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Incident-energy and polarization-dependent resonant inelastic x-ray scattering study of La CuO 2 4 L. Lu,1 J. N. Hancock,2 G. Chabot-Couture,1 K. Ishii,3 O. P. Vajk,4 G. Yu,5 J. Mizuki,3 D. Casa,6 T. Gog,6 and M. Greven1,2 1Department of Applied Physics, Stanford University, Stanford, California 94305 2Stanford Synchrotron Radiation Laboratory, Stanford, California 94309 7 3Synchrotron Radiation Research Unit, Japan Atomic Energy Agency, Hyogo 679-5148, Japan 0 4NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 0 5Department of Physics, Stanford University, Stanford, California 94305 2 6CMC-XOR, Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439 n (Dated: February 6, 2008) a J We present a detailed Cu K-edge resonant inelastic X-ray scattering (RIXS) study of the Mott 1 insulator La2CuO4 in the 1-7 eV energy loss range. As initially found for the high-temperature 1 superconductor HgBa2CuO4+δ, the spectra exhibit a multiplet of weakly-dispersive electron-hole excitations, which are revealed by utilizing the subtle dependence of the cross section on the incident photon energy. The close similarity between the fine structures for in-plane and out-of-plane ] l polarizationsisindicativeofthecentralroleplayedbythe1scoreholeininducingchargeexcitations e - within the CuO2 planes. On the other hand, we observe a polarization dependence of the spectral r weight,andcarefulanalysisrevealstwoseparatefeaturesnear2eVthatmayberelatedtodifferent t s processes. The polarization dependence indicates that the 4p electrons contribute significantly to . the RIXS cross section. Third-order perturbation arguments and a shake-up of valence excitations t a arethenappliedtoaccountforthefinal-energyresonanceinthespectra. Asanalternativescenario, m we discuss fluorescence-likeemission processes dueto 1s→4p transitions intoa narrow continuum - 4p band. d n PACSnumbers: 74.25.Jb,74.72.-h,78.70.Ck,71.35.-y o c [ I. INTRODUCTION usuallyidentifiesenergy-lossfeatureswiththeexcitations of the valence electrons. When viewed as a second-order 3 v optical process, there exists a close connection between With the advent of third generation synchrotron 1 the initial absorption and final emission stages in RIXS. sources, inelastic X-ray scattering has emerged as a 1 Accordingly, the spectra simultaneously depend on both powerful probe of momentum- and energy-dependent 3 the incident and final photon energies.22 7 charge and lattice dynamics. This development 0 has led to new insight into low-density metallic 6 electrodynamics1, valence fluctuating compounds2, H O 2 0 molecular correlations3,4, phonon dynamics5, and the In the present work, we investigate these energy de- t/ Mott physics of correlated electron systems such as pendences as well as the polarization dependence of the ma the lamellar copper oxides6,7,8,9,10,11,12,13,14 and the cross section in La2CuO4, the best-characterized lamel- manganites15,16. Resonant inelastic X-ray scattering lar copper oxide. This Mott insulator is the parent com- - (RIXS) provides a considerable advantage over ordinary pound of the original high-temperature superconductor d n inelastic scattering since, at resonance, the inelastic sig- (La,Ba)2CuO4,andithasbeenthesubjectofanumberof o nal is significantly enhanced17. In the lamellar copper priorRIXSstudies.10,12,23 Exploitingtheincident-energy c oxides, this resonance condition can be readily met by sensitivity,weareabletoidentify additionalchargeexci- v: tuning the incoming photon energyto the vicinity of the tationfeatures. Wedemonstratethatthefinestructureis i Cu K edge. The RIXS cross section sensitively depends presentforphotonpolarizationbothparallelandperpen- X on the incident photon energy and on the nature of the dicular to the CuO2 planes, and suggest that the subtle r intermediate states12,18,19,20. differences between the two polarization conditions can a beexplainedintermsofmodelsinwhichthe4pelectrons Ifviewedasatwo-stageprocess,theintermediatestate play a significant role. in