energygapforahexagonalGQDwitharmchairedgesisproportionalto1 = p N or1/ L from hundredtomillionatomnanostructures,where N isthenumberofhexagonalunitsalong anedgeand L representsthesizeofthequantumdot. Besidessize,theedgestructurealsoplaysanimportantroleinthepropertiesofquantum dots,whichcanalsobeseeninfromFigure1.4:thesize-dependenceofthebandgapofGQDs withzigzagedgesandarmchairedgesarequitet.ThebandgapofthezigzagGQDs decaystozeromuchfasterthanthatofarmchairGQDsasthesizeoftheGQDincreases. 10 Figure1.4:(Left)Bandgapof ˇ - ˇ transitionsonchemicallyderivedGOcalculatedbyDFT asafunctionofthenumberofaromaticrings.ReproducedfromRef.[3].(Right)Size- dependenceofthebandgapofGQDswithzigzagandarmchairedges( N isthenumberof hexagonalunitsalonganedge).ReproducedfromRef.[4]. AsillustratedinFigure1.2,thegraphenelatticehastwosublatticesdenotedasAandB. Inthemiddleofagraphenesheet,3carbonatomsfromsublatticeAand3carbonatoms fromsublatticeBformafullybenzenoidring.Becauseofthesymmetricstructure,there arenolocalizeddoublebondsinthering.Ontheedgesofthegraphenesegment,though, theremightbenonbondingstatesdependingontheedgetype.Twolimitingmotifsforthe edgestructureofgraphenearezigzagandarmchair,asillustratedinFigure1.5.Armchair edgesconsistofpairsofAandBatoms,whilezigzagedgesareterminatedbyeitherAorB sublattices.Witharmchairedges,allthe ˇ -bondscanbesimultaneouslywithedge statesconsistingofstandingwaves[5].Withzigzagedges,thesymmetryofthepseudospin isbrokenatthezigzagedge.Thus,armchairedgesmaintainpseudospinsymmetryand arenonmagnetic,whilezigzagedgessupportlocalized,spin-polarizededgestates.Theedge- statespinplaysatpartinthemagnetismofagraphenesegment[57,58].Asshown 11 inFigure1.6,hexagonalGQDswithzigzagedgesandarmchairedgeshavetenergy statesandbandstructures.Moreover,atriangularGQDwithzigzagedgestructureshowsa shellofdegeneratelevelsattheFermilevelresultinginzero-energystates[59].Carrierscan beopticallyexcitedfromthevalencebandtothezero-energystatesorfromthezero-energy statestotheconductionband.Withthedevelopmentofbottom-upsynthesis[43],people cannowpreciselycontrolgraphenenanoribbonsandgraphenequantumdots,whichalso proceedtheexperimentalstudyofthesematerial. Figure1.5:Illustrationsofthesublatticesymmetryofarmchair(left)andzigzagedges (right).Theup-anddown-arrowsrepresentspins.ReproducedfromRef.[5]. Poorsolubilityandone-atomthicknesscanresultinaggregationofGQDs.Itcanhappen assphead-to-tailmoleculararrangements(J-typeaggregation[60])forzigzagedge GQDsorasapilingupasfordiscs.Aggregationcanlimitthepotentialapplicationsof GQDs.Chenetal.observedJ-typeaggregationbyTEM[60],andthehighconcentration 12 Figure1.6:Tight-bindingspectraof(a)armchairhexagonal,(b)zigzaghexagonal,and(c) zigzagtriangularGQDs.ReproducedfromRef.[6]. ofGQDsresultedinashiftoftheemissionpeaktolowerenergy.Topreventaggregation, peoplehaveattachedligandstotheedgesofGQDs.Kastleretal.performedadetailed experimentalcomparisonoftheoftattachedligandsandtsolvents ontheelectronicandopticalpropertiesofthreehexa-peri-hexabenzocoronenes(HBC)[61]. Yanetal.attachedmultiple1,3,5-trialkyl-substitutedphenylmoietiestotheedgeofGQDs. TheseligandsactedlikeacagesothataggregationamongGQDswassuppressed.This structurealsoincreasedGQDsolubilityinorganicsolventsliketoluene[51]. GQDsalsoshowgreatpromiseinsolarcellandphotovoltaicapplicationsbecauseof 13 theirsize-tunableopticalresponse.Theinjectiontimeofelectronsandholesinperovskite solarcellsbasedonCH 3 NH 3 PbI 3 wasmeasuredtobe0.4to0.6ns,whilethehotcarrier coolingtimewasonlyabout0.4ps[62].Thisindicatesthatalargeamountoftransferred energyiswastedbecauseofthefastcoolingprocess.Toincreasethepowerconversion ,Zhuetal.insertedanultrathinGQDlayerinaperovskitesolarcell[63];faster electron/holeextraction(90ps)wasobservedinTAmeasurementsthanthe280psextraction timeobservedinsolarcellswithouttheGQDlayer.TheGQDlayerincreasedthepower conversionfrom8.81%toover10%.Inanotherwork,Gaoetal.provedthatGQDs canformaheterojunctionwithcrystallinesiliconforhighlytsolarcellapplications[7]. Therewasalargejunctiongapbetweenn-typeSiandGQDsallowedcarrierstobetly separated.Asaresult,thecrystallinesiliconsolarcellwithGQDsshowedabetterexternal quantumandpowerconversionthandidasolarcellwithoutGQDlayers asillustratedinFigure1.7.Moreover,suchcrystallinesilicon/GQDsolarcellsshowedgood stabilityastheymaintainedhighafterhalfayearofstorage. 1.5C132andC168GQDs TheGQDsusedinourexperimentsconsistof132or168 sp 2 -hybridizedcarbonatoms inthecoreandarerespectivelylabelledasC132andC168.AsshowninFigure1.8,C168 hasatriangularformandC132hasatrapezoidalformmissingacornercomparedwith C168.TheseGQDsaresynthesizedbyasolution-chemistryapproachdevelopedbyourcol- laboratorsintheDepartmentofChemistryatIndianaandbasedonoxidativecondensation reactionsdevelopedbyKlausMullen'sgroup[64].Thesolution-phasesynthesisofC168is illustratedinFigure1.9,where\R"represents2,4,6-triakylphenylgroups,whichcovalently 14 Figure1.7:Improvedpowerconversion(left)andexternalquantum (right)ofsolarcellwith/withoutGQDlayers.ReproducedfromRef.[7]. attachtotheedgesoftheGQD.Thesephenylgroupstwisttoanout-of-plane tionduetostericconstraintstherebycagingtheGQDsinalkylgroupsandreducingthe aggregationbetweenGQDs,aswellasincreasingtheirsolubility. ThelongestedgeoftheGQDswehavestudiedisabout2.4nm.Thelineardispersionof graphenecanbedescribedby E = ~ v f q k 2 x + k 2 y ,where v f = c= 300and c isthespeedoflight invacuum.RozkhovandNorihavededucedanalyticsolutionsoftheSchrodingerequation foranelectroninatriangulararmchairgraphenedot[10].Theirsolutionyieldsat energyinC168of1.6eV.TheUV/vis(VarianCary50BioUV-VisibleSpectrophotometer) ground-stateabsorptionspectrumoftheGQDsdissolvedintolueneisillustratedinFigure 1.8.Fromthemeasurement,thelowest-energyfeatureintheabsorptionspectrumofC168 isatabout1.7eV. C132andC168weresynthesizedbyYanetal.withthegoalofdevelopingasensitizer 15 Figure1.8:(Left)StructureofC132andC168GQDs[8].Theblueareaisthetwo- dimensionalgraphenesheet,andtheblackstructuresareligandstopromotesolubilityand preventaggregation.(Right)Solidlines:Ground-stateabsorptionspectraofC132andC168 GQDs.Dashedline:photoluminescencespectrumofC132excitedat3.1eV inorganicphotovoltaics[51].WilliamShockleyandHansQueisser[9]calculatedthemaxi- mumtheoreticalofasolarcellusingasinglep-njunction.Theirresultisshown inFigure1.10.SincethebandgapofC168is1.7eV,suchGQDsmighthavepotentialas sensitizersforsolarenergyconversion. OneoftheareasofinterestintheseGQDsiscarrierinteractions,e.g,excitonandbiex- citonbehaviors.Biexcitonsareboundstatesoftwoexcitons.Whenabiexcitondecays radiatively,ittypicallyproducesafreeexcitonandaphoton.Intransientabsorptionmea- surements,anexcitoncanbeformedwiththepumpphoton.Afterthis,aprobephoton isabsorbedbythesysteminasingle-to-biexcitontransition.Therefore,understandingthe 16 Figure1.9:SynthesisofC168GQD.ReproducedfromRef.[8]. formationanddecayofbiexcitonswillhelpusinterpretthedynamicsofsingleexcitonsand spectroscopicmeasurements.BiexcitonbindingwillbediscussedinChapter3,andbiexci- tonAugerrecombination(non-radiativerecombinationofanelectron-holepairinwhichthe energyandmomentumistransferredtoanothercarrier)willbediscussedinChapter4. AfewtheoreticalandexperimentalreportsontheseparticularGQDs(C132andC168) precedeourwork.StefanSchumacheruseddensityfunctionalandteraction- basedelectronicstructuremethodstocalculatethelow-energyelectronicstructureofC168 [65].HisresultsindicatethatthelowestelectronicexcitationsinthesingletmanifoldofC168 areopticallydark,whichhelpsusbetterunderstandthetransitionsfromourexperimental absorptionspectrumwhenweinterpretourdata.PawelHawrylak'sgroupcalculatedthe 17 Figure1.10:Detailed-balancelimitoftheofasinglep-njunctionsolarcellcal- culatedasafunctionofbandgapbyWilliamShockleyandHansQueisser[9].Theupper horizontalaxisshowsthebandgapineV. electronicstructureofthesematerialsbasedonatight-bindingmodelcombinedwithHartree- Fockandteractionmethods[15].Theyfoundoutthelowestenergystateis darkandthebrightexcitonsoflowestenergyistwo-folddegenerate,whichismuchhelpful inourdatainterpretation.ThegroupofLiang-shiLiperformedinitialstudiesofwhatwas understoodasalong-livedtripletstateinC168[66]andmeasurementsofslow(100-300 ps)dynamicsintransientabsorptionthattheyinterpretedintermsofaphononbottleneck oftheGQDs[67],whichalsomotivatefurtherexplorationoftherelaxationmechanismsin theseGQDs. 18 Figure1.11:(Left)C168GQDsynthesizedbyYanandLi[8].(Right)C168 ac :GQDwith armchairedges[10]. Tight-bindingcalculationsareagoodplacetostarttolearnabouttheelectronicstructure ofGQDs.RozhkovandNori[10]performedsuchcalculationsontriangularGQDswith armchairedges.AlthoughourGQDshavemixededgetypesofarmchairandzigzag,the symmetryofanarmchairGQDwith168carbonatoms(C168 ac )isexactlythesameas thatofC168asshowninFigure1.11.Therefore,weexpectthatthecalculatedelectronic structureofC168 ac shouldbeagoodestimateoftheelectronicstructureofC168. Wecanunderstandthedegeneracyofthesingle-particlestatesofC168byconsidering theallowedwavevectorsinC168 ac .Thelow-energyelectronicstructureinC168 ac canbe determinedbyquantizingtheallowedwavevectors,asshowninFigure1.12.Thesymmetry ofthetriangulargraphenelatticerevealsthat,allavailablestatesintheBrillouinzonecan beobtainedfromthestatesinonesixthoftheBrillouinzoneby 2 ˇ= 3rotationsaround theoriginoraboutthelines k y =0and k y = p 3 k x .Theallowedwavevectors consistoflinearsuperpositionsofthetwoeigenvectors ~ K 1 and ~ K 2 : 19 Figure1.12:Theallowedvaluesof ~ k n;m ofatriangulararmchairGQDof60Catoms.The allowed ~ k n;m occupyonesixthofthegrapheneBrillouinzone.Everycirclerepresents oneorbitalstate(i.e,twostatesincludingspin),whileeachopencirclerepresentsahalfstate [10]. 20 ~ k n;m = n ~ K 1 + m ~ K 2 : (1.6) Foratight-bindingcalculationinvolvingonlynearest-neighborhopping,thecorresponding eigenenergiesare " n;m = t ˆ 3+2cos 2 ˇn 3 N +2cos 2 ˇm 3 N +2cos 2 ˇ ( n + m ) 3 N ˙ 1 = 2 (1.7) where N =8forC168 ac and t isthenearest-neighbourhoppingenergy.Weuse t =2.7eVto estimatetheenergiesofavailablestatesinTable1.1,inwhichthecolumnslabelled\ n "and \ m "arethecotsofthetwoeigenvectorsandthecolumnlabelled\Symmetry"lists theMullikensymbolsindicatingthesymmetryoftheirreduciblerepresentationstowhich eachstatebelongs(here,thereareonlyone-dimensional,\A",andtwo-dimensional,\E", representations). Table1.1:TheoreticalelectronandholestatesinC168 ac n m Eigenenergy/eV Symmetry Degeneracy 8 7 0.726 E 2 9 6 1.157 A 1 7 7 1.335 A 1 8 6 1.438 E 2 9 5 1.702 E 2 7 6 2.050 E 2 AsthereisnotaperfectwaytocharacterizetheGQDs,wealsoneedtobeawareof thattheopticalresponsethatwemeasureinourexperimentsmightarisefromimpurities. Riesenetal.performedtheoreticalcalculationsandopticalspectroscopyofC132[11].The intensityofthetwomainphotoluminescence(PL)peaksofC132(at ˘ 670nmand ˘ 750 nm)isstronglydependentontheexcitationwavelengthasshowninFigure1.13.Thatthe 21 excitation-wavelengthdependenceistforthetwopeaksisstrongevidenceforthe presenceoftchemicalspeciesinthesample.UndertheassumptionthatthePLpeak at ˘ 670nmmaybecausedbyimpurities,theyalsoperformedphotobleachingexperiments toexploretheoriginofthetwoPLfeatures.Bycomparingthemeasuredof C132inheptanebeforeandafterbleachingindichloromethane,theyconcludedthatthe670 nmemissionbleachesatatratethanthe ˘ 750nmemission,whichthat the ˘ 670nmPLpeakisgeneratedbyanimpurity. Figure1.13:Fluorescencespectraforvariousexcitationwavelengths ex ofC132intoluene. (reproducedfromRef.