Spin-orbital physics in the optical conductivity of quarter-filled manganites

Using finite-temperature diagonalization we investigate the optical conductivity $\sigma(\omega)$ and spin-orbital dynamics in the CE phase of half-doped manganites. We find $\sigma(\omega)$ characterized by a broad spectrum with pronounced optical gap due

Spin-orbitalphysicsintheopticalconductivityofquarter- lledmanganites

JanBa la1,2andPeterHorsch1

2

Max-Planck-Institutf¨urFestk¨orperforschung,Heisenbergstrasse1,D-70569Stuttgart,Germany

MarianSmoluchowskiInstituteofPhysics,JagellonianUniversity,Reymonta4,PL-30059Krak´ow,Poland

(Dated:February2,2008)

Using nite-temperaturediagonalizationweinvestigatetheopticalconductivityσ(ω)andspin-orbitaldynamicsintheCEphaseofhalf-dopedmanganites.We ndσ(ω)characterizedbyabroadspectrumwithpronouncedopticalgapduetochargeorderinginducedbyCoulombandfurtherneighborJahn-Tellerinteractions.Withincreasingtemperaturetheconductivityshowsasigni cantchangeoverawideenergyrangewithacharacteristicshifttowardslowerfrequencies.InthelowtemperatureCEphaseweobservein-gapabsorptionduetocombinedorbiton-spinexcitations.

PACSnumbers:75.10.-b,71.70.Ej,75.30.Et,75.40.Mg

1

arXiv:cond-mat/0503011v1 [cond-mat.str-el] 1 Mar 2005

Thelargevariationofphysicalpropertiesofmangan-itesasfunctionofdopingortemperatureoriginatesfromacomplexinterplayofspin,orbitalandchargedegreesoffreedom,aswellastheinteractionwiththelattice2].AnimportantexampleforthecontrolofmagneticorderduetotheorbitaldegreeoffreedomistheCEphaseatquarter lling.Hereferromagnetic(FM)zig-zagchainsarestaggeredantiferromagneticallyandtheoccupiedegorbitalsareorientedalongFMbondsinthe(a,b)planes[3][Fig.1(a,b)].Althoughthestructurere ectsthecoop-erativeactionofantiferromagnetic(AF)superexchangeandFMdouble-exchange(DE),theseinteractionsalonedonotguaranteeitsstability[4,5].TheAF-CEphasewasobservedincubic(Nd,Pr)1/2(Sr,Ca)1/2MnO3andinlayeredLa1/2Sr3/2MnO4LaSr2Mn2O7man-ganites.YetCE-typeorbitalcorrelationshavealsobeenobservedintheabsenceofantiferromagnetism,namelyintheFMmetallicphaseofNd1/2Sr1/2MnO3andalsoatsmallerholedopingandarebelievedtobeconnectedwiththemechanismofcolossalmagnetoresistanceAnimportantfeatureoftheCE-phaseisalargeopticalgapandabroadabsorptionmaximumat1–2eVshiftingtowardslowerfrequencieswithincreasingtemperature[11,12,13].Remarkablyasimilarspectralshiftisob-servedinphoto-excitationexperiments,whichrevealul-trafastresponsetimesofsuchMottinsulatingstructures[2,14].

Ouraimistoinvestigatetheelectrondynamics(opti-calconductivity,elementaryexcitations)intheCEspin-orbitalstate.Itisevidentthatarealunderstandingofthisphaseisonlyachievedifalsotheexcitationspec-traareconsistentwithexperiment.Thestrongvariationofσ(ω)withtemperatureTwasnotaddressedinprevi-oustheoreticalwork[15,16].Ourstudyemploying nitetemperaturediagonalizationisdesignedtoexplorealsoquantities,suchastheorbitalexcitationspectra,thatsofarcannotbemeasuredbyexperimentandinvestigatetheire ectonobservableslikeopticalconductivityandspinexcitations.Ourresultsfortheopticalconductivityreproduceforthe rsttimetheexperimentaltrendsonasemi-quantitativelevel,andpredictorbitalexcitations

inthechargegapoftheAF-CEphase.