RIXS is the same as the final state of x-ray absorption: for example, in Cu K-edge RIXS, a localized core hole is created through 1s → 4p photoexcitation. The 1s core hole interacts strongly with the valence electron This paper is organized as follows. After the discus- system, generating a strong response that corresponds sion of the experimental details in the next section, we to the many-electron bound states of the local, nascent presentourresultsforout-of-planeandin-planepolariza- core-hole potential.21 In RIXS, the relaxation of these tioninSecs. III andIV,respectively. SectionVcontains highly excited states leads to the emission of photons a discussion of our data, and we summarize our work in and leaves the valence system in an excited state. One Sec. VI. 2 h tor and a Si(400) secondary monochromator were used to obtain an incident energy resolution of 220 meV. A bent Ge(733) analyzer crystal situated at the end of a 2 m four-circle diffractometer arm selected the energy of the photons scattered from the sample, which were then collected by a solid state detector. In this geometry, the polarization vector of the incident photon was always parallel to the CuO planes, with a typical angle 2 ◦ of ∼ 32 with respect to the tetragonal a axis, i.e., the planar Cu-O bond direction. The overall energy resolution was about 400 meV [full width at half maximum (FWHM)], as determined from the energy width of the elastic line. Out-of-plane polarization measurements [Fig. 1 (b)] were performed at beamline 9-ID-B at the Advanced Photon Source in a vertical scattering geometry. The reciprocal lattice vectors (3,0,0) and (1,0,0) were chosen as reference zone centers to reduce the contribution from the elastic tail, because Bragg scattering at these reflections is forbidden. The setup employed a Si(111) primarymonochromator,aSi(333)secondarymonochro- mator, and a spherical diced Ge(733) analyzer crystal with a 1 m radius, and yielded an overall energy resolu- tionofabout300meV(FWHM). Fordatacollectedthat were measured at the reduced wave vector (π,0), or ab- solutemomentumof(1.5,0,0),weusedaprimarySi(111) andasecondSi(444)channel-cutmonochromatorincon- junction with a diced analyzer on a 2 m diameter Row- land circle. This configuration can provide at best an energy resolution of 110 meV, but we chose wide aper- ture slits in front of the detector to obtain a significant signal boost and a comparable resolution of ∼300 meV for better comparison with measurements at other wave vectors. FIG.1: (Color online) Thetwoscattering geometries usedin Data were taken at ambient temperature on the same this studyand discussed in the text. single-crystallinesampleinbothpolarizationgeometries. La CuO undergoes a tetragonal-to-orthorhombicstruc- 2 4 tural phase transition at ∼ 530 K associated with the staggeredtilting of the CuO6 octahedra.24 We note that II. EXPERIMENTAL DETAILS our crystal is twinned, and hence we do not distinguish betweenthe twoinequivalentplanarorthorhombicdirec- Thefocusofthisworkisonthecollectiveelectronicex- tions. The crystal was grown in an image furnace at citations of La2CuO4 in the 17 eV range using incident StanfordUniversity. As-growncrystalsareknowntocon- photonenergiesin the vicinity ofthe Cu K-edgeabsorp- tain excess oxygen, and hence hole carriers. In order to tion threshold. We have measured the RIXS response assure that the sample was free of any carriers it was atseveralhigh-symmetrypositionsinthe Brillouinzone, annealed for 24 h in Ar flow at 950◦C. This reduction covering a fine mesh of incident and scattered photon treatment resulted in a Néel temperature of T ∼ 320 N energies around the Cu K edge. K, as determined from a measurement of the magnetic Twosetsofmeasurementsweretaken,onecollectedat susceptibility.25 theAdvancedPhotonSourcewithincidentphotonpolar- Figure2showsthex-rayabsorptionspectra(XAS)for ization vector perpendicular to the CuO planes (E||c), eachpolarizationconditionasmeasuredby totalfluores- 2 and the other at SPring-8 (Japan) with incident photon cence