[11]) 22 1.6Excitont Excitons,boundstateofanelectronandahole,wereproposedasquasiparticlesby YakovFrenkelin1931[68].Anexcitonisahydrogen-likequasiparticlewithneutralcharge. Becauseofthesmallemassesofband-edgecarriersandthelargeindexofrefraction ofmostsemiconductors,anexcitonismuchmoreweaklyboundandcorrespondinglylarger thantheelectronandprotonofahydrogenatom.Anexcitoncanbegeneratedbyphoton absorptioninasemiconductor.Inmostbulksemiconductors,theenergybetween boundandunboundelectronandhole(thebindingenergy)isusuallysmallsothatthe thermalenergyatroomtemperatureexceedstypicalexcitonbindingenergiesofbulkcrystals. Therefore,atroomtemperature,excitonicinbulkmaterialsoftencanbeneglected. Manyofthebetweenbulkandreduced-dimensionalsystemsareduetothe dimensionality-dependenceofthedensityofstates(DOS).Inatwo-dimensionalsystem,an electronisconstrainedtomoveinaplane,andinaone-dimensionalsystemlikecarbon nanotubes,anelectroncanmoveonlyalongtheaxisofthetube.Figure1.14(reproduced fromRef.[12])showstheidealdensityofstates N ( E )forsystemsoftdimensionality. Thedensityofstatesscaleswithenergyas N ( E ) / E 1 = 2 inthreedimensions,anditchanges to N ( E ) / E 0 intwodimensions,whichresultsinastep-likeshapeasshowninFigure1.14. Intwo-dimensionalsystems,theenergyofthetransitionsbetweenquantizedenergylevels canbeestimatedbytight-bindingcalculations[69]asfollows: E n = E g E b ( n + 1 2 ) 2 ;n =0 ; 1 ; 2 ::: (1.8) where E g isthebandgapenergyofthebulkand E b isthebindingenergyofthebulk. 23 Figure1.14:Summaryofthedensityofstatesforabulkcrystal(3D),aquantumwell(2D), aquantumwire(1D),andaquantumdot(0D)(reproducedfromRef.[12]). Besidesthefromthedensityofstates,theCoulombforceisinverselyproportional tothedielectricconstant " .Inreduced-dimensionalsystems,manyoftheelectriclines betweenelectronsandholesareoutsideofthesystem.Theedielectricconstantis thendeterminedlargelybytheenvironment[70,71],notthesystemitself,resultinginweaker dielectricscreeningandasmallerdielectricconstant.Therefore,thebindingenergymaybe morepronouncedinareduced-dimensionalsystem,whichhasbeenvinexperiments onSWCNTs.Forexample,semiconductingSWCNTsof0.8nmdiameterhaveane excitonradiusof1.2nmandbindingenergyofabout400meV[72]. Manyreduced-dimensionalsystems(e.g.,SWCNTandsemiconductorquantumdots) 24 havebeenwellstudiedoverthepastfewyears[73,74,75,76].The sp 2 -hybridizedcarbonin graphenehastwoinequivalentenergybandsnear K and K 0 oftheBrillouinzone(Figure1.2). Theone-dimensionalnatureofSWCNTsenhancestheexcitonicBoththeoretical calculationandexperimentalmeasurementsshowthatthebindingenergyinSWCNTsis largeandcanbesimilartothebandgap[72,77,78,79,80]. 1.7BiexcitonbindingandAugerrecombination 1.7.1Introductiontobiexcitonbinding Whentherearemultipleexcitonsinasystem,anexcitoncanbindwithanotherexciton toformabiexciton(XX).Thebindingenergyofabiexcitonistheenergybetween biexcitionenergyandthesumoftheenergiesoftheotherwisefreesingleexcitonsthatbind toformthebiexciton E XX =2 E X E XX .Figure1.15showstheexcitonandbiexciton dispersioninthecaseofpositivebiexcitionbindingenergy,whichappliestooursystem. E XX canbe ˘ 0forweaklyboundbiexcitons[73,81]ornegativeforbiexcitonrepulsion [82]. Ifabiexcitonisweaklybound,alowerdrivingforceisrequiredtounbindthebiexciton. Muchresearchhasbeenfocusedonincreasingthenumberofcarrierstogetmoret nanocrystalsolarcells[13,83,84].Inphotovoltaicresearch,peopleareinterestedinbiexciton studiesthatmightleadtopotentialapplicationsifthereisantwaytogenerate, separate,andtransportcarriersinthesematerials. 25 Figure1.15:Illustrationofexcitonandbiexcitiondispersionforpositivebiexcitonbinding energy 1.7.2IntroductiontoAugerrecombination Augerrecombinationisanon-radiativeprocessinwhichoneelectron-holepairrecombine bytransferringtheirenergyandmomentumtoanotherelectronorhole.Augerrecombina- tionusuallyhappensrapidly(afewps[75,85])innanometerscaleparticlesasthecarrier interactionsareenhancedandmomentumconservationconstraintsrelaxed[86].Thein- verseprocesscanalsooccur.Asemiconductorcanabsorbahigh-energyphoton,creating ahigh-energyexcitonfromwhichmultipleexcitonscanbecreatedbytransferringenergy andmomentumamongcarriers.Thisprocessiscalledmultipleexcitongeneration(MEG)or carriermultiplication(CM).AschematicenergyleveldiagramshowingAugerrecombination 26 andcarriermultiplicationisillustratedinFigure1.16[13,87,88]. Figure1.16:Energy-leveldiagramillustratingcarriermultiplicationandAugerrecombina- tion(reproducedfromRef.[13]). 27 Chapter2 Equipmentandmethodsforthestudy ofGQDs Weusettime-resolvedspectroscopymethodstoinvestigateexcitonsinGQDs. Transientabsorption(TA)istheprimarytechniqueinthestudyofGQDsdiscussedinthis dissertation.ThesetupoftheTAexperimentisexplainedinthischapterasaresome thedetailsoftheTAexperiment,suchashowtodeterminekeyparameters.Besidesthe TAmeasurement,wealsousedupconversionofphotoluminescence(uPL)toexploreexciton dynamicsintheGQDs.Theresultofthesemeasurementswillbepresentedinlaterchapters wheretheexperimentaldataisanalyzed. 2.1Comparisonofmeasurementtechniques Time-resolvedabsorptionandemissionareoftenusedforexploringthedynamicsof chemicalandbiologicalsystemsinthepicosecondandfemtosecondtimedomains[89].Ina transientabsorptionmeasurement,aninitialpulse,the\pump"excitesthesample.Another timedelayedpulse,the\probe",measurestheabsorptionofthesample.Thepumpexcites someoftheelectrons,andreducestheprobabilityofobsorbingaprobephoton. Inourexperiment,transientabsorptionisaconvenienttechniquewhichprovidesuswith detailedinformationaboutttransitions. 28 Asfortime-resolvedemissionmethods,therearemanyttechniques,suchas single-photoncounting(TCSPC),upconversion,streakcamera,andKerrgating,whichare suitablefortsituations.TSCPCisagreatmethodwith20-30psinstrumentalreso- lutionwhenthesignaliscoupledwithastable,highrepetitionratelaser[90].Whencoupled toaspectrographwithtwo-dimensionaldetection,streakcamerasprovidemultichannelde- tectionwith2-10psresolution[91,92,93].Weareoftenfocusedondynamicsinthefew hundredfsrangewhichrequiresbettertimeresolutionthanTCSPCandstreakcameracan provide.BothupconversionandKerrgatingcanprovidesub-picosecondresolution.We performedbothKerrgatingandupconversion.However,theKerrgatingmethodhasalarge backgroundsignalfromlong-livedemissionmakingithardtogetdecentsignal-noiseratio intheexperiment.Weeventuallychoseupconversiontoperformtime-resolvedemission measurementsinourstudy. 2.2Transientabsorptionmeasurement Transientabsorptionspectroscopycanbeusedtostudymanyprocesseslikephotoinduced chemicalreactions,thebehaviourofelectronsthatarefreedfromamoleculeorcrystalline material,andthetransferofexcitationenergybetweenmolecules[94].Thechangeinab- sorbanceofasampleataparticularwavelengthorrangeofwavelengthsismeasuredasa functionoftimeafterthesystemisexcitedbythepump.Inatypicalexperiment,boththe lightforinitialexcitation(pump)andthelightformeasuringtheabsorbance(probe)are generatedbyapulsedlaser. OurTAmeasurementsareperformedbyexcitingthesampleswitheitherhigh-energy photons(3.1eVphotonsgeneratedbydoublingtheoutputofan1kHzTi:sapphire 29 laser(Spectra-PhysicsPRO-XP)producing ˘ 100fspulses)orlower-energyphotons thataregeneratedbydoublingtheoutputsignalpulsesofacustombuilt,betabariumborate (BBO)-basedopticalparametric(OPA)producingtunablesignalfrom0.78to1.05 eV.Theprobepulsesconsistofasupercontinuumgeneratedbyfocusing ˘ 1 Jofthe 800nmfundamentalina1-mm-long c -cutsapphirecrystal.Thecombinationofahalf- waveplateandthinpolarizer(TFP)beforethesapphirecrystalallowstuningof thebeampowerfocusedinthesapphirecrystalforoptimizingthestabilityofthewhite lightcontinuum(WL).ThecontinuumispartiallycompressedwithapairofBrewster-angle fused-silicaprisms.Theanglebetweenthecolinearlypolarizedpumpandprobeis ˘ 7 : 5 . Broadbanddetectionisperformedwithacharge-coupled-device(CCD)spectrometer(Ocean OpticsUSB2000+,600lines/mmgrating,50 mentranceslit,2nmresolution)synchronized toanopticalchopper(NewFocus3501)thatmodulatesthepumppulsesonandwitha12 msperiod.Forsingle-wavelengthdynamictracesandmeasurementswherethespectrometer isinsensitive,detectionisperformedwithanSiorInGaAsphotodiodeatthe outputofaspectrometer(ActonSP300i,typicalresolution ˘ 5nm)andgatedbyaboxcar integrator(StanfordResearchSystemsSR250).Thefrequencyoftheopticalshopperisset tohalftherepetitionrateofthelasersothateveryotherpumppulseexcitesoursample. 2.2.1OPA-derivedpump Intransientabsorptionexperiments,weneedtobeabletovarythepumpphotonenergy. ForthiswebuiltanopticalparametricHerewechoosetype-IIphasematching (signalandidlerorthogonallypolarized)inourOPAsothatwecanselectsignal(theshort wavelength)oridler(thelongwavelength)simplywithapolarizer.InourOPA,weusea supercontinuumastheseedforparametricinthepassthroughtheBBO 30 crystal.Afterthat,thesignalfromthepassisbackthroughthesame BBOcrystalataslightlytangleandbyanother800nmbeam.Inmost cases,weneedtunablevisiblelight,sowepasstheinfraredbeamthroughanotherBBO crystaltogeneratethesecondharmonicofthesignaloridlerormixitwiththeresidual 800nmbeamfromthesecondpasstoproducevisiblesum-frequency.Thisvisibleoutputis thensentthroughacompressor,whichconsistsoftwotriangularBrewster-angleprismsand amirror,totemporallycompressthevisiblelight. 