Many-body Effect in Optical Properties of Monolayer Molybdenum Diselenide - Xiao et al. - Unknown - Unknown

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Many-bodyEffectinOpticalPropertiesofMonolayerMolybdenumDiselenideKeXiao1,‡,TengfeiYan2,‡,*,QiyeLiu1,SiyuanYang1,ChimingKan1,RuihuanDuan3,ZhengLiu3,XiaodongCui1,**1DepartmentofPhysics,UniversityofHongKong,HongKongSAR2GraduateSchoolofChinaAcademyofEngineeringPhysics,China3SchoolofMaterialsScienceandEngineering,NanyangTechnologicalUniversity,SingaporeSupplementaryInformation1.Structureandphotosofthedeviceusedintheexperiment.2.ComparisonofPLandreflectionspectrumatchargeneutral.3.Reflectionspectraatvariousgatebias.4.TemperatureandexcitationintensitydependenceofPL.5.AbsorptionspectrumasafunctionofgatebiasaccordingtoK-Krelation.6.ChargecarrierdensityandFermienergycalculation.7.Interactionstrengthdifferencebetween1sand2sstates.8.Typicalphotoluminescencedatafittingresultunderdifferentgatebias.S1

11.Structureandphotosofthedeviceusedintheexperiment.FigureS1.(a)Schematicstructureofthedeviceusedintheexperiment.Greenrectanglesrepresenth-BN,whichcovermonolayerMoSe2onbothsides.TheorangepartisMoSe2andthepurplepartsaregraphite.(b)MicroscopicimageforatypicalsampleonSiO2/Sisubstrate,reddashedlinerepresentstheshapeofmonolayerMoSe2.(c,d)Atomicforcemicroscopeanddarkfieldopticalmicroscopeimageofthesamesample,respectively.2.ComparisonofPLandreflectionspectrumatchargeneutral.FigureS2.PhotoluminescenceandreflectionspectrameasuredatVG=0V.S2

23.Reflectionspectraatvariousgatebias.FigureS3.ReflectionspectraofmonolayerMoSe2collectedatvariousgatevoltages.Thesampleischargeneutralat??~10?.4.TemperatureandexcitationintensitydependenceofPL.FigureS4.(a)Temperature-dependentPLspectraofmonolayerMoSe2withfeaturesnearAexcitonandBexcitonplottedinleftandrightfigure,respectively.(b)Excitationpower-dependentPlspectranearBexciton.S3

35.AbsorptionspectrumasafunctionofgatebiasaccordingtoK-Krelation.FigureS5.AbsorptionspectrumofmonolayerMoSe2asafunctionofgatebiascalculatedbyK-Krelations.6.ChargecarrierdensityandFermienergycalculation.Thelateraldimensionofgraphene-BN-MoSe2-graphenesamplestructureistypicallyintheunitofmicrometer.Thethicknessofh-BNwhichactsasdielectriclayeris?~25??,waysmallercomparedtothelateraldimension.Thestructureisthusconsideredasaparallel-platecapacitor.Thechargedensity?inducedbyagivenvoltageis???/??.ThequantumcapacitanceofMoSe?2?(?)isneglected2?=?duetoexistenceofopticaldoping.TheFermienergyofcarriercouldbecalculatedinthesimplestparabolicapproximationofbandstructure.???ℏ2??=?(?)=?∗,inwhich?(?)isdensityofstatesintwo-dimension.Wetaketheeffectivemasscalculatedinref.[Kormányos,A.,etal.(2015).2(2):022001].?1=0.58?,?2=0.5?and?1????ℎ=―0.6??.Thespinsplitinvalencebandenergyisconsideredtobeintheorderof200???.Thecurrentstructurecannotaffordforthegatevoltagelargeenoughtoprovidethedopingholedensitytofillthespinsplitvalenceband.Thespinsplitoftheconductionbandis~20???equalingtotheelectrondensitydopedwithagatevoltageof~6.3?.S4

4FigureS6.(a)ElectronFermi-Diracdistributionat77Kcalculatedatdifferentgatevoltage.(b)Electrondensityattheupperspinsplitconductionband(blue),lowerband(orange)andtotal(yellow)asfunctionsofgatevoltage.(c)Densityofelectronsintheconductionbandasafunctionofenergy(E-EC)plottedatvariesgatevoltages.(d)Densityofelectronsintheupperspinsplitconductionbandasafunctionofenergy(E-EC)plottedatvariousgatevoltages.7.Interactionstrengthdifferencebetween1sand2sstates.Theattractionforcebetweenanexcitonandanelectroncanbeconsideredclassically:adipolelocatedintheelectricalfieldinducedbyanelectron,whereadipolemomentis℘=α?,isexcitonpolarizability,α?istheelectricalfieldinducedbyanelectron.[Efimkin,D.K.andA.H.MacDonald(2017).PhysicalReviewB95(3):035417.]Perturbationtheoryisusedtodeterminethepolarizabilityof1sstateand2sstate.α?1=―???(S1)For1sstate,sincethesstatewavefunctionhasacentralsymmetry,thefirstorderofperturbationiszero:?1=⟨0,0│?1│0,0⟩=0.Thesecond-ordertermcanbewrittenas:|⟨0,0│?│?,?⟩|2|⟨0,0│?│?⟩|21?11?2=∑?,??00?―????+∑??00?―??(S2)?Thefirsttermisvirtualtransitionsfrominitialstatetootherstate,thesecondtermdescribesvirtualionizationtransitions.Weget?2?2?2?1??1?2?2=―0.1973?00=―2?(S3)?with?2?2??1?=0.400(S4)??Toestimationof2sstate,nondegenerateperturbationtheoryisstillusedsince2sand2pstatesarenotS5

5degeneratetakingscreeningeffectintoaccount.Weassumetheenergydifferencebetween2sand2pstateisabout50meV.Asamatteroffact,for2sstate,theexcitedstatessuchas3sstate,3pstates,etc.mergewiththecontinuumandonlythesecondtermsurvives.|⟨1,0│?│?,?⟩|2|⟨1,0│?│?⟩|22?11?2=∑?,??10?―????+∑??10?―??(S5)?Weget??2?22?2?2??2=―2?with?2?=11?10(S6)???=2(S7)4???Bysimpleregularization:?1??21?1?(?)=―()(S8)???24??(?2+?2)2?1???1??21?1?=????(?=0)=2(4??)?2(S9)?1?Theratioof1sstateto2sstatebindingenergyisabout2~4takingthescreeningeffectintoγconsideration.[He,Keliang,etal(2014).Physicalreviewletters113(2):026803.],[Goryca,M.,etal(2019).Naturecommunications10(1]:1.]isintroducedtoaccountfortheratio?betweentheBohrradiusof2sstateand1sstate,herewetakeγ=3,?=3.9.??1?0012?2???1?=?1??2≈28?1?0?2≈5.4(S10)8.Typicalphotoluminescencedatafittingresultunderdifferentgatebias.FigureS7.Typicalphotoluminescencespectrafittingunderdifferentbias.S6