yield. For each polarization, there are two peaks, polarization parallel to the CuO planes (E⊥c). The at 8991 and 8998 eV for E||c, and at 8995 and 9002 eV 2 measurements with in-plane polarization [Fig. 1 (a)] for E⊥c. The lower of these resonances is usually iden- were taken in horizontal scattering geometry at beam- tified with a transition into a well-screened state,”6,8,10 line BL11XU at SPring-8,with the tetragonalreciprocal a many-body excitation which effectively screens the 1s lattice point G=(3,0,0) as Brillouin zone center, and a core hole and has significant 1s4p3d10L character. The ◦ scattering angle of ∼65 . A Si(111) main monochroma- higher resonance is identified with a transition into a 3 FIG.2: (Coloronline)RIXSspectraplottedversusfinalpho- ton energy along with the absorption spectra monitored by totalfluorescenceyield forphotonspolarized (a) parallel and (b) perpendicular to the CuO2 planes. The shaded areas in- dicatetheincident-energyrangesprobedinthepresentstudy. AllspectraweretakenatQ=(3,0,0)andatdifferentincident energies corresponding to the shaded ranges. FIG.3: (Coloronline)(a)RIXSsignalversusenergytransfer for out-of-plane polarization at three representative incident energiesatamomentumtransferof(3,0,0),whichcorresponds poorly screened state,6,8,10 another bound state of the to the two-dimensional zone center (0,0). The lines are the many-electron system in the core hole potential which result of a fit, as discussed in the text (reproduced from12). haspredominantly1s4p3d9 character. Foreachpolariza- (b) Contour plot of all zone-center scans, taken in 500 meV tion,theresonancepeaksareseparatedbyapproximately increments of Ei. 7 eV. The resonance energies differ by about 4 eV be- tweenthetwogeometries,whichmaybeprimarilydueto the larger Cu-O distance for the negatively charged api- cal oxygens, rendering the 4pz electronic orbitals lower ductor HgBa CuO (Hg1201).12 This study revealed 2 4+δ in energy than their 4p counterparts.26,27 The spectra σ additional features in the 2-5 eV range, an observation presented in this paper were taken in the vicinity of the that necessitates a new interpretation of the charge dy- well-screenedconditionforboth polarizationgeometries. namicsinthesematerials. Forexample,a∼2eVfeature was identified in Hg1201. It was argued in Ref. 12 that this feature is not likely a d → d excitation, but rather III. INCIDENT ENERGY DEPENDENCE, indicates the presence of a remnant charge-transfer gap OUT-OF-PLANE POLARIZATION even at optimal doping in this model superconductor. The presence of an additional feature at ∼3 eV, which RIXS spectra obtained in early work in the soft x- was only identified through inspection of multiple spec- ray regime exhibited a clear incident and final pho- tra obtained with different photon energies, constrains ton energy dependence.22 The incident-photon-energy the dispersion of the 2 eV feature to be less than 500 dependence was recently employed in the hard x- meV. The same approach was applied to La CuO , and 2 4 ray regime (at the Cu K edge) in a study of both preliminary data revealed charge-transfer features that La CuO andthesingle-layerhightemperaturesupercon- areremarkablysimilartothoseinHg1201,12aresultthat 2 4 4 is qualitatively differentfrom priorwork onthe Mott in- sulatorsLa CuO 10 andCa CuO Cl 9 Thesmalldisper- 2 4 2 2 2 sion of the 2 eV feature was further confirmed by subsequent measurements.23,28 Our primary focus here is to investigate in greater detail the incident-energy and polarization dependence of the inelastic cross section near the absorption threshold in La CuO . 2 4 The molecular orbital excitation at ∼ 7 eV was stud- iedindetailinRef. 11andismostprominentlyobserved near E = 8998eV, an incident photon energy for which i the lower-lying charge-transfer excitations in the 2-6 eV range do not resonate. We therefore limit our attention to the incident energy range E = 8989-8994eV [shaded i areainFig. 