2.2.1.1OptimizationoftheOPA TomaximizethepowerfromtheOPAwetakethefollowingsteps[95].Beginningwith thepass,weadjustthe800nmwaveplatebeforethethinpolarizerandtweak thetranslationstagethatholdsthelenswhichfocusesonthesapphirecrystaltogenerate anintense,stableWLseed.Theindicatorofagoodseedisauniformcentralwhitespot surroundedbyaredring.Adjustingthecompressorofthe800nmlaserisalsoneeded tostabilizetheWL.AftergettingastableandintenseWL,weadjustthelasttwoturning mirrorswhichtheWLhasnotpassedyetandDM1whichthe800nmpumpofthe passtomakethe800nmbeamgothroughthesamepathastheWL.Withsomeadjustment ofthetranslationstageforthe800nmbeam,the800nmbeam(thepump)andthe WL(theseed)temporallyoverlap.Whenspatiallyandtemporallyoverlapped,thepump andtheseedgenerateinfraredlight.Throughacascadedsecond-orderprocess,the infraredlightdoublesormixeswiththepumpintheBBOcrystal,andproducesvisiblelight withadistinct,crystal-angle-dependentcolor.Wealsoadjustthepositionandtheangleof theconcavemirrorbehindthesapphirecrystaltooptimizethespatio-temporaloverlapof 31 Figure2.1:SetupoftheopticalparametricThepoweroftheincoming800nm beambeforetheOPAentranceis ˘ 0.5to1W. 2representsthehalfwaveplate.BS representsa20/80beamsplitterwith80%ofthepowersenttothesecondpass.TFP representsathinpolarizer.DMrepresentsdichroicmirrors.Theboxeswithdashed linesrepresenttranslationstagesintheOPAsetup.AllthebeamsthatdonotpasstheTFP havelinearpolarizationparalleltothetablesurfaceandthebeamaftertheTFPhaslinear polarizationverticaltothetablesurface.TypicallywecangetafewhundredmWoutof theOPAwiththepowerdependingonoutputwavelength. 32 theseedandthepump.Afterthepasshasbeenoptimized,weadjusttheangleofthe concavemirroronthetranslationstageandtwoturningmirrorssothatthesecond-pass800 nmbeamfollowsthesamepathastheinfraredbeamfromthepass.Afterthat, wetweakthetranslationstagethatholdstheconcavemirrorthatandcollimatesthe IRfromthepasstocreatethesameopticalpathlengthfortheandsecondpasses. Theprocessaboveyieldsahigh-powersignalwithverticalpolarizationandanidler withhorizontalpolarization.Thereisalsoweakvisiblelightfromsecondary,non-phase matched,second-orderprocesses(frominfraredlightdoublingormixingwith800nm).We eliminatethevisiblelightbyusinganappropriatelongpassandsendingtheinfrared lightthroughanotherappropriatelyorientedBBOcrystal(BBO2markedinFigure2.1) todoublethefrequencyormixwith800nmlightinasum-frequencyprocesstoandget lightfrom470nmto730nm.InthisBBOcrystal,theverticallypolarizedsignalgenerates horizontallypolarizedvisiblelight. Intheexperiment,whenthesurfaceoftheBBOcrystalisperpendiculartotheincident beam,thepoweroftheinfraredlightismaximum.Thisisconvenientforinitiallyestablishing spatio-temporaloverlapoftheseedandpump. ThevisiblelightiscompressedbyapairofBrewster-angleprisms.Theprismanglesare optimizedbyrotatingaprismaboutanaxisnormaltotheopticalplanesothatthelateral displacementofthetransmittedlightisminimized.Wethentweaktheverticalknobofthe mirror(M6inFigure2.1)togettheslightlyhigherthanincidentbeam sothatitbypassestheturningmirror(M8inFigure2.1)whichsendsthebeamintothe compressor.ThisWLthenservesastheprobeinourTAexperiment. 33 2.2.2DatacollectionandLabviewVIs AsstatedatthebeginningofSection2.1,weusetheCCDspectrometertoperformbroad- bandphotondetectionandweuseanSiorInGaAsphotodiodeattheoutputof aspectrometertoperformsingle-wavelengthphotondetectionwheretheCCDspectrometer isinsensitive.Theexperimentaldetailsareintroducedinthefollowing. 2.2.2.1BroadbandmeasurementbyCCD WeuseanOceanOpticsUSB2000+CCDarrayspectrometerthatcandetectphotons withwavelengthfrom340nmto1032nm.AsdiscussedatthebeginningofSection2.1,we needtodeterminetheinprobetransmissionofasamplebetweenwhenitisexcited andwhenitisunexcitedtogetthechangeoftheabsorptioncot.Asweuseanoptical choppertoturnthepumponandsuchmeasurementrequiresprecisesynchronization betweentheCCDandthechopper. Theidealscenarioisoneinwhichwecancollectourentireprobespectrumonashot- by-shotbasiswhilesimultaneouslychoppingalternatepumppulsesonandThiswould minimizetheimpactofpulse-to-pulseandalsoallowustoselectivelythrowout pulsesinwhichtheprobeenergyshowsabnormallylarge(e.g.,duetopulses droppingoutofthecontinuumgenerationorprobescatteringbysmallparticleswhena samplesolutionisstirred).Becausethelaserinourlabhasarepetitionrateof1kHz,we wouldneedtobeabletoperformafullcycleoftriggeredacquisitionandreadoutin1ms.As describedbelow,thisisnotpossiblewiththeUSB2000+,andwemustcarefullysynchronize slowerchoppingofthepumpwithslowerCCDintegrationrates. 34 ThelimitsontheCCDtimingareshowninFigure2.2[96].Theminimumtriggercycle is2.909mswithaminimumintegrationtimeof1msinhardwareedgetriggermode.In otherwords,forourmeasurement,thetimethatthespectrometerisnotintegratingsignal isabout1.909mspercycle. Figure2.2:USB2000+hardwareedgetriggermodetimetable Ifthemeasurementisperformedwithaperiodof3ms,thedevicecancollectonly1 pulseandwilldrop2pulsesineachcycle.WhentestedwiththeLabviewprogramdescribed below,thesystemcannotstablyacquiredatasorapidly.Thissuggeststhatthedevicecannot reachtheminimumsptriggercycleof2.909ms.Eventhoughthisisthebestwayto reducetheimpactofonsinglepulses,one-thirdofdatacollectionisa highpricetopay.Inordertobalancebetweendataandcollection,we 35 choosethecollectionperiodtobe6ms,whichmeansthatwecanget4pulsesbutonlylose 2pulseswithineachmeasurement.Inpractice,wechoosetheintegrationtimetobe3.8ms. Thischoiceoftheintegrationtimewillbediscussedlater.However,whenchoosingtheexact integrationtime,weshouldalsoconsiderthesignallevelfromasinglepulserelativetothe saturationintensityofthedetector.Ifthesignallevelissohighthatsaturationoccursina singlepulse,thenusingtheminimumintegrationtimethesystemcanhandleispreferred. Figure2.3:Timingofspectrometertrigger Figure2.4showsthebasiclayoutofalltheconnectionsbetweenthemeasurementdevices. Apulsedelaygenerator(StanfordResearchSystems,ModelDG535)istriggeredbythelaser SYNCHpulse(aTTLoutputsequencefromthelasersynchronizedwiththelaseroutput) andissettoproduceanoutputwithaperiodofT=6ms.Theopticalchoppercontroller receivesthepulsegeneratoroutputandchopsthebeamathalfthepulsegeneratoroutput frequency.Thus,thepumpischoppedwithaperiodof12ms.Finally,theUSB2000+ spectrometerissynchronizedbytheoutputofthepulsegeneratorandproducesspectra every6ms. 36 Figure2.4:Layoutofopticalpathsandelectricalconnections Table2.1:22-pinaccessoryconnectorpinoutdiagramwhenfacingthe22-pinAccessory ConnectoronthefrontoftheverticalwalloftheUSB2000+ 20 18 16 14 12 10 8 6 4 2 A2 19 17 15 13 11 9 7 5 3 1 A1 TocollectdatafromtheUSB2000+,weuseaLabviewprogramtosettheparameters (numberofscans,delays,etc.),readdata,andwritedatatoTheprogramisbased onaLabviewPlugandPlay(project-style)InstrumentDriver[97].Anewdriver (NI-VISA)isrequiredforusingthisLabviewprogram.Thisdriverisincompatiblewiththe SpectraSuitesoftwarefromOceanOptics,anddriversneedtobereinstalledwhenswitching betweentheLabviewprogramandSpectraSuite.IntheLabviewprogram,theVISAsource namefortheconnectionwillindicateaUSBconnection. Weadjustthedelayinthepulsegeneratorsothephasebetweenthelasertriggerpulse andUSB2000+triggersequenceyieldsstabledataacquisition.Thephasebetweenthe USB2000+triggersequenceandthechoppersequencecanbeadjustedbyadjustingthe phasecontroloftheopticalchopper.Bymonitoringtheintensityofscatteredpumplight collectedbythespectrometer,wecantellwhetherthepulseisfullyblockedorclippedby 37 Table2.2:anddescriptionsofpin6and7,whichareusedtoconnectwiththe USB2000+triggersource[1] Pin Function Input/Output Description 6 Ground Input/Output Ground 7 ExternalTriggerIn Input TTLinputtriggersignal Figure2.5:LocationofUSB2000+Accessoryconnector thechopperblade.Ifthespotsizeofthebeamisnottoobigcomparedtothebladewidth ofthechopper,thenduringthe12msperiodofthechoppersequence,wecanmakethe 6mscorrespondtothechopperONstate(notblockingthepumpbeam)andthenext6ms tothechopperOFFstate(blockingthepumpbeam),asillustratedinFigure2.6. Figure2.6:Timingdiagramforchopperandspectrometertrigger Ingeneral,ifwewishtointegrate N pulsesperacquisitionperiodandwelosetwopulses 38 duetoreadoutandidletime,nottoobigmeansthatthepumpbeamshouldhaveadiameter, d ,describedas d 3 w N +2 ,where w isthechopperslotwidthandthefactorof3iscalculated fromour6mschopperontimeand1mspulseseparation.Inthepresentcase, N =4sothe beamdiametermustbelessthan1/2ofthebladewidth.Ifthiscriterionisnotmet,pulses atthebeginningand/ortheendoftheONandOFFintegrationtimesmaybeclippedby theedgeofthechopperandmightnotbeentirelyONorOFF.Onemustthatnone oftheNpumppulsesofinterestshouldbeclippedbythechopper.Onewaytothis istomonitorthescatteredpumpspectrumandreducetheintegrationtimeinincrements ofonelaserperiod.Iftheintensityoftheresultingspectrumdecreaseslinearly,thismeans thatthechopperisnotclippingthepumppulsesofinterest.Ifclippingisobserved,the phaseofthechoppermustbeadjusted. SupposethatwewanttoperformaTAmeasurementwitheachpumpON/OFFcycle consistingofspectrafrom4pulseseachwithpumpONorOFF.TimecountingintheLabview programisusedtothatthetimeweactuallyusedmatchestheexpectations.Based onthedevicespwemaycapture4pulsesoutof6byusinganintegrationtime from3.1+ msto5.0- msaslongasthephasebetweenthespectrometertrigger sequenceandthelasertriggerpulseisappropriatelyset.Wesetthetriggersequenceslightly earlier(0.15ms)thanthelasertriggerpulsesequence,andsettheintegrationtimetobe3.8 ms.AccordingtotheclockintheLabviewprogram,ittakes6msforeachacquisition,as expected. Inreality,whentheLabviewTAprogramisrunning,onecannotrelycompletelyonthe presumedtimingdescribedabove.Forunknownreasons,thedetectormaymissonelaser pulseduetothecommunicationbetweenthedetectorandthecomputer,sothedetermination ofthepumpONandpumpOFFsignalwilloccasionallyintheLabviewprogram 39 (typicalprobabilityof ˘ 0.3%).Toaccountforthis,weuseareferencelightsource(an inexpensive3eVdiodelaser)togothroughthechopperatthesamelocationasthepump. ThisensuresthatourreferencebeamhasthesameON/OFFstatusasthepump.We scatterpartofthisthisreferencelightintotheCCDtoserveasamonitoroftheON/OFF statusofthepump.Theintensityofthereferencebeamshouldjusthavetwot levels,maximumwhenunblockedandzerowhenblocked;theCCDshouldnotyieldany intermediatesignallevels.Wemeasuretheintensityofthesetwolevelsofthereference beambeforeeverymeasurementandsetathresholdforsortingeachspectruminLabView accordingtowhetherthepumpisONorOFF. 