Orbitaldegeneracyprovidestheoptionforthesystemtoloweritsdimensionality,whichisessentialfortheap-pearanceoftheCEstructureOurstudyexploresanovelaspectoforbitalphysicsinmanganites,namelytheroleoffurtherneighborJahn-Teller(JT)interactions,andtheconsequencesonthedynamics.We ndthator-bitalpolarizationinducedbythisinteractioniscrucialfortheappearanceofCE-antiferromagnetismatlowT,andleadstoaspinresponseconsistentwithAF-CEorder.WeconsideranextendedDEmodelwithchargeor-der(CO)triggeredbyCoulombrepulsionandfurther-neighborJT-interaction[5].AForderalongthecaxisandegorbitalelongationinthea/bdirectioncon nestheelectrondynamicsto(a,b)planesandthuswecon-sideratwo-dimensional(2D)model.Assumingin niteon-siteCoulombrepulsionbetweenegelectronsonthesamesite(U→∞)themodelhastheform ξζ

H= Sctijdiξσdjζσ+H.c. JHi·si

ij ξζ,σ

+JAF

ij

cSci·Sj+V

ij

i

ninj+HOO.(1)

The rsttermdescribesthemotionofegelectronsun-dertheconstraintthateachsitecanbeoccupiedatmost =d(1 niξσbyoneegelectron,d¯′).¯)σ′(1 niξσiξσiξσ¯(¯Heretheindexξσ)denotestheegorbital(spin)orthog-onaltoξ(σ),respectively.Thehoppingmatrixelements

22

tξζijintheorbitalbasis{|x ,|z }with|x ～x yand|z ～3z2 r2havethestandardform[18].Thesec-ondterminstandsforHund’scouplingbetween

electronspinsi=ξσσ′diξσσσσ′diξσ′andthet2gcore

c

spinSci(S=3/2).ThethirdtermrepresentstheAFsuperexchange(JAF>0)betweennearest-neighbort2gspins,whilethefourthtermistheinter-siteCoulombre-pulsionwhichfavorscheckerboardCO(V>0).Finally

HOO=2κ

′

jij′

(1 ni)Tjj′,(2)

Using finite-temperature diagonalization we investigate the optical conductivity $\sigma(\omega)$ and spin-orbital dynamics in the CE phase of half-doped manganites. We find $\sigma(\omega)$ characterized by a broad spectrum with pronounced optical gap due

accountsforthecooperativeJT-interactionandde-scribestheorbital-orbital(OO)interactionbetweenfur-therMn3+neighborsmediatedbytheMn4+ionin-between[5][Fig.1(c)].Hence, jij′boringMn3+–Mn4+–Mn3+sitesalong denotesaorthreebdirectionneigh-whilethetwo-siteorbitaloperator,Tjj′,inthebasishastheformTjj′=TjzTjz′+3TjxTjx√{|x ,|z }′

2

σ(nixσ nizσ)

andTix=

1

Nω

Re

∞

dte

iωt

jx(t)jx ,(3)

wherethex-componentoftheie ξζ ij ξζ,σtijδijx currentd iξσd jζσ H.c. operatorisjx=andδij

xdenotesthex-componentofthevectorconnectingsitesiandj.WeevaluatethisKuboformulausingtheexactdiagonaliza-tiontechniquefor nitetemperaturesdevelopedbyJakliˇcandPrelovˇsek[21]forcupratesandlaterusedalsoformanganites[5,18].Inordertomaketheproblemnumer-icallytractable,wesimulatedthecorespinsbySc=1/2spinsandscaleJHtopreservethesplittingbetweenhighandlowspinstates.Thedatareportedherewascalcu-latedforJH=15t.Althoughweexploitedthetransla-tionalsymmetrytosavestorageandcomputationcalculationsweretoasmall√

time,8clusterasonlySz

tot= restrictediSzisubspaces(Si=Sic+si)canbetreatedseparately,whileasimilarsymmetryintheor-bitalsectorisabsent.

ThesubtleroleofHOOistoenhanceCEtypeor-bitalpolarizationtherebysupportingtheAFdecouplingofFM-chainsatlowT(Fig.1a),whileforsmallκ′DEdominatesinalldirectionsleadingtoaFMchargeandorbitalorderedstate[5].InFig.2weshowthefre-quencyandtemperaturedependenceofσ(ω)emergingfromgroundstateswithAF-CE[Fig.2(a)]andFM-CE[Fig.2(b)]spin-orbitalorder,respectively.Acharacteris-ticfeatureofσ(ω)isabroadabsorptionmaximumfoundfor2t ω 5twhichshiftstowardslowerfrequenciesby ω 0.5tastemperatureincreasestoT 0.2t.Furtherincreaseoftemperatureleadstothemeltingoftheorbitalorderandtheclosingofthechargegap(ω 2t)withfurthershiftofthehighfrequencybroadstructure.Theshapeofσ(ω)anditstemperatureevo-lutionissimilartotheexperimentaldata,e.g.,forthecubicLa0.5Ca0.5MnO3[13]andlayeredLa1/2Sr3/2MnO4[11]compound.Assumingt=0.4eV[18,20]one nds

2

FIG.1:(Coloronline)SketchoftheAF-CEspin,orbital,andlatticeorder.In(a)large(small)spinsindicatesiteswith(without)anegelectron,respectively.ShadingindicatesdirectionofDEcarrierpropagation.In(b)fullsymbolsrep-resentoccupiedegorbitalsandin(c)fullcirclesindicatetheJT-shiftedOions,andthearrowthedisplacementofMn4+.