2(a)]andtoenergytransfersbelow7eVfor out-of-plane polarization. A fine step size of ∆E = 500 i meV was chosen, allowing us to demonstrate the high sensitivity ofthe RIXS crosssectionto the incident photon energy. Figure3(a)showsrepresentativelinescansatthezone center, taken at three different incident energies with out-of-plane polarization. These data resemble previous work10, yet closer inspection reveals additional features. At low incident energy, the most distinct feature is that at 2.25 eV. As the incident energy is increased, this sharp feature gradually weakens relative to those at higher energy. Another new feature at 3 eV, which was FIG.4: (Coloronline)Incident-energydependenceoftheE||c notobservedinpriorwork10,canbediscernedatallthree RIXS spectra for (a) (π/4,0), (b) (π,0), (c) (π/2,π/2), (d) incident photon energies. The feature at ∼ 4 eV, also (π,π). Thelinesaretheresultoffits,asdiscussedinthetext. seen in previous zone-center data10, becomes dominant atE =8992eV.Finally, a comprehensiveanalysisofall i data12revealsasecondnewfeatureat∼5eV.Wediscuss below how the systematic center-of-mass shift with incident energy suggests a modulation of the inelastic cross uum due to transitions to continuous unoccupied states. section through final photon-energy-dependent denomi- Asimultaneousleast-squaresfitofallspectraateachmo- nators. mentum transfer results in the lines in Figs. 3 (a) and 4 Figure3 (b) showsa contourplotconstructedfromall and allowed us to extract the peak positions plotted in line scans at the zone center. This mode of representa- Fig. 5. tion is similar to the “RIXS plane” of incident photon energy versus energy transfer in Ref. 29. By extend- Although the different components are not as eas- ing the energy-transfer spectra into the incident-energy ily distinguishable as at the zone center, the relative dimension, features at ∼ 2.25, 3 and 4 eV are readily strengths of the four features identified below 6 eV ex- apparent. hibit distinct dependences on momentum transfer and Figure 4 shows additional results at high-symmetry incident energy, which may be indicative of different ex- points of the two-dimensional Brillouin zone. Using the citation symmetry. The 5 eV component is most pro- fit procedure defined in Ref. 12, the spectra at each mo- nounced at (π,π) and at relatively high incident ener- mentum transfer are fitted simultaneously to obtain the gies. Away from the zone center, the relative weight of peak positions that determine the energy transfer of the the2eVfeaturequicklydecreases. Whilethe 2and3eV correspondingchargeexcitations. Specifically,weassume features are still separable and comparable in strength that the peak positions do not vary with incident energy at (π/4,0), the 2 eV component is only barely visible andthat eachenergytransferfeature is representedby a (at low incident energies) at (π,0), the 3 and 4 eV fea- Lorentzian line shape. The number, energy-transfer po- tures maintain comparable weight along [π,0]. In con- sitions, and energy widths of the features are considered trast, along [π,π], the overall response away from the to be shared parameters for all spectra at the same mo- zone center appears to be dominated by the (approxi- mentum transfer. At different incident energies, on the mately Lorentzian-shaped)response just above 4 eV. As other hand, the spectral weight of each component is al- for the fine structure, the momentum dependence away lowed to vary. We also use spectra on the energy gain fromthe zone center is againconsistentwith whatis ob- side (not shown in Fig. 4) for background subtraction, served in Hg1201, although that study was carried out and allow for linear slopes to approximate the contin- with in-plane polarization.12 5 FIG. 5: (Color online) Dispersion of the charge excitations forbothin-planeandout-of-planepolarization along thetwo FIG. 6: (Color online) Five line scans at the zone center ob- high-symmetry directions studied in the present work. The tained with in-plane polarization. dashed lines indicate sinusoidal fits to the dispersion of the three lowest-energy features discerned with out-of-plane polarization. Previous