2.2.2.2Single-wavelengthmeasurementbyphotodiodes Forsingle-wavelengthdynamicstracesandmeasurementswherethespectrometerisin- sensitive,detectionisperformedwithanSi(ThorlabDET36A)orInGaAsphoto- diode(ThorlabDET10D)attheoutputofaspectrometer(ActonSP300i,typicalresolution ˘ 5nm)andgatedbyaboxcarintegrator. Weapplytdetectordevicestocoveraslargeadetectionrangeaspossible.Our CCDspectrometercandetectsignalfrom340nmto1032nmwith2nmresolution.The InGaAsphotodiodecandetectsignalfrom1200nmto2600nm.Andthe Sidetectorhasadetectionrangefrom350nmto1100nm.Thedataacquisitionisperformed byadataacquisitioncard(NationalInstrumentsPCIe-6321)whichcollectsthevoltagesignal fromtheboxcar(whichreadsthephotodiode'ssignal)andlaserSYNCH.Thecollected informationisthentransferredtothecomputertobefurtherprocessedandanalyzed.The frequencyoftheopticalchopperforthepumpishalfofthelaserrepetitionrateandour Labviewprogramisdesignedtorecordthesignalfromeverylaserpulse.Theintensitiesof 40 thesortedsignalsare I on = I 0 e ( 0 ) l and I = I 0 e 0 l ,where istheabsorption cot, isthechangeoftheabsorptioncot,and l isthepathlengthofthe beamoflightthroughthematerialsample.Fromthesetwoquantities,weobtainthechange from l =ln( I on I ). ForTAmeasurements,wehavedevelopedtLabviewVIstointegrateparameter setup,dataacquisitionanddataprocessing.AwchartoftheLabviewdataacquisition andprocessingVIisshowninFigure2.7.TomaketheLabviewVIsmoret,weprocess dataacquisitionanddataprocessinginparallel.Fordynamicsmeasurements,weacquire datafromthemeasurementdevice(CCDorphotodiode)atonedelayposition,storethe dataandprocessthedatafromlastdelaypositionwhichisstoredpreviouslyatthesame time.ForthemeasurementofaTAspectrumatacertaindelaywhenusingasingle-channel detector,theideaissimilartothedynamicmeasurement.wecollectandstoredataatone wavelengthandthedataareprocessedatthesametimeaswecollectdataatthenext wavelength. 2.2.3TAsetupoptimization TheofourTAmeasurementisshowninFigure2.8. 2.2.3.1TAsetup Weusea250mmmechanicalstagetoadjustthetemporaldelayoftheproberelativeto thepump.Onereasonthatwedelaytheprobeisthatthespotsizeofthepumpis largerthanthatoftheprobe.Ifthereisasmallchangeofprobepositionorsizebecauseof thevariabledelay,ourpumpandprobearestillspatiallywelloverlapped.Also,wechooseto delaytheweak800nmbeamintheprobepathbeforewegeneratetheWL,becausethe800 41 Figure2.7:FlowchartofLabviewVI(dynamicmeasurement) 42 Figure2.8:ofTAmeasurement nmbeamfromtheisthebestcollimatedbeaminoursetup.Thisthenminimizes delay-dependentintensitychangesassociatedwithimperfectcollimation. parabolicmirrorsratherthanlensesareusedinTAsetuptofocustheWLinto thesampleandthenrecollimatethelightafterthesample.parabolicmirrorshelpus tominimizethedispersionintroducedintheWLprobe,minimizechromaticityinfocusing, andalsoeliminatesphericalaberration.ThisisespeciallyusefulwhenweuseaCCDdetector tomeasurethetransientabsorptionoverabroadspectralrange. Whenchoppingathalfthelaserrepetitionrate,adjustmentofthechopperphaseis straightforward.Ifweplaceawhitebusinesscardbehindthechopperandlookattheblades fromthebacksidewhiletheyareilluminatedbythelightscatteredfromthecard,theblades willappearstationary.Thephaseisthenadjustedsothatthestationarywindowbetween bladesiscenteredonthetransmittedbeam.Alternatively,wecansetthechopperfrequency 43 totherepetitionfrequencyofthelaser.Thephaseisadjustedtothecenteroftherange wherethepumpiscompletelytransmitted.Whenwegobacktohalfthelaserrepetition frequency,thechopperwillbeperfectlyphasedforchoppingalternatepulses. Sometimes,weobserveaslightinthepumponandsignalsatnegative delays,i.e.,beforethepumparrives.Onepossibilityisthatthecreatedbythe wallsofthecuvettetheprobebeamwhichgeneratesanextraweakprobewith slightlylongerdelay.Anotherpossiblesourceofsuchaisphotoluminescence, whichappearsasaconstantcontributiontotheofpump-onandsignals. Thiscanbeaccountedforbytakingthebetweenpositive-andnegative-delay signals. 2.2.3.2AlignmentprocedureofourTAsetup Sincetheprobeandpumpbeamsmayslightlyshiftday-to-day,weneedtoalignthesys- temperiodically,whichincludesoptimizingthestabilityoftheWLcontinuum,overlapping thepumpandtheprobeatthesamplepositionandsomeotherprocedures. ThepoweroftheTi:sapphireregenerativedriftsovertime.Thiscancause instabilitiesintheWLintensityandspectralshapebutcanbecompensatedbysmallchanges inintensityorspatialmodeofthe800nmpulseatthesapphirecrystal.Thesechanges canbedonebyadjustingthefocusingordelay-stageexitiris.Typicalpulse-to-pulseWL are0.4%to0.6%(standarddeviation). ApinholepositionedatthefocusoftheWLprobeisusedtooptimizetheoverlapofthe pumpandprobeandensurethatthedelaystageiswellaligned.WesendtheWLsignal throughthepinholeandmakesurethatthereisnegligiblechangeintheratioofthepower throughthepinholetothepowerwithoutpinholeatanydelayposition.Whenusinga 44 1-mm-pathlengthcuvette,thesamplestageismovedabout0.3mmclosertotheWLsource tocompensatethecuvettethickness.Totheoptimizedoverlappositionforoursample, wetheWLdelaystageformaximumTAsignalandthenadjustthesamplepositionto furthermaximizethesignal. Itiscrucialtowellaligntheprobebeamintothedelaystagesothatthelighthittinga smallspectrometerslitdoesnotmoveduringtheexperiment.Ifthealignmentofthedelay stageisperfect,weshouldgetthesamefractionofpowertransmittedbythepinholeatany delayposition. Inordertodeterminethepumpandprobespotsizesanddeterminetheexcitation ences,werecordthefractionofWLandpumppowertransmittedbythepinhole.We estimatespotsizesbyassumingthattheyarecircularGaussianspatialmodesforwhichthe intensitypassingthroughacircleofradius r is I ( r )= I 0 exp( 2 r 2 d 2 ).Thespotsizecanbe describedas: d = D ˆ ln 1 P PH P noPH 1 2 (2.1) where D isthediameterofthepinholeand P PH and P noPH arethepowermeasuredwith andwithoutthepinhole. Usually,thespotsizeoftheprobe(WL)inourexperimentiscomparabletothe100 m diameterofthepinholeweuse.Thediameterofthepumpisfrom0.6to2mmdependingon theavailablepumppoweranddesiredintensitiesforaparticularmeasurement.Weadjust theWLfocusingastightlyaspossibleandmakethespotsizeofourpumplargeenoughso thatintheabsenceofabsorptiontherewouldbearoughlyconstantovertheprobe path. Giventhespotsizeandintensityofthepumpandthecrosssectionofoursample,we 45 canestimatetheaveragenumberofphotonsthatareabsorbedperpulsebyeachquantum dot: = 4 P noPH ˇd 2 R E ˙ (2.2) Inthisequation, R istherepetitionrateofthelaser, E istheenergyofapumpphoton, and ˙ isthecrosssectionofthesampleatthepumpwavelength.FortheGQDsinour experiments,thereportedcrosssectionsat400nmforC132andC168arerespectively 2 : 0 10 16 cm 2 and3 : 78 10 16 cm 2 [8]. WhenthepumpisgeneratedbytheOPA,toobtainamorestableWLandmaximum powerfromtheOPA,tweakingthecompressionofthemaybealsoneeded.The moststableWLandmaximumpowerOPAmayrequiretcompressions,whichmeans thatweneedtobalancebetweenthetwobyadjustingthecompression. Thereareafewsubtledetailsthatneedtobekeptinmindtoensuretheaccuracyofthe measurements.Forexample,thepumpshouldnotbetoostrongatthesampleposition.This isbecauseahighintensitypumpcancausearefractiveindexchangeinthesamplesolution thatresultsinadelay-dependentshiftoftheprobepositionorspatialmode.Iftheshift istlylarge,itmayproducechangesinspectrometer-slittransmissionunconnected tothequantumdotresponsemayoccurleadingtoinaccurateassessmentsofpump-induced changesinprobetransmissionbythesample.Weuseanachromaticlenstofocustheprobe intothedetectortogetthesizeassmallaspossibleacrossthevisiblespectrum,sowecan reducethefrombeamshiftsifthereareany.TheresolutionoftheTAexperiment isdeterminednotonlybythepulsedurationbutalsofactorslikecrossinganglebetween thepumpandprobe.Whenthepumpandtheprobearecolinear(zerocrossingangle),the temporalcross-correlationbetweentheoverlapofthesetwobeamsisminimum.Inreality, 46 thereisanangle( ˘ 7 : 5 )betweenthepumpandtheprobebeamsinourTAexperiment, butitonlyreducesourresolutionto150fs,whichisacceptableinourexperiment. 2.3Upconversionofphotoluminescence InaTAexperiment,thechangeinabsorptionpresenceofelectronsand/orholes inthestatesinvolvedinagiventransition,so dependsonthesumofelectronsand holes.Incontrast,emissionoccursonlywhenanelectron-holepairrecombines,sothe photoluminescencesignalscalesastheproductofthenumberofelectronsandholesin thestatesbetweenwhichemissiontakesplace.Bycombiningtheresultsfromthesetwo experiments,wegetaclearerviewoftheelectronicstructureandtherelaxationpathways ofoursampleundertconditions. Thetechniqueweusefortemporallyresolvingtheemissionisupconversionofphotolu- minescence(uPL),whichwasdescribedindetailbyJagdeepShah[98].Upconversionisa nonlinearprocessinwhichonePLphotonandonegatephotonannihilateandgeneratelight atthesumfrequencyofthetwoannihilatedphotos.IncontrasttoTA,whichhasmanydif- ferentcontributions,theupconversionmeasurementdoesnotincludeanyinducedabsorption contributions,andthemethodisbackground-free,thusallowingformorestraightforward idenoftherelevantstates. AsshowninFigure2.9,the800nmbeamfromthelaserpassesawedgetogeneratea weakThebeamthatgoesthroughthewedgeissplitbya50/50beamsplitter (BS).Afterthebeamsplitter,oneofthetwobeamsgoesthrougha250mmmechanicaldelay stagecontrolledbycomputerandthenaneutraldensity(ND)wheelforadjustingthe power.EventuallythisbeamissenttotheBBOcrystal(typeII,theta=27.2degree,8 8 0.6 47 Figure2.9:SetupofuPLmeasurement mm)usedforupconversion.TheotherbeamissentthroughaBBOcrystaltoproduce400 nmlightbysecondharmonicgeneration.The400nmbeamservesastheexcitationbeam inourexperiment.ABG39eliminatestheresidual800nmbeamfromthedoubled light.The400nmpumpintensityisadjustedwithaNDAGalileantelescopeisused tosettheproperspotsizeoftheexcitationbeamonthesample.Theopticalchopperin the400nmbeampathissynchronizedwiththelaserpulsesandsettohalftherepetition 48 rateofthelaser.Afterexcitation,thesampleluminescencesinalldirections.Weusealarge (2 00 indiameter)concavemirrorwithfour-inchfocallengthtocollecttheluminescenceand redirectittomeetthe800nmgateintheBBOupconversioncrystal,whichistunedfor phasematchingoftheupconversionprocess.Thedistancefromthesampletotheconcave mirrorisabout6inchesandthatfromtheconcavemirrortotheBBOcrystalisabout12 inches.Intheend,severalturningmirrorssendtheuPLsignaltoaspectrometer(SpectraPro 2300i,PrincetonInstruments)onwhichathermoelectricallycooledCCDarray(Princeton Instrumentsmodel1024HER)ismounted.ThelastmirrorinfrontoftheCCDismounted onagalvanometerthatissynchronizedwiththeopticalchopper,soastothepump-on (signal)and(background)signalstotheightsattheentranceslit(andsoon theCCD)forbackgroundsubtraction.BecausetheGQDsemitat670nm,theupconverted lightisclosetothesecond-harmonicat800nm.Therefore,oneofthemainsourcesof backgroundnoiseisthesecond-harmonicgeneration(SHG)fromthegate800nmpulse intheupconversioncrystal.Furtherbackgroundisduetocascadedparametricgeneration andupconversionproducedbythegate.Thesearecriticalissuesparticularlyinstudyingthe graphenequantumdotsbecauseoftheirweakemission,whichmakesgalvanometricdetection essential.Ashortpass(NewportSP385)justinthefrontofthespectrometerallowsus toreducethebackground,especiallythatfromSHGofthegate.Enclosureofthedetection allowsustofurtherreducethebackground. Themixingprocessistonlyifphase-matching.Weadjusttheangleofthecrystal togetphasematchingfortphotoluminescencewavelengths. SincetheluminescencefromC132graphenequantumdotsisintherangeofabout650 nmto820nm,theupconvertedlightisintherangeofabout358nmto404nmandweak andsocannotbeseenbyeye.Inordertoaligntheopticsaftertheupconversioncrystal, 49 weremovetheBG39fromthepumppathandreplacethesamplewithsomething scatterthelight.Thisprovidesan800nmbeamthatfollowstheexactsamepathasour luminescencesignaldoes.Eventhoughthecrystalanglemaybeslightlytfrom theangleatwhichtheluminescenceisupconverted,thereislittleinthebeam direction.The800nmpulseisupconvertedto400nmattheappropriateangleforphase matching,andthiscanbeseenbyeye.Byutilizingthisupconverted400nmbeam,wecan alignthelast3turningmirrorsbeforethespectrometerandthenreturntothe forupconvertingluminescencefromthesample.WhenadjustingtheangleoftheBBO crystalbacktotheangleforupconversionofsamplephotoluminescence,wemayslightly translatethepositionofthecrystalaswelltoensurethatthesetwobeamsarestillspatially overlappedwellintheBBOcrystal. 