0.12

(a)

0.08)

ω0.080.04

(σ0ρ0.041.01.5

0.00

0.12

)

ω0.08(σρ0

0.040.00

ω/t

FIG.2:(Coloronline)Temperaturedependenceoftheopticalconductivityσ(ω)obtainedinthephasewith:(a)AF-CEorderinthegroundstate,and(b)aFM-CEgroundstateforJAF=0.04t.Otherparameters:(a)V=0.2t,κ′=0.15t;(b)V=t,κ′=0.05t.ThespectraaregeneratedusingaLorentzianbroadeningΓ=0.2t.Inset:Mid-gapabsorptionbroadenedwithΓ=0.05t.

thepositionofabroadpeakstructureinFig.2centeredclosetotheexperimentalvalue1.2eVmeasuredinsingle-layerLa1/2Sr3/2MnO4[11].Asubtletestistheabsolutescaleofσ(ω)whichiscontrolledbythesumrulethatlinksσ(ω)andthekineticenergyofthemodel[18,22].

Forthemaximumoftheabsorption(σmax≈0.1ρ 1

ndwithρ= a/e2[18]andthelatticeconstant0)we0a～5.5 A,σmax≈500( cm) 1.Observedvaluesforσmaxare～1000( cm) 1inLa1/2Sr3/2MnO4[11,12]and～700( cm) 1incubicLa0.5Ca0.5MnO3[13].

ThemainabsorptioninFig.2describesexcitationsacrossthechargegap( corbitally 3(Vordered+2κ′)incaseofper-fectCO)whichinthestateinvolveegelectrontransfermainlybetweenbridge(groundstate)andcorner(excitedstate)siteswithinthesamezig-zagchain.Tounderstandtheroleofhoppingtitisusefultoconsideranembeddedthree-sitemodelwithanegelec-tronbeingeitheratthebridge(|B )oratoneofthecorner(|CLtionatωc ,|CR )sites.One ndstheopticaltransi-non-bonding ( c+2t2/ c,betweenbonding～||CL |CR )/

√B and

Using finite-temperature diagonalization we investigate the optical conductivity $\sigma(\omega)$ and spin-orbital dynamics in the CE phase of half-doped manganites. We find $\sigma(\omega)$ characterized by a broad spectrum with pronounced optical gap due

3

entparities.WithincreasingTmagneticexcitationsareincludedinthethermodynamicaveragewithreducedef-fectivehopping,whosevalueisdictatedbytheDEmech-anismforcorrelatedegelectrons.Thusthereductionofthesecondorderterm2t2/ cforthesestatesresultsinaspectralshifttowardslowerωc.FurtherreductionofωciscausedbymeltingoforbitalandCOleadingtogaplessorbitalexcitations[18]andareductionof c.

IntheFMspinstateσ(ω)ischaracterizedbytwobroadpeaksandawell-developedchargegap[Fig.2(b)].Here,spin uctuationsaresmallandσ(ω)isdominatedbycharge-orbitonexcitations,i.e.,thepurechargeexcita-tioniswellseparatedfromtwofurtherpeakscorrespond-ingtotheadditionalexcitationofoneandtwoorbitons.Ontheotherhand,intheAF-CEspinorderedstatespin uctuationsaresubstantialandinter-sitechargetrans-ferisaccompaniedbymagnonandorbitonexcitationsleadingtoabroadenedspectrumabovethechargegapaswellastosubstantialin-gapabsorption.ThelatteriscompletelyabsentintheFMstateatlowtemperature.Togainadeeperinsightintothestructureofσ(ω)intheCEspin-orbitalphase,wehavecalculated[usingasimilar nite-temperatureprocedureasforσ(ω)]thedy-namicalspin,orbital,andspin-orbitalresponsefunctions:

Aq(ω)=

1

Using finite-temperature diagonalization we investigate the optical conductivity $\sigma(\omega)$ and spin-orbital dynamics in the CE phase of half-doped manganites. We find $\sigma(\omega)$ characterized by a broad spectrum with pronounced optical gap due

=

jxGS=t-+α++__tβ-+β

+

+

__t’β-

2

+

...

+

...