results10, obtained with out-of-plane polarization, are shown for comparison. comparablespectralweightalong[π,0],andremaincom- parableupto(π/2,π/2)along[π,π]. Between(π/2,π/2) and(π,π),however,the3eVfeatureisquicklysuppressed IV. INCIDENT ENERGY DEPENDENCE, betweenand,asinthe caseforout-of-planepolarization, IN-PLANE POLARIZATION the 4 eV spectral weight becomes dominant. Differencesbetweenthespectraobtainedinthetwoge- Figure 6 shows line scans at the zone center for ometries are also noticeable. First, with in-plane polar- the complementary in-plane-polarized (SPring-8) exper- ization, the intensity of the 2 eV feature relative to that iment. As indicated in Fig. 2 (b), the incident photon of spectral features with higher energy transfer is much energy was chosen to lie in the range 8993 8999 eV. smaller than for the out-of-plane polarized experiment. The fine structure revealed through the incident energy However, the 2 eV feature is still observable even up to dependence of the spectra is very similar to that found (π,π),reachingamaximumatanincidentenergyof8997 for out-of-plane polarization. Differences in the configu- eV,slightlyabovetheabsorptionthreshold(8995eV)for ration between the two experiments make a direct com- in-plane polarization. Second, for in-plane polarization, parison of signal levels difficult. Overall, there are still as we increase in incident energy, the center of mass of four major features present at ∼ 2, 3, 4 and 5 eV. The thespectracontinuouslyshiftstohigherenergytransfers. close similarity of the spectra supports the intuitive no- Forout-of-planepolarization,ontheotherhand,itpeaks tion that the spherically-symmetric 1s core hole poten- between 4 and 5 eV. tial dominates the generation of the valence excitations As for the E||c data, all spectra acquired at the same in both cases.9,30,31,32,33 momentumtransferweresimultaneouslyfitassumingthe There also exist similarities between the two polar- same set of peak positions. The results of the fits are ization geometries in the momentum dependence of the shownbythelinesinFigs. 6and7andthepeakpositions multiplet structure. Figure 7 shows line scans with fits are compared to those for out-of-plane polarization in atfourreducedmomentumtransfervaluesawayfromthe Fig. 5. While there is an overall good agreement, the 2 zonecenter. Asintheout-of-planepolarizedexperiment, eVfeatureshavesignificantlydifferentexcitationenergies thestrengthofthe2eVfeaturedecreasestowardthezone for the two polarization conditions. We note that the boundary. Inthepresentcase,itisnolongervisuallyob- dispersions of all four features summarized in Fig. 5 are servable at (π,0). Also, the 3 eV and 4 eV features have weak. Theseobservationswillbediscussedinmoredetail 6 latter findings were argued to be consistentwith embed- dedmolecularclustercalculations.36 Indeed,ourdatare- vealthattheactualexcitationenergiesofthe2meVfea- ture differ by as much as 300 meV for in-plane and out- of-plane polarization conditions. While the former excitation has no discernible dispersion, the latter appears to disperse by 100 150 meV toward the zone boundary. In addition to these differences, we also find that the spectral weights of these two low-energy features ex- hibitratherdifferentmomentumdependences,especially along [π,π]. We note that it is not likely that the 2 eV features are d →d excitations, since the latter lie below 2 eV and are expected to be much weakerat the K edge than at the L and M edges.22 In an attempt to understand the differences between the two 2 eV features,we consider a possible photon po- larizationeffect. When the polarizationvector of the in- cidentphotonlieswithintheCuO plane,the4pelectron 2 in the intermediate state is in the 4p orbital [see Fig. 8 σ (b)] and overlapswith the 2p electrons. The repulsive x,y Coulomb interaction between O 2p and Cu 4p therefore tends to suppress the O2p ⇒Cu3d charge transfer. On the other hand, for out-ofplane polarization, the 4p or- z bital is oriented orthogonally to the 2p electrons [see x,y Fig. 8 (c)] and