2.4Samplesstudiedinthisthesis C132andC168aresynthesizedbyLiang-shiLi'sgroup[8].WereceiveddryGQDsand dissolvedtheGQDsinanhydroustoluene.FortheexperimentsIpresentinthisdissertation, allGQDsolutionswerepreparedinagloveboxwithanitrogenatmosphere.BothC132 andC168GQDsweredissolvedin99.7%water-freetoluene,andthesolutionsweresealed inair-tightfusedsilicacuvetteswithpathlengthsof1mmor10mm.Weusetheground- stateabsorptionandphotoluminescencespectratoregularlymonitorthesample.Ground- stateabsorptionspectraaremeasuredbyaUV-visiblespectrophotometer(VarianCary50 BioUV-VisibleSpectrophotometer)andthephotoluminescencespectrumismeasuredbya sp(PTIQuantaMaster300). 50 Chapter3 BiexcitonBinding 3.1Introduction 3.1.1Biexcitonbindinginquantumsystems TheCoulombinteractionisinverselyproportionaltothedistancebetweentwocharges andtothedielectricconstant.Withreducedspatialseparationofelectronandhole,the bindingenergyofanexcitoncanincreasequickly[99].Similarly,areductioninthee dielectricconstantinlower-dimensionalmaterialscanalsoenhancethebindingenergy.The- oreticalstudieshaveshownthatquantumttlyincreasesthebiexciton bindingenergyofbulksemiconductors[100,101,102,103].InbulkGaAs,thebindingenergy ofanexcitonis ˘ 4.9meV,whiletheexcitonbindingenergyinGaAs-Al x Ga 1 x Asquantum wellscanbethreetimesashighasthatinbulk( ˘ 13meVwithGaAswellthickness42 A)[104].Similarly,thebiexcitonbindingenergyrisesfrom0.5meVinbulkGaAs[105]to2.2 meVinGaAs-Al x Ga 1 x AsquantumwellswithGaAswellthickness100 A[103].Insemicon- ductorquantumdots,tinthreedimensionsincreasesthebindingofthebiexciton evenmore[106,107].Comparedtoothersemiconductingsystems,carbon-basedquantum dotsaremadeofrelativelylightatoms.Thisresultsinevenfurtherreducedscreeningand asmallerdielectricconstant.Thebindingenergyofasingleexcitoninsingle-walledcarbon nanotubes(SWCNTs)ismeasuredtobeaslargeas0.4eV[72,80].Thebindingofbiexcitons 51 inSWCNTsisfoundtobecorrespondinglystrong.In(9,7)SWCNTs(diameter=1.1nm [108])ingelatin,thebiexcitonisboundby105meV,whichis ˘ 40%ofthesingle-exciton bindingenergyof250meV[109]. Figure3.1:(A)Experimental(blue)andtheoretical(black)groundstateabsorptionofC168. (B)Calculatedband-edgesingletexciton(X)andbiexciton(XX)states(blacklines)derived fromthedegenerateHOMOandLUMOstates.Greylinesshowexcitedexcitonstates accessiblefromX 1 ; 2 .Dipole-allowedelectronictransitions,whichcorrespondtoachange in m of 1,fromthegroundstateandfromthelowestsingletexcitonstatesareshown respectivelybysolidredandbluearrows.Dashedarrowsindicateopticallydarkelectronic transitions.[14] Wedescribedthetheoreticalcalculationoftheenergiesoftelectronictransitions inGQDsinChapter1usingasingle-particlepicture.Tobetterunderstandtheelectronic structureofsuchGQDs,IsilandPawelHawrylak[15]performedmoredetailed 52 calculations.Becauseofthethree-foldrotationalsymmetryofC168,theelectronicstates canbelabelledbythequantumnumbersm=0, 1withopticallybrighttransitionsoccurring onlybetweenstateswhere 1,asdemonstratedbybothtight-bindingandab-initio calculationsinFigure3.1(B). Figure3.2:Single-pairexcitationwithtotalangularmomentum 1(opticallybright exciton)and(opticallydarkexciton)(reproducedfromRef.[15]). Becauseofthetwo-folddegeneracyoftheHOMOandLUMOstates,therearefour entband-edgesingletexcitonsasshowninFigure3.2.Thecalculatedabsorptionspectrum isrepresentedinFigure3.1(A)bythesolidblackline,andtheexperimentalabsorption spectrumisrepresentedbythesolidbluelineinthesameforcomparison.Theenergy 53 statesfromthetheoreticalcalculationaremarkedinFigure3.1(B)bysolidblacklinesand greylines.Thelowest-energystates(X 1 andX 2 ),whichwedenotecollectivelyasLX,are two-folddegeneratearound1.7eV.Basedonthecalculation,theLXstatesareoptically darkintheabsenceofcouplingtophonons[15,65].Whencoupledwithphonons[11],LX statescanyieldtheabsorptionshoulderat1.7eVshowninFigure3.1.Themainpeakat 2.1eVintheexperimentalabsorptionspectrumisassociatedwiththetwo-folddegenerate band-edgebrightsingletexcitons(markedasX 3 andX 4 inthe performedinteraction(CI)calculationsofthelow-energysingle- andbiexcitonelectronicstatesofC168.Theelectronictight-bindingHamiltonianofthe GQDdescribinginteractingelectronsin p z carbonorbitalsisexpressedasfollows: ^ H = N X i;l =1 X ˙ ˝ il c + i˙ c l˙ + 1 2 X i;j;k;l X ˙;˙ 0 h ij j V j kl i c + i;˙ c + j;˙ 0 c k;˙ 0 c l;˙ (3.1) where c + i;˙ representsthecreationoperatorforanelectronofspin ˙ inthe i th orbital. Intheexpression,thetermistheone-electrontight-bindingHamiltonian,andthe secondterm, h ij j V j kl i ,describesthescreenedelectron-electroninteractions, V ( ~r ~ r 0 )= 2 = ( j r r 0 j )[15].Withthisexpression,theHFcalculationisperformedbyrotatingthe c + i;˙ siteoperatorsintoHFoperators b + j;˙ ,where j areHFstates.Thegroundandexcited statesoftheGQDarethenexpandedinmulti-pairexcitationsoutoftheHFgroundstate j 0 i , j v i = k 0 j 0 i + P i;j;˙ k v (1) ij b + i˙ b j˙ j 0 i + P i;j;k;l P ˙ 1 ˙ 2 k v (2) ijkl b + i˙ 1 b + j˙ 2 b k˙ 1 b l˙ 2 j 0 i + ::: .TheCI Hamiltonianmatrixisbuiltinthespaceofmulti-pairexcitationsanddiagonalizednumer- ically.TherearemanyavailablestateswithhighenergyinGQDs.Wefocusonthestates withthelowestenergies,whicharetheonescontributingtothelow-energyexcitonsand 54 biexcitonsthatweprobe,byincludingonlystatesbelowaenergy.Toaccountfor AugercouplinginCIcalculations,HFstateswithinanenergywindowof3 E g arekeptin thecalculation.Aof E XX < 3 E g forXXenergiesisusedtoreducetheCI-Hilbert space.ThetunnelingmatrixelementsinEquation3.1arechosenas ˝ =4 : 2eVfornearest neighborsand ˝ 0 = 0 : 1eVfornext-nearestneighbourstogivebrightsingletexcitonsat 2.128eV,closetothestrongabsorptionpeakmeasuredat2.1eV.Meanwhile,thedielectric constant,whichdeterminesthebright-singlet-dark-tripletsplitting,waschosenas =5, similartotheexpectedvalueforextendedgrapheneintoluene[71],toscreening bysigmaelectronsandsurroundingSimilartothecharacterizationofsingle-exciton states,biexcitonstatesarecharacterizedbytotal0 ; 1. Extrapolatedband-edgesingletexciton(X)andbiexciton(XX)statesderivedfromthe degenerateHOMOandLUMOstatesareillustratedinFigure3.3.Thetransitionfrom X 1 ; 2 toXX 1 3 isorbitallyforbiddenbutbecomesallowedduetoelectron-phononcoupling. StatesXX 4 7 canbeformedbyabsorptionof ˙ +or ˙ photonsfromtheorbitallydark X 1 ; 2 states.XX 8 ; 9 statescanbeformedfromthegroundstatebyabsorptionof( ˙ +, ˙ +)or ( ˙ , ˙ )photonsandX 10 canbeformedbyabsorptionof( ˙ +, ˙ )photons.TheXX 8 10 statesquicklycooltothelowest(dark)states,sothattheyareonlyaccessiblebeforethis cooling. 3.1.2Experimentaltechniquesformeasuringbiexcitonbinding Photoluminescenceandtransientabsorptionmeasurementsaretypicalmethodsusedin thestudyofexcitonsandbiexcitonsinquantumsystems.Inaphotoluminescence measurementtheamplitudeofthephotoluminescenceisproportionaltotheaverageproduct 55 Figure3.3:Predictedexciton(X)andbiexcitonstates(XX)inGQDs.Dipole-allowed electronictransitions,correspondingto= 1,arelabeledwitharrows.Red(blue)arrow represents=+1(-1)correspondingto ˙ +( ˙ )photonpolarization[14]. 56 ofthenumberofelectronsandholesperGQD.Hence,aphotoluminescencesignalcanbe detectedonlywhencarrierspopulateboththevalenceandconductionbands.Inprinciple, wemayobtainahighersignal-to-noiseratio(SNR)thanthatinTA,andwithfewerstates involvedinthegenerationofphotoluminescence,itmightbeamorestraightforwardmethod thanTAtocharacterizetheexcitontransitions. Wesetupanupconversionofphotoluminescence(uPL)measurementasdescribedin Chapter2.GQDsarepumpedat3.1eV,andphotoluminescenceisdetectedatt photonenergies,asshowninFigure3.4. Figure3.4:Excitation,cooling,andopticaltransitionsinvolvedinphotoluminescencein GQDs 57 uPLmeasurementdatafromC132areshowninFigure3.5withapumpof4.2 mJcm 2 perpulse.Thedynamicsdataarenormalizedbythelong-delaysignals PL (t= Figure3.5:NormalizeddynamicdataofC132pumpedat3.1eVandprobedat1.68eVand 1.84eV 100 1000ps)ofeachtrace.Thesmallbetweentheprobesignalsat1.68eVand 1.84eVonlyexistsforthefewps,whiletheinstrumentresponsetimeis ˘ 2psshown inFigure3.5.AsdiscussedinChapter1,wewerenotabletocompletelyseparatethesetwo peaks(670nmand750nm)fromeachotherinourmeasurementwiththeneededtemporal resolution,anditispossiblethattheemissionwemeasureinouruPLisfromimpurities[11]. 58 ConsideringbothC132andC168havelowphotoluminescencequantumyields(lessthan2% and0.2%respectively),impurityemissionmaybethedominantfeatureinsuchcase.InTA measurements,wecanchoosetheprobeenergytofocusonGQDtransitionswithgreater oscillatorstrengthsothattheimpuritieshavelessimpactonthedata. 3.2Experimentandresults 3.2.1Experimentalsetup TheexperimentalfemtosecondTAsetupwasdescribedinChapter2.C168intoluene wasexcitedat3.1eVandprobedwith ˘ 130fstemporalresolutionusingeitherabroadband continuum(forphotonenergies ~ ! probe 1.1eV)ortheoutputofaBBO-basedOPA(for 0.5eV ~ ! probe 1 : 05eV).TheformerwasmeasuredwithaCCDspectrometerwith2 nmspectralresolution,whilethelatterwasmeasuredwithamonochromatorandInGaAs photodiodewith5nmspectralresolution. 3.2.2Experimentalresultsanddiscussion Figure3.6showsatypicaldelay-dependenttransientabsorptionspectrumobtainedwith aCCDdetectorwhenexcitingC168at3.1eVwithapumpof0.79mJcm 2 per pulse.Thereislittletonochangeintheshapeoftheabsorptionspectrumafterthe ˘ 10ps.Afterthattime,themainchangeinthespectrumisadecreaseinamplitude,which weattributetotheresidualcoolingofthelattice.ForprobingtheBXelectronicstructure, wefocusonlongdelays,whenthesystemhasunquestionablycooledtothelowest-energy singletexcitonstates. 