General, Strong Impurity-Strength Dependence of Quasiparticle Interference - Hong, Lihm, Park - 2021 - Unknown

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pubs.acs.org/JPCCArticleGeneral,StrongImpurity-StrengthDependenceofQuasiparticleInterference,∥,∥Seung-JuHong,*Jae-MoLihm,*andCheol-HwanPark*CiteThis:J.Phys.Chem.C2021,125,7488−7494ReadOnlineACCESSMetrics&MoreArticleRecommendations*sıSupportingInformationABSTRACT:Quasiparticleinterference(QPI)patternsinmomentumspaceareoftenassumedtobeindependentofthestrengthoftheimpuritypotentialwhencomparedwithotherquantities,suchasthejointdensityofstates.Here,usingtheT-matrixtheory,weshowthatthisassumptionbreaksdowncompletelyeveninthesimplestcaseofasingle-siteimpurityonthesquarelatticewithansorbitalpersite.Then,wepredictfromfirst-principles,averyrich,impurity-strength-dependentstructureintheQPIpatternofTaAs,anarchetypeWeylsemimetal.ThisstudythusdemonstratesthattheconsiderationofthedetailsofthescatteringimpurityincludingtheimpuritystrengthisessentialforinterpretingFourier-transformscanningtunnelingspectroscopyexperimentsingeneral.■3INTRODUCTIONScanningtunnelingmicroscopy(STM)playsakeyroleinJs(;)qkωρ=−∑∫d(;)(iikωρkq;)ωi=0(3)nanosciencebecauseitdirectlyprobesthesurfacetopographyandelectronicdensityofstateswithasub-nanometerwhere1,2resolution.Inparticular,Fourier-transformscanningtunnel-ingspectroscopy(FT-STS)hasbeenwidelyusedtoexaminethe1surfaceelectronicstructuresinmomentumspace.3Theρa(;)kkω=−limTrIm+σωaG0(;i)ηπη→0(4)experimentalresultsofFT-STSareinterpretedastheconsequenceofquasiparticleinterference(QPI)inducedbyisthespindensityalongthea-thdirection(a=1,2,3)withσaasimpuritiesordefectsonthesurface.Theoretically,aQPIpatternthePaulimatrix.Inoueetal.usedsurface-projectedSSPto5isdefinedastheFouriertransformoftheperturbationtotheexplainthemeasuredQPIpatternsinTaAs.TheSSP4approximationgoesbeyondtheJDOSapproximationbylocaldensityofstates(LDOS)inducedbytheimpurity.The5−9forbiddingthescatteringsbetweenoppositelypolarizedpureQPIpatternsofvariousmaterialsincludingWeylsemimetals,10−149,15,16spinstates.Still,SSPisindependentofimpurity-specifichigh-Tcsuperconductors,andtopologicalinsulatorsDownloadedviaKINGABDULLAHUNIVSCITECHLGYonMay14,2021at16:55:21(UTC).Seehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.arebeingactivelystudied.properties.OneoftenanalyzestheobservedQPIpatternsbycomparingTheBornapproximationisalsofrequentlyusedtosimulate3theQPIpattern.UndertheBornapproximation,onetakesintothemtothejointdensityofstates(JDOS)accountonlythefirst-ordereffectofthescatteringpotentialonJ(;)qkωρ=−∫d00(;)(kωρωkq;)theGreenfunction.Consequently,theQPIpatterncomputed(1)withintheBornapproximationisindependentofthestrengthofHere,thescalarimpuritiesuptoanoverallprefactor.18Inthispaper,weusetheT-matrixmethodtoinvestigatethe1effectofthestrengthofsimplenonmagneticscalarimpuritiesonρ0(;)kkω=−limTrIm+G0(;ωηi)πη→0(2)theQPIpattern.Contrarytocommonbelief,theQPIpatternisthesurfacedensityofstatesatwavevectorkandenergyωintheabsenceofimpuritieswithG0thesurfaceGreenfunction.Received:February15,2021SincethereisnoreferencetothepropertiesoftheimpuritiesinRevised:March14,2021eq1,theJDOSapproximationneglectstheimpuritydependencePublished:March30,2021oftheQPI.AnothercommonlyusedapproximationoftheQPIpatternis17thespin-dependentscatteringprobability(SSP)©2021AmericanChemicalSocietyhttps://doi.org/10.1021/acs.jpcc.1c014107488J.Phys.Chem.C2021,125,7488−7494

1TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleFigure1.Structureofthesystemsexaminedinthispaper.(a)Single-orbitalsquarelattice.(b)As-terminatedsemi-infinitesurfaceofTaAs.dependsverysensitivelyontheimpuritystrength.Thus,thewheretheTmatrixisdefinedasapproximationsthatdonotcapturesuchimpurity-strength−1dependence,suchasJDOS,SSP,andtheBornapproximation,X()ωω=[−ZMK0()Z](9)mayfailtodescribeeventheQPIarisingfromsimplewithMtheidentityoperator.Fortheimpuritypotentialdefinednonmagneticimpurities.WefirstshowthattheQPIpatternsineq7,theTmatrixbecomesofasimplesquarelattice[Figure1a]dependdramaticallyontheimpuritystrength.AfterfullyunderstandingthephysicsofthisX(,;)RR′=ωδTRR,0δ′,0simplestsystem,welookintoTaAs[Figure1b],anarchetype−1WeylsemimetalandfindthatitsQPIpatternalsodepends=[−VPTTPK0(,;)00ωδPVT]RR,0δ′,0significantlyontheimpuritystrength.(10)■wherethereal-spaceGreenfunctionisdefinedasMETHODSTheGreenfunctionofasemi-infinitesurfaceintheabsenceof1−−ik()RR′K0(,;)RR′=ωω∫d(keG0k;)impuritiesisdefinedasΩBZ(11)+−1K00()(ωω=−−i0L)(5)withΩBZastheareaofthe2DBrillouinzone.TosimulateSTS,weprojecttheGreenfunctiontothewhereL0istheHamiltonianofthepristinesemi-infinitetopmostatomiclayer,assumingthatonlytheLDOSofthesurface.TheGreenfunctionofthepristinesurfaceisblocktopmostatomiclayerismeasured.Thechangeinthesurfacediagonalwithrespecttothein-planewavevectors.TheGreenLDOSinducedbytheimpurityreadsfunctionforawavevectorkis1+−1Δρ=(;)RωlimIm∑⟨wwiiRR|ΔK(ωη−i)|⟩G00(;)(kωω=−−i0TLTkk)(6)πη→0+i∈topmostwhereTkistheprojectionoperatorontothesubspacewithwave(12)vectork.where|wiR⟩isalocalizedorthogonalbasisfunction,suchastheNow,letusintroduceascalarimpuritywithpotentialZ.InWannierfunction.TheQPIpatternΔρ(q;ω)istheFourierthispaper,weworkusingthetight-bindingdescriptionofthetransformofΔρ(R;ω)HamiltonianandconsiderthecasewheretheimpuritypotentialishiftstheonsiteenergyofallorbitalsofthetopmostatomintheΔρ(;)qkω=−lim+∑∫d⟨wikq−|centralin-planeunitcell.Concretely,theimpuritypotential2πη→0i∈topmostmatrixelementfororbitalsmandninthein-planeunitcellsR†andR′is((i)Δ−−KKωηΔ−|(i)ωη)wik⟩(13)ZmnRR,T′′=VP()mn,δδRR,0,0(7)IfthesystemisinvariantunderaC2rotationwithrespecttoz[Figure1],eq13reducestowithVastheimpuritystrength.Here,PTisaprojectionoperator1totheorbitalsinthetopmostatom,sothat(PT)m,nis1ifm=nΔρ(;)qω=limIm+andmisanorbitalthatbelongstothetopmostatom,and0πη→0otherwise.∑∫d(k⟨|wwiikq−Δ−|Kωηi)k⟩UsingtheT-matrixformalism,thechangeintheGreeni∈topmost(14)3functioninducedbytheperturbationZreadsUsingeqs8,10,and14withtheGreenfunctionofthepristineΔKKKK()ωωωω=−=()00()()()()XωK0ω(8)surfacecomputedfromtheiterativemethod,19onecan7489https://doi.org/10.1021/acs.jpcc.1c01410J.Phys.Chem.C2021,125,7488−7494

2TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleFigure2.(a)JDOS,(b,c)absolutevalueoftheQPIpattern,and(d)absolutevalueoftheQPIpatterncomputedwithintheBornapproximationforthesingle-orbitalsquarelatticewitht=−1eVatω=3.5eV.Thedashedcurvesarecircleswithradiusq=2kc.efficientlycomputetheQPIpatterninducedbyalocalizedpotentialimpurity.Now,asthesimplestexample,letusconsiderthesquarelatticewithones-likeorbitalpersiteandthenearest-neighborhoppingwithhoppingintegralt.Theenergydispersionreadsϵ=[kx2cos()cos()takk+ay](15)whereaisthelatticeparameter.Inthissingle-orbitalcase,theTmatrixbecomesacomplexnumberandsoisG0(k;ω).TheQPIpatternbecomes1Δρ=(;)qqωlimIm[TΠ−(;ωηi)]+πη→0(16)whereΠ=(;)qkωω∫dGG00(;)(kk−q;)ω(17)■RESULTSANDDISCUSSIONFigure2a−cshowsthecomputedJDOSandQPIpatternsofthesingle-orbitalsquarelatticewitht=−1eV,respectively.Thequantitiesarecalculatedatω=3.5eV.Atthisenergy,theconstant-energycontourisapproximatelyacirclewithradiuskc=0.225π/a.ComparingtheJDOSandtheQPIpatterns,wefindthattheintensitiesoutsidetheq=2kccircleareclearlydifferent:theJDOSiszerooutsidetheq=2kccircle,whiletheQPIpatternsarenonzero.Moreimportantly,wefindalargedifferencebetweentheQPIpatternsforV=1eVandV=3eV.WithintheBornapproximation,theTmatrixisreplacedbytheimpuritypotentialV.Therefore,thecomputedQPIpatternFigure3.(a)RealandimaginarypartsofT[eq10]vsV.(b)AbsolutebecomesindependentoftheimpuritystrengthVexceptforaand(c)signedvaluesoftheQPIpatternsforvariousvaluesofV.(d)Realand(e)imaginarypartsofΠ[eq17].Alltheresultsarefortheproportionalityconstant.TheQPIpatterncomputedwithinthesingle-orbitalsquarelatticewitht=−1eVatω=3.5eV.ThedashedBornapproximation[Figure2d]ismoresimilartotheexactQPIverticallinesin(b−e)indicateq=2kc.WenotethatΔρ(q;ω)isreal-patternforsmallV[Figure2b]thanthatforlargeV[Figure2c]valued[eq14].sincetheBornapproximationbecomesmoreaccurateforweakerperturbations.However,theQPIpatternofastrongerimpurity[Figure2c]considerablydeviatesfromtheBornapproximation.UsingtheobtainedTmatrix,wenowaimtounderstandtheVTounderstandtheimpurity-strengthdependenceoftheQPIdependenceoftheQPIpatterns[Figure2b,c].InFigure3b,wepatternsshowninFigure2,weexaminetheTmatrix.Figure3ashowstherealandimaginarypartsofTasafunctionofV.Also,showtheabsolutevalueoftheQPIpatternsalongthelineqy=0.weplotTatV=±∞,whichcanbeusedtosimulateavacancyatThisseeminglycomplicateddependenceoftheQPIpatternontheimpuritysite.Atsmall|V|,therealpartofTcanbeVcanbesimplyunderstoodbycomparingthesignedQPIapproximatedasVanditsimaginarypartisnegligible,indicatingpatterns[Figure3c]andtheΠfunction[eq17andFigure3d,e]theadequacyoftheBornapproximation.However,atlarge|V|,20(seealsoFigureS1).ThecomplementaryfeatureofReΠandbothReTandImTaresizable.TheQPIsignalisproportionalImΠisthekeytounderstandingtheimpurity-strengthtotheimaginarypartofTΠ[eq16].Thus,onlytheimaginarypartofΠcontributestotheQPIpatternatsmall|V|,whilebothdependenceoftheQPIpattern.Usingeq16,onecanwriteImΠandReΠcontributeatlarger|V|.theQPIsignalas7490https://doi.org/10.1021/acs.jpcc.1c01410J.Phys.Chem.C2021,125,7488−7494

3TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleFigure4.(a)SSPand(b−e)absolutevaluesoftheQPIpatternsfortheAs-terminatedTaAssurfacewithasingleon-siteimpurityforωattheFermilevel.Vrepresentstheon-sitepotentialshiftofthesurfaceAsporbitals.1potentialactsonmultipleorbitalsofthetopmostAsatom,theTΔρ(;)qqω=limImRe[TΠ(;ωη−i)πη→0+matrixisnowamatrix,notacomplexnumber.Hence,asimpleanalysislikeeq18isnotpossible.However,asinthesquare+Π−ReTIm(;qωηi)](18)latticecase,thepositionsofthepeaksaremainlydeterminedbythepristineGreenfunctions.Also,theimpurity-strengthApparently,theQPIsignalisalinearcombinationofReΠanddependenceoftheTmatrixistheoriginofthecomplexImΠwithcoefficients−1/πImTand−1/πReT,respectively.impurity-strengthdependenceoftheQPIpatternsshownclearlyForV=1eV,since|ImT|<|ReT|[Figure3a],ImΠdominatesinFigure4.thequalitativefeaturesoftheQPIpattern.However,forV=3Inreality,thereareusuallymanykindsofimpuritiesintheeV,wefind|ImT|≫|ReT|[Figure3a],andthustheQPIFourier-transformedregion.Thus,itisnecessarytotakeaccountpatternisalmostexclusivelydeterminedbyReΠ.Thisanalysisofmulti-impurityeffectsinQPIpatterns.TodealwithmultipleclearlydemonstrateswhythesignedQPIpatternsforV=1eVimpurities,onemustconsidertwodistinctfactors:multipleandV=3eV[Figure3c]resemble−ImΠ[Figure3e]and−RescatteringsfromdifferentimpuritiesanddifferentpositionsofΠ[Figure3d],respectively.Inbrief,thepositionsofthepeakstheimpurities.Forthefirstpart,wenotefirstthatmultiplearedeterminedbyΠortheGreenfunctionG0(k;ω)[eq17],scatteringsfromasingleisolatedimpurityarewelltakencareofwhiletheirintensityandshapearedeterminedbyT.bytheT-matrixmethod.TheassumptionofneglectingmultipleInpassing,wenotethatref21investigatedtheimpurity-scatteringsfromdifferentimpuritiesiswellacceptedbythestrengthdependenceoftheQPIpatternsoftheparent13,15,26communityifthedensityofimpuritiesisnottoohigh.compoundsofiron-pnictidesuperconductors,magneticmateri-Forthesecondpoint,theeffectofthedifferentpositionsofalswithaspindensitywaveorder.Thesubjectsandfindingsofmultipleimpuritiescanbetakenintoaccountbyconsideringtheourworkandref21aretotallydifferentasinthatwork,thepeaksphasefactoreiq·Ri,whereRisthepositionofthei-thimpurity.intheenergypositionofthespin-densitywaveareofcrucialiForexample,inref27,theauthorsconsideredtheimpurityimportance.WealsonotethatthefindingsinourworkarepositiondependenceoftheQPIpatternsbymultiplyingthisrelevanttogeneral,evennonmagneticsystems.phasefactoreiq·RiwiththesingleimpurityQPIpattern.Inordertoseetheeffectofreal-spacemaskingperformedasapost-processingoftheexperimentaldata,15weappliedthereal-WhenwecalculatethetheoreticalQPIpatterns