the Coulomb interaction has a limited effect on the charge-transfer process. That difference in intermediatestatesmayexplainwhyweobserveasecond (local) “2eV” component for in-plane polarization, since the component that involves a nonlocal charge-transfer process from a neighboring CuO plaquette to the cen- 4 tral core hole site could be spectroscopically suppressed FIG.7: (Coloronline)RIXSspectrawithE||abat(a)(π/2,0) for in-plane polarization. The lower of the two 2 eV fea- (b) (π,0) (c) (π/2,π/2), and (d) (π π). tures may be intrinsically weaker, which could explain why we are unable to discern it for out-of-plane polarization. As an alternative scenario, consistent with the identificationof two distinct features,a detailed analysis in the next section. of the scattering configurations reveals that the optical- limit Ramanefficiencies setby geometryarequite different in each case. It is possible that the suppression of V. DISCUSSION the higher energy feature for the plane-polarized experiment reflects details of the symmetry of this excitation. Further experimentation is required to resolve this pos- The presence of a ∼ 2 eV resonance feature has been sibility. The above is consistent with earlier suggestions discussedinconnectionwithexcitationsacrossthecharge transfergap.7,9,10,12,22 InLa CuO ,thisexcitationisob- basedonCuK-edgeXAS.37Especiallyforin-planepolar- 2 4 ization, the ∼2 eV excitation still maintains its strength served only for transitions into well-screened states, in along [π,π], whereas it is weakened at the same location which the 1s core hole is screened by a valence elec- for out-of-plane polarization. Therefore, it indeed ap- tron from a neighboring CuO plaquette, leaving a dou- 4 bly occupied Cu+ ion (3d10) and a hole on the neigh- pears that these two low-energy excitations have a distinct physical origin, consistent with the high-resolution boring plaquette. It has been suggested that the nonlocal hole can form a Zhang-Rice singlet,34 which can EELS work.36 propagateefficientlythroughtheantiferromagneticback- The magnitude of the dispersion will be an impor- ground, and that this singlet could form a strong bond tant factor in the eventual determination of the origin with the Cu+ quasiparticle and become even more dis- of the charge excitations. Our La CuO data reveal 2 4 persive as a bound exciton.35 However, high-resolution charge-transfer features that are remarkably similar to electron-energy-loss spectroscopy (EELS) on the related those for Hg1201,12 a result that is qualitatively differ- Mott insulatorSr CuO Cl suggeststhe existence ofan- ent from prior work on La CuO .10 As summarized in 2 2 2 2 4 other charge-transfer excitation, with slightly lower en- Fig. 5, below 4 eV we identify one additional branch ergy,thatinvolvesonly the localCuO plaquette. These at ∼3 eV for both polarization conditions. For exam- 4 7 chargeexcitationsin the higher-energy(transfer)region, a more complex approach appears to be necessary. An initial suggestionconcerningthe 4 eV excitationinvoked anexcitonicstateofunspecifiedorigin,10 yetmorerecent work12,32,42 suggests that the higher-energyspectral features ought to be described in terms of a multiband picture. Considering the involvement of bonding and non- bonding oxygen 2p and 2p orbitals, as well as charge- σ z transfer processes through local and nonlocal screening channels, there exist many candidate modes. Eskes and Sawatzky43 also find that triple-band physics, including the Zhang-Rice triplet states, as well as d3z2−r2-orbitals are relevant up to about 7 eV in binding energy. Fur- therexperimentation,includingsymmetryanalysis,isre- quired to resolve the physical origin of the high-energy spectral features. We will now discuss the photon energy and polarization dependence of the observed higher-energy features. Figure 9 (a) shows zone-center contour plots of incident energyversusenergy transfer. The out-of-planedata are fromRef. 