59 Figure3.6: L asafunctionofwavelengthanddelayforC168excitedat3.1eVatan intensitycorrespondingto ˘ 1.2.Thescalecorrespondstothedataatdelayst 5.0 ps.Thedataintherightpanel(t > 8ps)aremultipliedby3.Theblackcurvesindicate the L =0contours[14]. Forinterpretingtheexperimentaldataatlongdelays,wefocusontheband-edgebiex- citons.High-energyexcitonswillrelaxtolower-energyexcitonssothatonlyX 1 ; 2 isleftat longdelays.ThecalculatedsingletexcitonsandbiexcitonsinFigure3.1(B)canbeclassi- byexaminingwhetherthesingle-excitonstatesfromwhichtheyareprimarilyderived includetwo(XX 1 3 ),one(XX 4 7 ),orzero(XX 8 10 )LXexcitons.Followingthe 1 selectionrule,theallowedtransitions(solidbluearrows)areindicatedinFigure3.1(B). Thebiexcitonbindingenergyisthebetweenthebiexcitonenergy, E XX i ,and thesumoftheenergiesofsingle-excitons(X j andX k )thatprimarilyformthebiexciton: XX i =( E X j + E X k ) E XX i .Whenthereisapositivebindingenergy(asinoursample), biexcitonbindingcausesthetheenergyoftransitionsfromasingle-excitonstatetobiexciton 60 statetobelowerthanthatfromthegroundstatetotheexcitonstatecorrespondingtothe secondexcitonaddedtoformthebiexciton.Thetheoreticalcalculationdescribedabove yields XX 1 3 =56-82meV, XX 4 ; 5 =142meVand XX 6 ; 7 =104meV. Datafor t =100psinFigure3.6areshownbybluecirclesinFigure3.7.Bleaches oftheground-statetransitions( ˘ 1.7and2.1eV)areaccompaniedbyinducedabsorption immediatelytothered( ˘ 1.45and1.95eV).Suchpatternsarecharacteristicsignaturesin TAofbiexcitonbinding[106].Measurementsunderthesameconditionsbutatlowerprobe energies(0.5eVto1.1eV)areusedtoprobeintrabandtransitions,whicharealsoillustrated inFigure3.7. ThepopulationofX 1 ; 2 opensnewtransitionsX 1 ; 2 ! XX X 1 ; 2 +X n ,whichrequiresless energythanthe0 ! X n transitionbecauseofthebiexcitonbinding.However,induced absorption(IA)below1eVprovidesclearevidenceofintrabandtransitions.Thisraisesthe possibilitythatsuchtransitionsmaycontributetoourTAsignalinthevisibleregion(1.3 to2.5eV)aswell. Inthebiexcitonbindingmeasurements,weusedhighintensitysothatessentiallyall theprobedGQDsareexcited.ForthoseGQDsthatareexcited,therearethreet contributionstotheTAspectrum.Eachtransitionisbleachedtlybecauseoftheir tdegeneraciesandoccupations.ThisisillustratedinFigure3.9a.Whenan electronoccupiesanexcitedstate,thetransitionprobabilitydecreasesduetostate Becauseofthetwo-foldorbitallydegenerateHOMOandLUMO,the0 ! X i transitioncan occurfromeitheroftwoavailablestatesinthevalenceband.Accountingforthet twospinsassociatedwitheachorbitalstate,wehave4possibletransitionsasshownon theleftofFigure3.8.Afterexcitation,onestateofconductionbandhasbeenoccupied whichleavesonly2possibletransitionsasshownontherightsideofFigure3.8.Asa 61 Figure3.7: L atdelayt=100psforC168at =1.2and ~ ! pump =3.1eV withTAspectrumofintrabandtransitions. 62 Figure3.8:PossibletransitionsofgivingrisetoX 3 andX 4 (left)andX 1 ; 2 ! X 1 ; 2 +X 3 ; 4 i.e.,X 1 ; 2 ! XX 4 7 (right)showingtheofstateonthetransitionstothelowest brightsingletexcitons result,theprobabilityoftheX 1 ; 2 ! X 1 ; 2 +X i transitionis1/2thatofthe0 ! X i transition inthesingle-particlepicture(theratiomaybetwhenincludingcorrelations).The spectrumisshiftedasillustratedinFigure3.9bbecauseofbiexcitonbinding.Westartour modelfromaof5Gaussianpeakstotheground-stateabsorptionspectrumasillustrated inFigure3.9ainred.ThegreencurveinFigure3.9cshowsthemodeledspectrumwithall threecontributionstotheTAspectrumconsidered.Thebetweentheground-and excited-stateabsorptionspectrayieldstheTAspectrum.Theoftheexperimental dataisillustratedinFigure3.9dbythesolidbluecurve.Thereisaslightmismatchatthe peak,butallthecharacteristicsoftheTAspectrumarewell. ToclarifyourinterpretationoftheexperimentalTAspectrum,Isilcalculatedthe spectrum LX 0 ,whichisshownbytheblackcurve(50mVgaussianbroadeningadded)in Figure3.10.Inducedabsorptionat ˘ 1.95eVisprimarilycausedbyX 1 ; 2 ! XX 4 7 .Because oftheorbitallyforbiddennatureofthetransitionX 1 ; 2 ! XX 1 3 ,thereisnosimilarfeature 63 Figure3.9:Stepbystepanalysisoftcontributionson L inTAmeasurement, rightsideofeachspectrumillustratethetcontributiontotheabsorptionspectrum 64 inthecalculatedspectrumnearthe1.7eVpeak.Calculatedinducedabsorptionat0.65eV, 1.0eVand1.35eVroughlymatchthefeaturesseenintheexperimentat0.6eV,0.75eV and1.45eVandfurthersuggestthatcautionshouldbeexercisedininterpretingthe1.45eV featureintheexperimentalspectrumasasignatureofboundbiexcitonsX 1 3 . Figure3.10:Experimentalandtheoreticalabsorptionspectrafromthelowestsingletexciton state.Bluecirclesindicatemeasured L (t=100ps)ofC168.Theredbarsindicate calculatedtransitionsfromthestatesX 1 ; 2 accountingforintra+interbandtransitions.The blacklineisthetheoreticallycalculated,Gaussianbroadened = ( 1 ; 2 0 )assuming equallypopulatedX 1 ; 2 states.Toppanelshowssingletexcitons(lightgrey),band-edge excitons(colorcorrespondingtoFigure3.3)andhigherXXs(darkgrey)accessiblefrom X 1 ; 2 [14]. 65 ToquantitativelydeterminethebiexcitonbindingenergiesofXX 4 7 ,arou- tinewaswrittenwithMathematica.WemodeltheTAspectrumasfollows.Theener- giesoftheground-stateopticaltransitionsaredeterminedbythepositionsofthepeaks of [( d 2 0 ) = ( d ( ) 2 )][110]asillustratedbythebluelineinFigure3.11Suchaderivative Figure3.11:Groundstateabsorptionisplottedinblackdotsandthemodelispresented inredline.Thepeaksof [( d 2 0 ) = ( d ( ) 2 )]indicatesavailabletransitions,whichareshown bydashedlineinthe yieldstwofeaturesinthelowenergyshoulder,oneat1.68eVandasmalleroneat1.86eV, correspondingtoX 1 ; 2 ,whichisopticallybrightenedbycouplingtophonons[11];twointhe 66 peakaround2.1eV,oneat2.09eVandanotherofabouthalftheamplitudeat2.22eV, correspondingtoX 3 ; 4 ;andanotheraround2.6eV,correspondingtoexcitonsthatarenot derivedfromatransitionbetweentheHOMOandLUMOstates. ThespectralwidthsoftheX ! XXand0 ! Xtransitionsaresetasequalsincethemain sourceofadditionalbroadeningofX 1 ; 2 ! XX 4 7 relativeto0 ! X 3 ; 4 isexpectedtobe biexcitonAugerrecombination,where ˝ XX =0.3ps[111].Theamplitudeofthebleach( X i ) ofeachtransition0 ! X i andthebindingofthetwopairsofopticallybright m = 1)XX aresetasparameters.Tominimizethenumberoffreeparameters,weuseacommon valueofthebiexcitonbinding 1 3 )forXX 1 3 and 4 7 )forXX 4 7 sothattheyields anaveragebindingenergyforeachsetofstates.Thefunctioncanbewrittenas: ( ! ) L = L + 4 X i =1 X i [ g X i ( ! ) R i g X i ( ! + i )] (3.2) where L representsaspectrallyinducedabsorptiontakenas-0.05, g X i ( ! )aregaussians takenfromthetotheground-stateabsorptionspectrum, X i representsthebleachofeach transitionandR i istheratiooftheoscillatorstrengthoftheX 1 ; 2 ! X 1 ; 2 +X i transitiontothe 0 ! X i transition.Weassumethat X 1 = X 2 and X 3 = X 4 .AbestoftheTAspectrum inthespectralrangingfrom1.5to2.4eVisshownbythesolidbluecurveinFigure3.10with XX 1 3 =280 30meVand XX 4 7 =140 10meV.Weconstrain R 3 to0.59maximum inthebecauseIsil'scalculationsyield0.43and0.59forthetransitionstothetwo setsofbiexcitonsinthisrange.Theyields R 1 =0 : 22 0 : 01and R 3 =0 : 59 0 : 01.The valueof R 3 isconsistentwiththevalueof1/2expectedfrominasingle-particle pictureasdiscussedabove. 67 TotheoriginofthespectralfeaturesintheexperimentalTAspectrumshown inFigure3.10,ourcollaboratorcalculatetheTAspectrum ( LX ( ! ) 0 ( ! ))shownby thebluecurve.ThetransitionGS ! X 3 ; 4 givesrisetothepeakintheTAspectrumat 2.1eV.TheTAspectrumalsoinvolvesopticaltransitionsfromthephotoexcitedstateto excitedexciton,X ,andbiexciton,XX,statesshownbygreyandblacklinesinthetop panelofFigure3.10andyielding LX ( ! ).Inparticular,thedipat1.95eVisduetoX 1 ! XX 6 7 andX 2 ! XX 4 5 transitionsthatinvolveadditionofabrightX 3 ; 4 excitontothe photoexcitedLX.Thissupportstheinterpretationoftheexperimentalfeatureatthisenergy asduetoasingle-to-biexcitontransition.ThelowerpanelofFigure3.10showsnegative TAcontributionsatenergies E< 1 : 5eV,wellbelowtheXXtransitions,duetointraband excitationsofphotoexcitedelectronsandholesfromLXtoexcitedexcitonstatesX .While thecalculatedandmeasuredenergyrangesoftheX contributionareinqualitativeagree- ment,thebroadeningoftheexperimentalspectrapreventsdetailedcomparisonofenergies ofexcitedexcitonstatesX .ThepositivecontributiontotheexperimentalTAspectrumin Figure3.10at1.7eVcorrespondstoabsorptiontodarkexcitonstatesthatareprohibited inthecalculatedspectra,whichdonotaccountforelectron-phononcoupling. ComparedtoC168,C132missesacornerofthetriangularshapesoitislesssymmetric. OurcollaboratorsdidnotcalculatedthebiexcitonelectronicstructureofC132.Experimen- tally,weapplythesameTAmeasurementsmentionedabovetoC132.TheTAspectrum ofC132from0.4eVto2.3eVisshowninFigure3.12.OnthehighenergyendoftheTA spectrum,theoscillationpatternindicatesthepresenceofbiexcitonsbetween1.7and2.0 eV.WetriedtotheexperimentalTAspectrumofC132similarlyaswedidwithC168,and gotabiexcitonbidingenergyof0.20meVatthemainpeak.Withmorepeaks(transitions) involved,thegroundstatespectrumandTAspectrumofC132havemorestructurethan 68 Figure3.12: L atdelayt=100psforC132at =1.2and ~ ! pump =3.1eV withTAspectrumofintrabandtransitions 69 thoseofC168,whichmakestheinterpretationofexperimentaldatamoretenuousthanin thecaseofC168. 3.3Conclusion Theseexperimentsandtheoriesdemonstratetheinteractionofexcitonsingraphenequan- tumdots.ThedegeneracyofHOMOandLUMOstatesleadstoapairofbrightsinglet excitonsandorbitallydarksingletexcitonsandacorrespondingbandofstronglycorrelated biexcitonstates.ForC168,theexciton-excitoninteractionlowerstheenergyoftheXX 4 7 biexcitons,withvaluesexceedingthebindingenergyofthelowest-energyXX(105meV) measuredin(9,7)single-wallcarbonnanotubes[109].Thestronginteractionbetweencar- riersandrapidbiexcitonAugerrecombination[111]suggestthatGQDscouldbet materialsforcarriermultiplication.Thisallowgenerationofmultiplecarriersoccurringbe- foreexcesscarrierenergyislost,whichcouldbeusefulinphotovoltaic.Suchmultiple-carrier generationcouldmakesitpossiblefortheofasolarcelltoexceedtheShockley- Queisserlimit.Thestrongbindingofhigherbiexcitonstatesdemonstratedinourstudy highlightsthecentralroleofmany-bodyinreduced-dimensionalmaterials.Asstated above,thelesssymmetricC132hasnotyetbeenwellstudiedtheoretically.Furtherstudy onC132mayhelpusbetterunderstandhowstructureandsymmetrytheexcitonand biexcitonbindinginthesequantumsystems. 70 Chapter4 BiexcitonAugerrecombination InthisChapter,TAmeasurementsofAugerrecombinationarediscussedindetail.The studyofthisnonradioactiverecombinationhelpsusbetterunderstandthebiexcitonbe- haviorinGQDsandmayprovidemoreinformationwhenGQDsareusedasmaterialsin photovoltaics. 