FIG.5:AFMsegmentshowingthedominanttermsintheAF-CEgroundstatewithCO(α,β 1)inthelimitJH→∞(left).Whenapplyingthecurrentoperatorjx(right),onlythe rsttermwouldcontributeintheFMcase.Thesecondterm～βinjx|GS (describinganorbitalexcitation)is niteduetoquantum uctuationsinanAFenvironment.Heretandt′(t>t′)standfortwoe ectivehoppingsbetweencornerandbridgesites,whilefull(empty)symbolsstandforthesitesoccupied(unoccupied)byanegelectronwithtotalspin(arrows)S=1(S=1/2),and|0 ≡|S=1,Sz=0 .

standsforaboundstate[23,24]whileatω 1.5tastrongmaximumisfoundhavingacharacterofananti-boundstate.Thisindicatesthattheorbitalexcitations,althoughmuchmorerobustthanthespinones,playanactiverolealsoatalower-energyscale.Athigherenergiesω 1.5t,however,Cq(ω)andTq(ω)structurefactorsstronglydi er.

Asalientfeatureofσ(ω)isthein-gapabsorption(ω< c)intheAF-CEphase,Fig.2(a),andtheabsenceofsuchabsorptionfortheFM-CEcaseinFig.2(b)atlowtemperature(T 0.1t).Wearguethatthestructureatω tisanorbiton-likecollectiveexcitationwhichbe-comesinfrared-activewhenparityisbrokenbyAFcorre-lations.ThisisillustratedschematicallyinFig.5.IntheCOphasethegroundstate(GS)isdominatedbycon g-urationswithoccupiedbridgesitesbuthybridizedwithstatesreachedbyinter-sitehopping(～t)orchangedbycorespinsuperexchange(～JAF).AswedealwithdorbitalsthehoppingmatrixhasevenparityleadingtoevenGSasfaraschargedistributionisconcerned.Con-sequently,theGSchangedbythecurrentoperatorisoddwithrespecttochargeand(neglectingcorespins)onlyprocessesacrosstheopticalgapareallowed( rstlineofjxselection|GS inrulesFig.5).noInlongerthepresencestrictlyofhold,AF uctuationsasspin-exchangetheseleadstoanadmixtureofspinandspin-orbitalexcitations(secondandthirdlineofjxcancelbysymmetry.

|GS inFig.5)whichdonotThisorbiton-magnonmechanismforthein-gapab-sorptionissupportedbythefactthattheintensityofthestructureinσ(ω)atω t[seeinsetinFig.2(a)]increases(abovethebackgroundofotherexcitations)withdecreasingtemperatureinthespin-orderedcase(T 0.1t).Themomentumconservationpresentinop-ticalprocessescanbeful lledcombiningTwithq=(±π/2,±π/2)whichareq andSz

dominant qex-citationsintherespectivedynamicalstructurefactors[seeFig.3(a)andFig.4(a)].Thedecreaseofthestaticstructurefactor

4

Sq= ∞

∞dωSq(ω)atq=(±π/2,±π/2)withtempera-tureisapparentlycorrelatedwiththesuppressionofthein-gapabsorptioninσ(ω).

Insummary,opticalconductivityandspin-orbitaldy-namicshavebeeninvestigatedintheCEphaseofcharge-orderedmanganitesatquarter lling.Twocontributionsinourtheorydeterminethechargegap:(i)elasticforcesand(ii)Coulombinteraction.Thisisimportanttoex-plaintherecentlydiscoveredpico-secondswitchingofop-ticalspectra,whichcanbeunderstoodbytheinstanta-neousreductionoftheCoulombcontributionduetothesuppressionofCObytheultra-shortpulse,aswellasthefastrecoveryafterthepulse.WehaveshownthatthecooperativeJTcouplingbetweenfurtherMnneigh-bors,stabilizingtheCE-typecorrelationsinourmodel,canexplainsemi-quantitativelytheopticaldataanditstemperaturedependenceinhalf-dopedmanganites.Spinandorbitalexcitationsleadtocollectivemodesactiveinopticalexperiments,andareresponsibleforcharacteris-ticdi erencesbetweenAF-andFMCEphases.WewouldliketothankA.M.Ole´sforfruitfuldis-cussions.J.B.acknowledgesthesupportoftheMPIf¨urFestk¨orperforschung,StuttgartandbythePolishStateCommitteeofScienti cResearch(KBN)ofPoland,ProjectNo.1P03B06826.

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Using finite-temperature diagonalization we investigate the optical conductivity $\sigma(\omega)$ and spin-orbital dynamics in the CE phase of half-doped manganites. We find $\sigma(\omega)$ characterized by a broad spectrum with pronounced optical gap due

5

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