10andweretakenwithcoarserincidentenergy stepsize(1eVvs. 0.5eV),butspanawiderincidenten- ergy range than our data in Fig. 3 (b). The in-plane data (inset) are from our present study and were taken with incident-energy increments of 1 eV. As mentioned FIG.8: (Coloronline)(a)La2CuO4unitcell. InaCuK-edge above, the center of mass of the RIXS spectra shifts to absorption process,several4pstatescanbereached,depend- higher energy transfer as the incident photon energy in- ing on thepolarization of the incident photon relative to the creases. This variation is identified in Fig. 9 (a) as a crystal axes. (b) 4pσ configuration created by absorption of “streak” of intensity from 2 to ∼7 eV which, instead aEi ∼8996 eVphotonwithE||ab. (c)4pz configurationcre- ated by absorption of aEi ∼8991 eV photonwith E||c. The of extending horizontally as one would expect for reso- emissive process discussed in the text corresponds to relax- nancesassociatedwithfixedincidentphotonenergy,tilts ation from state (b) to state (c). towardthe upper right corner. We will first discuss that this streak of intensity can be interpreted in a shakeup picture, which uses third-order perturbation theory. As an alternative scenario for the slope-1 component of this streakofintensitybetween∼5and7eV,wewillthendis- ple, for out-of-plane polarization, simple fits to a sinu- cuss fluorescence-like emission processes due to 1s → 4p soidalform(showninFig. 5)yielddispersionsof120(30), transitions into a narrow continuum 4p band. 410(110),and490(70)meVforthe2,3,and4eVfeatures Pioneering work on La CuO (Ref. 7) utilized in- alongbothhigh-symmetrydirections. The5eVfeatureis 2 4 plane polarizationand revealedone single excitation be- dispersion-less within the experimental uncertainty, and tween ∼3 and 6 eV, and it was found that the peak po- we note a possible anomaly at (π, π). Below, we discuss sition varied nonlinearly with incident energy. These re- the possibility that the 5 eV feature may actually be the sults were interpreted in terms of a shake-up of the 3d resultofashake-upexcitationat7.2eV.The∼100meV electron system, and explained within third-order per- dispersion of the nonlocal 2 eV excitation is less than turbation theory. Following this treatment, which was the Zhang-Rice singlet bandwidth of 250 meV identified byangleresolvedphotoemissionspectroscopy.38,39,40This formulated in detail in Ref. 7,19,44, the scattering am- plitude in third-order perturbation theory is given by observation challenges the notion of an excitonic picture to explain the dispersionof the charge-transfergapexci- tation. Inprinciple,iftheelectronicstatesoftheelectron hf|b |mihm|H |nihn|b |ii and hole are asymmetric,41 the observed dispersion may A ∝ 2 c 1 (E −E −E −iΓ )(E −E −E −iΓ ) either represent the bandwidth of the upper Hubbard Xm,n m f,el f m n i,el i n band,iftheelectronismoremobile,oroftheZhang-Rice hf|b H b |ii 2 c 1 singlet band, if the hole is freer to move. However, it is ≈ (1) (E −E −iΓ )(E −E −iΓ ) ex,2 f 2 ex,1 i 1 difficult to reconcile the small electron-hole-pair dispersion of ∼ 100 meV with the relatively large Zhang-Rice where |mi and |ni denote different intermediate states singlet bandwidth, unless the observed behavior repre- containing a virtual 1s4p exciton, E and E are the in- i f sents a significant core hole effect. cident and final photon energies, E and E the ini- i,el f,el In order to understand the appearance of multiple tial energy andfinal energies of the electronsystem, and 8 b and b are absorption and emission operators19. A 1 2 simplification is made by defining the constants E ≡ ex,1 E −E , and E ≡ E −E , which can be con- n i,el ex,2 m f,el sideredincident andfinalresonantenergies,respectively, and the respective inverselifetimes Γ and Γ .7,19. With 1 2 this simplification, we ignore the details of the interme- diatestates. Wenotethatthisformulacontainsseparate denominators involving incident and final photon energies. The scattering intensity is given by I ∼ L(E −E ,Γ )L(E −E ,Γ ) ex,1 i 1 ex,2 f 2 ×|hf|b H b |ii|2δ(E −E −ω), (2) 2 c 1 f,el i,el whereL(E,Γ )representsaLorentzianfunctionwithhalf i width Γ . i To model