4.1Carrierrelaxationinquantumsystem Whencarriersinquantumdotsorotherquantumsystemareexcitedbyhigh- energyphotons,theyrelaxbyvariousmechanisms.Oneofthemechanismsisphonone- mission.Suchrelaxationin,forexample,PbSeandCdSequantumdots,isusuallyfast (subpicosecond)[112,113,114].Whenthehighestenergyphononhaslowerenergythan thebetweenquantizedenergylevels,phononemissionispossibleonlybysimul- taneousemissionofmultiplephonons.Theslowingofcoolingwhensimultaneousemission ofmultiplephononsisrequiredisknownasthephononbottleneck[115,116].Afteroptical excitation,carriersareinanonthermaldistributionthatthenthermalizesin10-150fs[16]. Thisisfollowedbyintrabandcarrier-phononscatteringbetween150fsand1ps.After1 ps,electronsandholesrecombineuntiltheequilibriumdistributionisrestoredasillustrated inFigure4.1.Peoplehavedoneexperimentalstudiesonultrafastrelaxationdynamicsin graphene[16,117,118,119,120].Amongthem,Winnerletal.excitedepitaxialgraphene 71 withtexcitationenergies[120].AnOptical-phononbottleneckisobservedinthe excitationrangefrom30-245meV.Carrierrelaxationbecomesslowerwhentheexcitation energyisbelowtheoptical-phononenergy. Figure4.1:Aschematicoftheprocessesbywhichopticallyexcitedelectronandholedistribu- tionsapproachequilibriuminepitaxialgraphene.Distributionatthetimeofexcitationnear theDiracpointshowsanintrinsicthermalpopulationofelectronsandholes.(Reproduced fromRef.[16]). Besidesphonon-assistedprocess,electronandholescanrecombineandgivetheirenergy toathirdcarrier,whichisAugerrecombination(AR)(illustratedascarriercoolingin Figure4.1).Augerrecombinationcanbetlytinquantumsystems 72 thaninbulk.Relaxationofcrystalmomentumconservationinnanoscalelattices[86,121] canleadtorapidARofmulti-excitons[75,122].Inbulksemiconductors,therateofARis tlytbetweendirect-gapandindirect-gapmaterials.Indirect-gapmaterials, ARisathree-carrierprocesswhileinindirect-gapmaterials,ARisafour-particleprocess involvingthreecarriersandemission(orabsorption)ofamomentumconservingphonon [123].Thetruncationoftheperiodiclatticeinnanocrystalleadstorelaxationoftranslational momentumconservation[86].InCdSenanocrystalquantumdots,theBXARlifetime decreasesfrom ˘ 1nstolessthan10psasthenanocrystalradiusdecreasesfrom5to ˘ 1nm[75].Insingle-walledcarbonnanotubes(SWCNTs)withdiametersof ˘ 1nmand lengthsof ˘ 400-500nm,theexciton-excitonannihilationtimeisonlyoftheorderof1 ps[85,109,124].BecauseourGQDshavesimilarsizeasthecircumferenceofcommon SWCNTsbutareonlyoneatomthick,theBXARlifetimeinGQDsabout2nminspatial extentmightthenbeexpectedtobemuchshorterthan0.1ps. 4.2Experimentanddataanalysis 4.2.1Experimentalsetup uPLmeasurementswereperformedasdescribeearlier.AsillustratedinFigure3.5,after thefewps,thereisbarelyanybetweenthesetwotraces.Inaddition,two relativelyslowdecayscanbeobserved( ˘ 10psand ˘ 280ps)duringthisprocessaswell. Weattributethecoolingofhotcarrierswithinlessthan2pstocarrier-phononscattering. This2pswindowmightonlybeanupperlimitbecauseit'sapproachingthelimitofour instrumentresponsetime(around1ps). WeappliedtheTAmeasurementsdescribedinChapter2toexplorethedynamicsof 73 GQDs.TheTAinstrumentresponseis ˘ 130fs.Residualdispersionacrosstheprobe spectrumwasdeterminedbyacquiringTAspectrainaBBOcrystalplacedatthesample position.Toaccountforthesolventresponse,identicalmeasurementswereperformedon acuvettewiththesolventalone.Theresultwassubtractedfromthedataforthe GQDsolution.MeasurementsonsamplesinwhichthesolutionwasstirredtoremoveGQDs fromtheexcitationvolumebetweenlaserpulsesshowednoindynamicsfrom staticsolutions,indicatingthatphotochemicalprocessessuchasthegenerationoflong-lived charge-separatedstatesdidnotdistorttheresults.Wefocusedonpumpexcitationwiththe lowestphotonenergiessincethelodynamicsofthemainabsorptionpeakare whenexcitinglow-energycarriers,makinginterpretationandanalysismorestraightforward. 4.2.2Dataanalysis Thepump-inducedsampleresponseisreportedas L ,where isthechangein absorptioncotand L isthesamplelength.AsdescribedinChapter2,weusedthe CCDtomeasureTAinthevisibleregionofthespectrum.RawdatafromtheCCDhasa wavelengthstepof0.3nmwhiletheresolutionofthespectrometeris2nm.Inthiscase, wewroteaLabviewprogramtoaveragedatain2nmbins.Thedatafromexcitationof C132at ~ ! pump =1.94eVareplottedinFigure4.2(a)and4.2(b)forpumpinthe multiple-andsingle-photonabsorptionregimesrespectively.Forfurtheranalysis,,wefocus onthelargebleacharoundthemainpeakintheground-stateabsorptionspectrumwhere thesignal-to-noiseratioisthehighest.Bleachingatenergiesabove ~ ! pump isaconsequence ofground-statedepletion.Figure4.3showstheevolutionof L upto100psinC132at ~ ! pump =2.34eVattexcitationEachtraceisnormalizedbythesignal atthelongestdelayshowninFigure4.4toweightlow-anddataequally.To 74 Figure4.2: L asafunctionofprobeenergyanddelayforC132pumpedat1.94eV andof(a)1 : 3 10 16 and(b)1 : 4 10 14 photonscm 2 perpulse.Thecolorscale correspondstothedatafort 5.0psin(a).Thedataintheotherthreequadrantsare multipliedbythefactorsshowntomatchthescales. highlighttherelaxationdynamics,wesubtractedthenormalizedsignalatthelongestdelay andpresenttheseinFigure4.3.Thedynamicsofthelowtraces(darkyellow andmagentalegendsinFigure4.4)areidenticalinshapewithintheuncertaintyofthe measurements.Thisisexpectedinthelinearexcitationregime.Whenabsorptionofmore thanonephotonperGQDperpulsecanbeneglected,thedynamicsonlysingle- excitondynamics.Atlowthesignaldecaysbyatotalof20%withaninitialdecay ofseveralpercentsonatimescaleoftheorderof0.1psfollowedbysub-10%decayson timescalesofafewpsandtensofps.Weattributethesedynamicstocoolingwithinthe 75 Figure4.3: ;t ) L versusprobedelayforC132pumpedat1.94eVandprobedat 2.34eV.Solidcurvesarethedescribedinthetext.Thedashedcurverepresentsthe instrumentresponsefunction. 76 low-energyvibronicmanifoldandsolvationdynamics. Figure4.4: ( t ) = ( t long )versusprobedelayforC132intolueneexcitedat ~ ! pump =1 : 94 eVandprobedat ~ ! probe =2 : 34eV : Astheexcitationincreases,newdynamicsemerge.Sp,theamplitude ofthesubpicoseconddecayincreasesbymorethan2ordersofmagnitudewhilethelong- delaysignalincreasesbyonlyafactorof9.WeanalyzedthedatapresentedinFigure4.4 byaglobalwithatriexponentialform: ;t ) L = A f e t=˝ f + A s 1 e t=˝ s 1 + A s 2 e t=˝ s 2 + A 0 (4.1) ThetriexponentialisexecutedinOriginwithitsbuilt-infunction.Toperforma 77 globalonourdata,wesharedtimeconstants( ˝ f , ˝ s 1 and ˝ s 2 )acrossalltracesofthe Thisyieldsthetimeconstants ˝ f =0 : 34 0 : 01ps, ˝ s 1 =2 : 5 0 : 1ps, ˝ s 2 =22 1 psandtheamplitudesplottedinFigure4.5(allreporteduncertaintiesarethestandard errorsoftheAccordingtostudiesofotherstronglynanoscalesystems,atlong delays,multi-excitonscanrecombineandonlysingleexcitonsremain[75,85,124].Hence, A 0 shouldbewellbythePoissonprobabilityforabsorptionofatleastonepumpphoton: A 0 = A 0 ;sat 1 X n =1 ; 2 ;::: P n ( ˙ 0 = A 0 ;sat [1 P n =0 ( ˙ 0 = A 0 ;sat (1 e ˙ 0 )(4.2) where A 0 ;sat isthesaturatedsingle-excitonresponseat100ps, P n ( ˙ 0 isthePoisson probabilityforabsorbing n photonsgivenanaveragenumberofabsorbedphotonsperGQD perpulseof n = ˙ 0 and ˙ 0 istheeGQDabsorptioncrosssectionatthepump wavelength.Theof A 0 (solidblackcurveinFigure4.5)impliesane ˙ 0 ( ~ ! = 1 : 94eV)of7 : 4 10 16 cm 2 .ThisistfromthevaluereportedbyXinetal.[8], whichis9 : 9 10 17 cm 2 .WecanestimatethecrosssectionofGQDsfromtheareaofthe lattice.ThelongestlengthofC168is2.4nm,accordingtowhichwecalculatedthearea ofthetriangular-shapeC168is2 : 49 10 14 cm 2 .SincetheareaofC132is3/4ofC168, usingthesameformula,wegettheresultof1 : 87 10 14 cm 2 fortheareaofC132.At4.1 eV,wherequantumtarelesspronounced,theabsorptionofgrapheneis ˘ 7%[125].ConsideringourexperimentusesalinearlypolarizedpumptoexcitetheGQDs andourGQDsareorientedrandomlyinsolution,1/3ofourdotsareorientedwiththeir two-dimensionallatticeperpendiculartothepolarizationofthepumpandsoarenotexcited intheTAmeasurement.Sothee ˙ 0 ofC132at4.1eVisexpectedtobe1 : 87 10 14 cm 2 0 : 07 2 3 =8 : 7 10 16 cm 2 .Fromtheground-stateabsorptionspectruminFigure 78 1.8,theratioofthecrosssectionofC132at1.94eVtothatat4.1eVis ˘ 0.31to0.05,so ˙ 0 ofC132at1.94eVisexpectedtobe8 : 7 10 16 cm 2 0 : 05 0 : 31 = 10 16 cm 2 ,whichis closetoourresult. Figure4.5:Amplitudes A i fromEq.4.1describingtheof ;t ) L ofC132inFigure 4.3.Theinsetshowsthequantity A f 0 inthetextandassociatedwithmultiexcitons andusesthehorizontalscaleofthemainCurvesshowthedescribedinthetext. A 0 islinearatlowinFigure4.5,andat 3 10 15 photonscm 2 per pulse,theamplitudeofthefastestdecayterm( A f ),issuperlinear.Thisisacharacteristic ofmulti-excitonAR. A f willthenhavecontributionsfrombothsingle-andmulti-excitons andcanbecharacterizedbytheform: A f = A f;X [1 P 0 ( ˙ 0 + A f;MX P > 1 ( ˙ 0 (4.3) 79 where ˙ 0 istheabsorptioncrosssectionofGQDsafterabsorptionofonephoton; A f;X is associatedwiththesmall-amplitudedecayofsingleexcitonsseenmostclearlyatlow asdescribedearlier;and A f;MX isassociatedwithrelaxationofmultiexcitons.Thedashed redcurveinFigure4.5showsaof A f byEq.4.3with A f;X , A f;MX ,and ˙ 0 as parametersand ˙ 0 bytheearlierof A 0 .Thebestyields ˙ 0 =(0 : 41 0 : 07) ˙ 0 . Becauseofthetwo-folddegeneracyoftheHOMOandLUMO,thestrengthofthetransition shouldbereducedbyhalf(inasingle-particlepicture)whenanelectron-holepairiscreated atthebandedge.Ourresultmatchestheexpectedvalueof0.5.Thecontributionof the A f;X termto A f isillustratedbythereddottedcurveandisseentobemuchlessthan themulti-excitoncontributionat > 5 10 14 photonscm 2 perpulse.Themulti-exciton contributiontothefastdecayishighlightedbytheinsetofFigure4.5,whereweplotthe points A f 0 = A f A f;X [1 P 0 ( ˙ 0 Theinsetalsoshowstheterm A f;MX P > 1 ( ˙ 0 (solidcurve).Themultiphotonshowsexcellentagreementwiththedata,andasexpected foraprocessassociatedwithabsorptionofmorethanonephoton, A f 0 isperfectlyaligned withaquadraticscalingwithpump(dashedcurveintheinset)inthelower range. Toruleoutnon-Augerprocessesbeingresponsiblefortherapiddecayobservedathigh ences,weanalyzedthedatabyalternativeapproaches.Oneapproachistousethesamepro- cedureasdescribedabovewithoutnormalizingthedata.Wesubtracted ( t =100ps) L fromthedataandperformedaglobaloftheresultingdatatoatriexponentialformwith sharedtimeconstants.Thisweightsdatamoreheavilythanlodata. ForC132,weobtained ˝ f =0 : 34(0 : 01)ps, ˝ s 1 =2 : 4(0 : 1)ps,and ˝ s 2 =22(1)ps.Another approachistothelowdataindependently,normalizethatatlongdelay,and subtractthenormalizedfromthenormalizeddataatotherThedata 80 arethentoatriexponentialformdecayingtozerowithglobaltimeconstants.ForC132, thisyields ˝ f =0 : 34(0 : 01)ps, ˝ s 1 =2 : 3(0 : 1)ps,and ˝ s 2 =18(1)ps.Thet proceduresproducevaluesoftheprimarytimescaleofinterest, ˝ f ; byonly0.02ps fromthevaluesobtainedfromtheanalysis.InthecaseofC132,neither ˝ s 1 nor ˝ s 2 changessubstantially. SimilarmeasurementswereperformedonC168.Figure4.6showstheTAdataforC168 at ~ ! pump =1.70eV(neartheabsorptionedge)withpumpinthemultiple-and single-photonabsorptionregimesrespectively.Single-wavelengthdynamicsandare Figure4.6: L asafunctionofprobeenergyanddelayforC168excitedat1.70eVat of2 : 0 10 15 photonscm 2 perpulse(a)and2 : 4 10 13 photonscm 2 perpulse (b).Thecolorscalecorrespondstothedatafort 5.0psinpanel(a).Thedatainthe otherthreequadrantsaremultipliedbythefactorsshowntomatchthescales. 