,wemustspacemaskingtoourcalculatedQPIpatterns.Wefindthatthereflectthepositionsofexperimentallyobservedimpurities.Forreal-spacemaskingstillpreservesmuchoftheimpurity-strengthmultipleimpurities,theresultantQPIpatternisgivenas20∑eiq·RiΔρwhereΔρdenotestheQPIpatternifonlythei-thdependenceoftheQPIpatterns(seeFigureS2).iiiWenowmoveontotheAs-terminatedsurfaceofTaAs.Weimpuritywaspresentattheorigin.ThisresultisasimplesumofusedtheQuantumESPRESSOpackage22,23fordensitythetermsarisingfromasingleimpuritysincetheinter-impurityfunctionaltheorycalculationsandtheWannier90package24,25scatteringscanbesafelyneglectedinmostexperimentally13,15,26toconstructtheabinitioWannier-function-basedtight-bindingrelevantcases.WethencancomparethisQPIpatternormodels.(SeetheSupportingInformationforcomputationalitsreal-spaceFouriertransformwiththeexperimentalSTM/FT-details20).WhencomputingtheJDOS[eq1],theSSP[eq3],STSresult.andtheQPIpattern[eq13],thesumoveratomicorbitalswasFurthermore,inrealexperiments,wecanbreakdownthelimitedtotheorbitalsbelongingtothetopmostAsatoms.overallQPIpatternsintocontributionsarisingfromeachTheSSPandtheQPIpatternsofTaAsareshowninFigure4.isolatedimpurity.Forexample,onecanapproximateΔρi(r)TheJDOS(seeFiguresS3andS420)andSSPhaveonlyminorwithaparametricformwithasmallnumberofparametersandfitdifferences.5However,asinthecaseofthesquarelattice(Figure∑iΔρi(r−Ri)totheexperimentalSTMdata.NotethatΔρi(r)2),theQPIpatterniswidelydifferentfromtheJDOSandSSP.willbethesameforthesamekindofimpurities.Moreover,thecalculatedQPIpatternsarestronglydependentAlthoughwerestrictedthetheoreticaldiscussiontotheR=R′ontheimpuritystrength.=0caseforsimplicity,theformalismcanbeextendedtothecaseTounderstandthisstrongimpurity-strengthdependenceofwithnonzeroimpuritymatrixelementsforR≠0orR′≠0.WetheQPIpatterns,weinvestigatethecorrespondingsignedQPIcalculateT(R,R′;ω)witheq11.IftherangeofVisfiniteinreal20space,thematrixequationfortheTmatrixisafinite-signals.FigureS5showsthatthepositionsofthepeaksofΔρdonotstronglydependonVandaredeterminedbythedimensionalmatrixalgebraandcanbeeasilysolvednumerically.electronicstructuresofTaAs,assimilarpeaksalsooccurintheThen,weperformthesumoverRandR′forthecomputationof20SSP(orJDOS;seeFigureS4).However,thesignedQPIQPIpatterns.WeperformedthiscomputationforanimpuritypatternsvarystronglywithV.Thesebehaviorsareverysimilartothataltersthenearest-neighborhoppingandforanimpuritythatthecaseofthesquarelattice[Figure3c].Sincetheimpurityshiftstheon-sitepotentialofthenearest-neighboringatoms.7491https://doi.org/10.1021/acs.jpcc.1c01410J.Phys.Chem.C2021,125,7488−7494

4TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleFigure5.(a)SSPand(b−e)absolutevaluesoftheQPIpatternsfortheAs-terminatedTaAssurfaceforωattheFermilevel.Here,theon-siteimpuritypotentialattheporbitalsofstrengthVwasappliedatasinglesurfaceAsatom.ThehoppingintegralfromtheporbitalsofthatatomtotheporbitalsoftheadjacentAsatomswasalteredby0.5V.20TheresultsareshowninFigures5andS6,respectively.Again,Equation22holdsforgeneralimpuritypotentials.UsingthewecanseethattheQPIpatternsdependmuchonthepropertiesBornapproximationtotaketheV→0limit,wefindoftheimpurity.Wenowprovideanintuitiveexplanationforthediscrepancy2VΔρ(;)qkω=∫ddω′betweenJDOS,SSP,Bornapproximation,andtheT-matrixπformalism.Wefirstconcentrateonthedifferencebetween1JDOS/SSPandtheBornapproximation.Forthesingle-orbitalImGG00(;kkω′−)Im(q;ω)ωω′−(23)case,thetraceineq2reducestoasingletermandthedensityofstatesisequaltotheimaginarypartoftheGreenfunction.Thus,Comparingeq23withJDOS[eq19],wecanseethattheoff-eq1canbewrittenasshellstatesareincludedintheQPIpatterns[Δρ(q;ω)]evenin1thelimitofV→0,whileonlytheon-shellstatescontributetoJG(;)qkωω=−2∫dIm(;)Im(00kωGkq;)JDOS.Thus,JDOS(orSSP)isnotthecorrectV→0limitofπ4,26QPIpatterns,andwemustalwaysconsidertheoff-shellstates(19)tocorrectlyinterpretQPIpatterns.Thispointisvisually(Weomittedlimforbrevity;thelimitisalwaysimplied).ThedemonstratedinFigure2a,d.TheJDOS[Figure2a]isnonzero+η→0onlyinsidethecircleinmomentumspacewithradiusq=2kc,equationforQPIundertheBornapproximationisobtainedwhiletheBornapproximationQPIintensity[Figure2d]isfromeq16,andtheresultishigheroutsidethecircle.VNext,letuscomparetheT-matrixformalismandtheBornΔρ(;)qkω=π∫dIm(;)([GG00kωωk−q;)]approximation.TheTmatrixT(ω)=V[I−G0(ω)V]−1isasensitivefunctionofV,especiallynearthezeroofthedenominator.ThissensitivityresultsinthestrongdependenceVÄÅÅ=−ÅÅÅÅ∫dRe(;)Im(kkGG00ωωkq;)ofQPIpatternsontheimpuritystrength.ThefrequencyπÅÇÉdependenceofT(ω)arisesfrommultiplescatteringeffectsasweÑÑ+−∫dIm(;)Re(kkGG00ωωkq;)ÑÑÑÑcaneasilyseefromtheseriesexpansionT(ω)=V+VG0(ω)VÑÖ+....SuchaneffectisignoredintheBornapproximationbecause(20)onlytheleadingordertermistaken,thatisT(ω)≈V.OnemightarguethatwhatonecaresaboutintheQPIIfthesystemisinvariantunderaC2rotationwithrespecttoz,patternsisjustthescatteringwavevectors,andthus,JDOS,SSP,wehaveorBornapproximationcangiveareliabledescriptionoftheQPI2()Tωpatterns.However,extractingscatteringwavevectorsfromQPIΔρ=(;)qkω∫dRe(;)Im(GG00kωωk−q;)patternsmaynotbetrivialorstraightforwardinsomecasessoπ(21)thatsimpleapproximationscanfailtodescribeeventhescatteringwavevectors.Forexample,letuspayattentiontoFromeqs19and21,weseethattherealpartoftheGreentheregionsnearq=0inFigure5.InthecaseofFigure5b,somefunctioncontributestotheQPIintensitywithintheBornscatteringvectorsareobscuredbythebroadintensityapproximation.DuetothisrealpartoftheGreenfunction,distributioncenteredatq=0.Also,thescatteringvectorsatscatteringsfromandtostateswhicharenotontheFermisurface6thecornersareweakened.ItisobvioustoseethatextractingthealsocontributetoQPIasopposedtoJDOS.scatteringvectorsisnotalwaysunambiguous,anditisquiteWecanfurthersimplifyeq21byusingtheKramers−KronigimportanttobeabletounderstandtheQPIpatternsthemselves.relationbetweenReG0(k;ω)andImG0(k;ω)Furthermore,ifwelookatFigure6,wecanfindthatevenin2()Tωthesimplestcaseofthesquarelatticewithansorbitalpersite,Δρ(;)qkω=∫ddω′themaximumintensityscatteringwavevectorsforV=1eVandπ1V=3eVaredifferent.ThedifferenceisΔq=0.070π/a,whiletheImGG00(;kkωω′−)Im(q;)(22)scatteringvectorfromthedensityofstatesisqc=2kc=0.450π/a.