81 plottedinFigure4.7.Thedatapresentedintheplotareanalyzedwiththesamemethods Figure4.7:(a) L forC168intolueneexcitedat ~ ! pump =1.70eVandprobedat ~ ! probe =2.21eVataseriesofexcitationfrom2 : 4 10 13 to2 : 0 10 15 photonscm 2 per pulse.SolidcurvesaretriexponentialdescribedbyEq.4.1,andthedashedcurveisthe instrumentresponsefunction. weuseforC132.Dynamicdatanormalizedbythesignalatthelongestdelayareshownin Figure4.8toweighlow-anddataequallyaswell. UsingthesameprocessasdescribedforC132,thetoEq.4.1yield ˝ f = 0 : 33 0 : 01ps, ˝ s 1 =2 : 9 0 : 2ps,and ˝ s 2 =39 3ps.Theamplitudes A 0 , A f , A s 1 ,and A s 2 areplottedinFigure4.9,andthe A f arefoundtobewelldescribedbythesamemodel weusedtodescribedC132. Similarly,weusedthesamethreeapproachesasweusedforC132toanalyze theC168data.Theprocedureoutlinedabovewithoutnormalizationyields ˝ f = 82 Figure4.8: ( t ) = ( t long )versusprobedelayforC168intolueneexcitedat ~ ! pump =1 : 70 eVandprobedat ~ ! probe =2 : 21eV : 0 : 31(0 : 01)ps, ˝ s 1 =2 : 1(0 : 1)ps,and ˝ s 2 =24 : 9(0 : 3)ps.Thethirdprocedureyields ˝ f = 0 : 32(0 : 01)ps, ˝ s 1 =2 : 3(0 : 1)ps,and ˝ s 2 =15 : 6(0 : 5)ps.Ingeneral, ˝ s 1 and ˝ s 2 fromt modelsdonotmatchwell.Thisiscausedbythetweightstothe datainthesetapproaches.tproceduresyieldvaluesof ˝ s 1 rangingfrom2.1to2.9psandvaluesof ˝ s 2 rangingfrom16to39ps. ˝ f barelychanges, though,becauseitispredominantlyaconsequenceofthegenerationofmultipleexcitons, whereasthedynamicsat ˝ s 1 and ˝ s 2 havecontributionsfrombothsingle-andmultiple- photonabsorption.Thevariationsintheslowdecaysmayresultfromtheincreasedexcess heatdepositedintheGQDlatticefollowingARinthemultiple-excitonregime. Figure4.9showsthedependenceoftheparameters A 0 ;A f ;A s 1 ; and 83 Figure4.9:Amplitudes A i fromEq.4.1describingtheof ;t ) L ofC168inFigure 4.7.Theinsetshowsthequantity A f 0 inthetextandassociatedwithmultiexcitons andusesthehorizontalscaleofthemainCurvesshowthedescribedinthetext. A s 2 derivedfromthetriexponentialfromEq.4.1tothedataofFigure4.7.The dependenceof A 0 is(solidblackcurve)bythePoissonformdescribedbyEq.4.2and correspondstoaneabsorptioncrosssectionof ˙ 0 ( ~ ! =1 : 70eV)=(2 : 1 0 : 3) 10 15 cm 2 .Aof A f totheformdescribedbyEq.4.3isshownbythedashedredcurve andcorrespondsto ˙ 0 =(0 : 6 0 : 15) ˙ 0 .Theratioalsomatchestheexpectedvalueof0.5 aswediscussedforC132.Thecontributionto A f ofthetermassociatedwiththe amplitude A f; X inEq.4.3isshownbythereddottedcurve,whichillustratesthatthe multiexcitoncontributionismuchgreaterthanthesingle-excitoncontributionat 1 10 14 cm 2 : TheinsetofFigure4.9explicitlyshowsthemultiexcitoncontribution 84 totheamplitudeofthefastdecaybyplottingthequantity A f 0 = A f A f; X [1 P 0 ( ˙ 0 ; i.e.,subtractingthesingle-excitoncontributiontothefastdecayrepresentedfrom A f .The solidcurveshownintheinsetisthecontributionofthemultiexcitontermintheofthe A f endence. BesidestheTAmeasurementsinthevisiblespectralrange,wealsoprobedthesamplein theinfraredrangeasdiscussedinChapter3.Thisprobestheintrabandtransitionsofthe GQDs.WeexcitedC168at3.1eVandprobedthesampleattheinducedabsorptionpeak (0.75eV).Figure4.10showstheexcitationdataofC168at ~ ! pump =3.10eVwithpump inthemultiple-andsingle-photonabsorptionregimesrespectively.Comparingwith theinterbanddataweshowedpreviously,wedidnotmuchinthefewps betweenmultiple-andsingle-photonabsorption.ItisunclearwhyAugerrecombinationis lesspronouncedatthisenergy.Furtherstudyisneededtobetterunderstandtheintraband transitionsoftheGQDs. 4.2.3DiscussiononAugerrecombinationofGQDs Theendenceof A f (fastdecayamplitude)supportstheassignmentofthe superlinearendentinitialTAresponsetoBXsdecayingwithanARlifetimeof ˝ AR;BX = ˝ f =0 : 3ps.InourGQDs,ARgeneratesahotexcitonthatrapidly(i.e., fasterthan ˝ f )coolstothelowest-energyexcitonstateanddepositsinthelatticeabout onebandgapofenergy.Afterthis,thelaststageofelectronicandvibroniccoolingor solventorientationproceedsonsimilartimescales( ˝ s 1 and ˝ s 2 )asobservedatlowexcitation TheenergyofthepumpphotonsintheTAexperimentsissettobeslightlyhigher thanthebandgapofthesample.InthecaseofabsorptionofasinglephotonbyaC132 85 Figure4.10: ( t ) = ( t long )versusprobedelayforC168intolueneexcitedat ~ ! pump = 3 : 10eVandprobedat ~ ! probe =0 : 76eV : GQD,only ˘ 0.2eVofexcessenergyisdissipatedduringrelaxationtothelowest-energy excitonstate.ARisanadditionalsourceofheatingofthecarriers,andconsequently,the amplitudes A s 1 and A s 2 shoulddependsuperlinearlyonasweobservedfromtheTA experiment. Thefactthatweobservedfastdecayamplitudesthathavesuperlinearendence isnotttoassign ˝ f toBXsdecaybyAR.Itispossiblethattwo-photonabsorption createsahigh-energysingleexcitoninsteadofabiexciton. ˝ f couldbethetimeconstant ofthecoolingprocessofthishigh-energysingleexcitontothebrightexcitonpeak(570nm 86 forC132)ortothelower-energyexcitonstates.ItisalsopossiblethatARoccursonatime scalebelowthetemporalresolutionofourmeasurementsandisfollowedbycoolingofahot single-excitonwiththetimescaleof ˝ f .FurtheranalysisoftheTAspectrumcanruleout thesealternatives.If ˝ f correspondstocoolingtothebrightsingletexcitons,thebleachof themainpeakshouldincreasewithtimeconstant ˝ f beforedecreasingonthetimescalefor coolingintothelower-energystates(X 1 andX 2 forC132andC168,respectively).If ˝ f was thetimescaleforcoolingfromthemainpeaktoX 1 andX 2 ,thenthebleachofthelatter shouldbuilduponthetimescale ˝ f ,whichwedidnotobserve. DespitethecommonlyobservedreductionofARlifetimeswithdecreasingnanoparticle size,furtherconsiderationsallowustounderstandwhyGQDBXARlifetimescanbesimilar toBXARlifetimesinSWCNTs.ARofdirectlygeneratedBXsin(9,7)SWCNTsoccursin 0.5ps[109].The\universality"ofBXARlifetimesinnanocrystalquantumdotsislargely duetorelaxationofmomentumconservation[86].ASWCNTretainslong-rangeperiodicity inonedimension,butin(6,5)SWCNTsthe\diameter"ofthesingleexcitonalongtheaxisof theSWCNTisonly ˘ 2.4nm[72],whichissimilartothespatialsize( ˘ 2 : 4nm)oftheGQDs studiedhere.Moreover,theBXlengthinSWCNTsispredictedtobesimilartothelength ofasingleexciton[126].ThesmallsizeoftheSWCNTbiexcitonmeansthatsubstantial amplitudeoveralargerangeofmomenta.Fromthisperspective,momentumconservation shouldnotbemuchmorerestrictiveforBXdecayinaSWCNTthaninGQDsof2-3nm (thelongestedgelengthforC132andC168is2.4nm).Otherprominentfactorsdetermining theBXARratearethestrengthofthescreenedCoulombinteractionandthesingle-exciton densityofstates(DOS X )attheenergyoftheBX.Coulombicinteractionsmaybesomewhat strongerinGQDscomparedtoSWCNTsbecauseoftheweakerscreeningassociatedwitha latticeofmorelimitedextent[127].WithrespecttoDOS X ,inSWCNTs,the E 22 transition 87 isneartwicetheenergyofthe E 11 resonance,butthereisnotsuchageneralenhancementin theGQDDOS X neartheenergyofthelowest-energyBX.Morepreciseassessmentsofthese factorswillrequiredetailedtheoreticalcalculations.Inparticular,thesimilarityinBXAR timesforC132andC168iscontrarytotheusualreductioninARlifetimewithreduction inquantumdotsize.Thismaybebecauseoftheinthedensityofstates (DOS X hot )inC132orC168and/orthesymmetriesofinitial(BX)and(X hot )states. ThelattermaybeimportantgiventhedistinctsymmetriesofC132andC168becauseit impactsthemagnitudeoftheARtransitionmatrixelementbetweenapairofstates[128]. 4.3Conclusion Insummary,wemeasuredBXARlifetimeof ˘ 0.3psingraphenequantumdotswith 132and168carbonatoms,theresultsofwhichsuggestthattheGQDshavethepotentialto enhancephotovoltaicthroughcarriermultiplication.Observationofmoderately fasterBXARinGQDssuggeststhatGQDsmaydemonstrateCMcomparable toorgreaterthanthoseofSWCNTs[129,130].Electrontransferfromrutheniumdyes [131,132]andPbSenanocrystalquantumdots[76]toTiO 2 occursontimescaleslessthan 100and ˘ 10fs,respectively,andhot-electroninjectionfromC132covalentlylinkedtothe TiO 2 (110)surfacehasbeenmeasuredtooccurwithatimeconstant < 15fs[133].This impliesthatthe0.3psBXARlifetimesmeasuredheredonotruleouttheextractionof multiplecarriers. 88 Chapter5 Summary 5.1Generalsummary Themethodsandresultsofthisresearchestablishafoundationforexploringelectronic structureinGQDsandotherlow-dimensionalmaterials.Thestudyofelectronicstructure inthisworkhasbeenrestrictedtoGQDsofC132andC168.Becauseofthemoresymmetric formofitslattice,C168hasaless-structuredspectrumandtransitionsthatareeasierto interpret.InChapter3,biexcitonbindingofC168ismeasuredopticallyandcalculated theoretically.Thebindingfromourmeasurementisrelativelystrongandiscomparablewith single-wallcarbonnanotubes.Chapter4reportsonmeasuredAugerrecombinationrates inC132andC168.Themodelmatchestheexperimentaldataquitewell.AstheAuger processisfast(0.3ps)intheGQDs,thissuggeststhattheinverseprocess,namelycarrier multiplication,canbet.AndelectroninjectionfromC132covalentlylinkedtothe TiO 2 (110)surfacehasbeenmeasuredtooccuronatimescaleof < 15fs,whichismuchless thanAugertimefromourstudy,fastAugerrecombinationmightnotpreventtheextraction ofmultiplecarriersfromaGQD.Combinedwiththefactthattheabsorptionspectrumof theGQDscoveringmostofUVandvisiblerangeofsolarspectrum,GQDsmaybegood candidatesforsensitizersinsolarcells. 89 5.2Futurework Thecompletionofthisworkopensthedoortothenextsetofexperimentsexploring GQDs.Therearestillmanyissuesworthfurthertheoreticalandexperimentalstudy.Based onouranalysis,theelectron-phononinteractionisimportanttofullyunderstandthestruc- tureandbehaviorsoftheGQDs.Theintensityofphotoluminescencefeaturesisstrongly dependentontheexcitationwavelengthbasedonRiesen'sexperiment[11]andMullenetal [67].Thisbehaviorisstrongevidenceofthepresenceofimpurityspecies.Furtherinsights maybepossibleifwecanseparateimpuritiesfromthemainspecies.Giventhefactthatthe studiedGQDshavelowPLquantumyield,itisalsoimportanttounderstandtheoptically allowedandforbiddenstatesinthesystem.Inourstudy,wefocusmoreonthesymmetric GQDC168asithasfewerdistincttransitionswhichcanbebetterdistinguished.Theless symmetricGQD,C132,requiresmoreinvestigationtounderstanditscomplicatedabsorp- tionspectrumandelectronicstructure.Thiscanalsohelppeoplebetterunderstandhow structureandsymmetrytheexciton/biexcitonbehaviorsinsuchalow-dimensional system. 90 BIBLIOGRAPHY 91 BIBLIOGRAPHY [1] USB2000+FiberOpticSpectrometerInstallationandOperationManual,docu- mentnumber270-00000-000-02-201107,OceanOptics,http://oceanoptics.com/wp- content/uploads/USB2000-Operating-Instructions1.pdf. 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