ωω′−Therelativedifferenceinthescatteringwavevectoris16%,Thus,thetransitionsinvolvingstateswithenergyω′differentwhichisnon-negligible.Therefore,foranaccurateextractionoffromωparticipateintheQPIbutdonotcontributetotheJDOSthescatteringwavevectors,wemustconsidertheimpurity-[eq19]andSSP[eq3].strengthdependence.7492https://doi.org/10.1021/acs.jpcc.1c01410J.Phys.Chem.C2021,125,7488−7494

5TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticlemethodbasedonfirst-principlescalculations,peoplemayfullyunderstandthephysicsofFT-STSandQPI.■CONCLUSIONSInconclusion,wehaveshownthattheQPIpatternstronglydependsonthescatteringpropertiesofimpurities,includingtheirstrength.Thisfindingholdseveninthesimplestcaseofasquarelatticewithnonmagnetic,scalaronsiteimpurities.Theimpurity-strengthdependenceisalsopresentintheQPIFigure6.AbsolutevalueoftheQPIpatternalongqy=0,whichisshownpatternsofTaAs.WewereabletodescribethecomplexinFigure2b,cforthe2Dsquarelatticewithans-likeorbitalpersite.impurity-strengthdependenceoftheQPIpatternsofbothsystemsfromaunifiedframework:thepristinesurfaceGreenfunctionsdeterminesthepositionofthepeaks,whiletheMoreover,peopleusuallyFouriertransformtheSTMpatternintensityandshapeofthepeaksaresignificantlyaffectedbytheoveralargeareawhichnecessarilycontainsdifferentkindsofTmatrix.OurfindingsthattheQPIpatternscanbecompletelyimpurities.Forexample,Figure1Bofref5showsseveraldifferentfordifferenttypesofimpuritiesaregeneral,withtheirdifferentkindsofimpuritiesinthetopographicimage.ThedI/profoundapplicabilityrangingfromthesimplesttoymodeltodVimageforthissurfaceisshowninFigure2Dofref5.Ifwecomplicatedtopologicalmaterials,andthus,itisimpossibletoFouriertransformthispattern,theQPIpatternswithdifferentfullyunderstandtheQPIpatternswithoutareferencetotheimpuritieswillmixup.However,asdescribedinourpaper,thetypesofimpuritiesinQPIexperiments.Therefore,ourfindingsQPIpatternisstronglydependentontheimpurities.ForamoreshouldgenerallybeusedinanalyzingtheresultsofFT-STSaccurateinvestigationoftheQPIpatterns,onemustdecomposeexperiments.individualQPIpatternsintocontributionsarisingfromeachimpurity.■ASSOCIATEDCONTENTAlthoughthetopographicimagesinFigure1Bofref5show*sıSupportingInformationthattherearethreedifferenttypesofimpuritieswithdifferentscatteringproperties,thisinformationisnotsufficienttoTheSupportingInformationisavailablefreeofchargeatpinpointtheimpuritiesfromfirst-principlescalculations.Wehttps://pubs.acs.org/doi/10.1021/acs.jpcc.1c01410.needmoreinformationonthechemistryofthesurfaceconditionDetailsoffirst-principlescalculationsandGreenfunctionandtheimpuritieswhichcanbeobtainedfromexperiencesorcalculation;real-spacemasking;Fourier-transformedfurtherexperiments,notnecessarilyconfinedtoSTS.LDOSandsurface-projectedJDOSforAs-terminatedAsasidenote,letuscomparethereportedexperimentalandTaAssurface;signedvaluesoftheQPIsignalfortheAs-theoreticalQPIpatterns,Figure4H,Kofref5.There,theterminatedTaAssurfacealongdifferentpathsintheintensitymapsaroundq=0arequitedifferentfromeachother:Brillouinzone;andSSPandtheabsolutevalueofQPIitisclearthatasimpleGaussianorLorentzianbroadeningofthepatternsfortheAs-terminatedTaAssurfacewithdifferenttheoreticalspectrumcannotreproducetheexperimentalone.impuritypotentialvalues(PDF)Furthermore,ifweconcentrateonthecorners,forexample,near(qx,qy)=(2π/a,2π/a),weseethattheQPIpatternisabsentinthetheoreticalspectrumunlikethatintheexperimentalone.■AUTHORINFORMATIONThisisduetotheneglectofUmklappscatteringsintheCorrespondingAuthorstheoreticalspectrumofref5.InTaAs,agoodpartofsurfaceSeung-JuHong−DepartmentofPhysicsandAstronomy,SeoulstatesarelocalizedneartheBrillouinzoneboundary(seeFigureNationalUniversity,Seoul08826,Korea;orcid.org/0000-20S3).Hence,thesmall-|q|Umklapp(q+G)scatteringis0002-9216-8657;Email:sjhong6230@snu.ac.krimportant.Thesefactorsarenotdirectlyrelatedtotheimpurity-Jae-MoLihm−DepartmentofPhysicsandAstronomy,SeoulstrengthdependenceofQPIpatterns.Still,theyreflectthefactNationalUniversity,Seoul08826,Korea;CenterforthatthecurrenttheoreticalunderstandingofQPIpatternsisCorrelatedElectronSystems,InstituteforBasicScience,Seoulratherincomplete.08826,Korea;CenterforTheoreticalPhysics,SeoulNationalTheabovediscussionsshowthatacompleteanalysisofQPIUniversity,Seoul08826,Korea;Email:jaemo.lihm@patternsisfarfromtrivial.Toreachanagreementbetweengmail.comtheoryandexperiment,weneed(i)theinformationonatleastCheol-HwanPark−DepartmentofPhysicsandAstronomy,roughlywhatkind(s)ofimpuritiesarepresentand(ii)first-SeoulNationalUniversity,Seoul08826,Korea;Centerforprinciplescalculationsstartingfromsuchknowledge.OurworkCorrelatedElectronSystems,InstituteforBasicScience,Seoulprovidesthetoolforthelatterpart(afirst-principlesmethod)08826,Korea;CenterforTheoreticalPhysics,SeoulNationalfortheunderstandingofQPIpatterns.However,westillneedUniversity,Seoul08826,Korea;orcid.org/0000-0003-theformerpart,theexperimentalinformationontheimpurities.1584-6896;Email:cheolhwan@snu.ac.krRequiredfirst-principlescalculationswilldependontheCompletecontactinformationisavailableat:quantityandqualityoftheexperimentallyobtainedinformationhttps://pubs.acs.org/10.1021/acs.jpcc.1c01410onimpurities.Byshowingtheimpurity-strengthdependenceoftheQPIspectra,ourworkinformsQPIexperimentalistsoftheAuthorContributionsimportanceofimpuritiesandmayleadthemtoconduct∥S.-J.H.andJ.-M.L.contributedequallytothiswork.experimentsonthetypesofimpuritiesandthedependenceofQPIspectraontheimpurities,insystemsnotlimitedtoTaAs.NotesThen,usingthosefurtherexperimentaldataandtheT-matrixTheauthorsdeclarenocompetingfinancialinterest.7493https://doi.org/10.1021/acs.jpcc.1c01410J.Phys.Chem.C2021,125,7488−7494

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