the shake-up process, we replace |hf|b H b |ii|2 with a sum of Gaussian functions, 2 c 1 each representing a distinct symmetry-allowed shake-up excitation with energy ∆ and heuristic inverse life-time Γ : s FIG. 9: (Color online) (a) Contour plot of La2CuO4 RIXS intensity: incident photon energy vs. energy transfer for E || (ω−∆)2 c (from Ref.10) and E || ab (inset; present work). The white G(ω)=exp − . (3) (cid:18) 2Γ2 (cid:19) dashed lines indicate the two trends with different slope, as s discussed in the text. (b) Calculations following third-order The single excitation observed in prior work on perturbation theory,as discussed in the text. La CuO (Ref. 7) was described with Γ = Γ = 2.38 2 4 1 2 eV, E = E = 8995 eV, Γ = 3.9 eV and ∆ = 6.1 ex,1 ex,2 s eV. Since we have been able to resolvea multiplet of ex- the value of excitation energy ∆, the two resonances citations rather than a single excitation, we apply third- mergeandbecomeindistinguishable. IfΓ orΓ is small orderperturbationtheory to the entire multiplet. In our 1 2 comparedto ∆ andΓ , one of the two resonancesnearly calculation,wefindthatfourvalenceexcitationswithen- s disappears, and the response either is either a circular ergies∆=2.3,3,4and7.2eVareadequatetorepresent region or a slope-one streak. By carefully choosing Γ the spectra for both polarization conditions. The result i andE (i=1,2)for eachcomponent, wewe were able of this calculation is shown in Fig. 9 (b). For this par- ex,i to emulate the main characteristics of the experimental ticular calculation, Γ to be 0.4, 1.0, and 1.1 eV for the s data. As seen from Fig. 9 (a), the lowest three exci- lower three excitations, respectively. We find that our tations are only strongly resonant at the well-screened data can be adequately represented if the 7.2 eV molec- state, while the 7.2 eV excitation is associated with the ularorbitalexcitationresonatesfortransitionsinto both poorly-screened state. Consequently, E should be dif- well-andpoorlyscreenedstates,withvariablecharacter- ex ferent for the latter. For the slope-1 streak component, istics. This excitation is represented by two Gaussians theincidentresonanceenergyisthesameasforthelower (withΓ setto3.9eV(well-screened)and1.9eV(poorly s three excitations, but we find it necessary to choose a screened)),andwethereforeconsideratotaloffiveGaus- slightly smaller E . sians for four excitations. ex,2 We note that the 5 eV feature discussed in Secs. III For out-of-plane polarization, we set Γ1 = Γ2 = 2 eV andIVwasnotconsideredintheabovecalculationssince and Eex,1=Eex,2 = 8990 eV for the lower three excita- it is most prominent at high incident energies and shifts tions. Forthe7.2eVcomponentthatresultsinthediag- with Ei. It therefore seems likely that this feature may onalstreak,wesetΓ1=1.7eV,Γ2=1.5eV,Eex,1=8990 actually be associated with a resonance of the 7.2 eV eV, and Eex,2 = 8989 eV, and for the other 7.2 eV com- molecular orbital excitation at the well-screened state. ponent we chose Γ1 = 1.0 eV, Γ2 = 1.5 eV and Eex,1= Due to the doubleresonance denominator, each compo- Eex,2 = 8998.5 eV. For in-plane polarization, we simply nent may, in principle, show two separate resonances for shifted all Eex values by 4 eV, and adjusted the relative E = E and E = E . For the final-energy reso- intensitybetweenthe threelow-lyingexcitationsandthe i ex,1 f ex,2 nance, as the incident energy changes, the peak position two 7.2 eV components. of the excitation shifts to ω = E −E so as to The above analysis has two important implications. peak i ex,2 maintain the same final energy. In the twodimensional Oneisthatthe“5eV”featureistobeviewedasashake- contour plot of Fig. 9, this manifests itself as a slope-1 up excitation at 7.2 eV. The second important impli- streak of intensity. cation is that this local molecular-orbital excitation not The inverse lifetimes determine the shapes of the res- onlyresonatesatthepoorlyscreenedstateatwhichcop- onant spectral features. When Γ , Γ , and Γ approach per has an open-shell configuration (1s3d94p), but also 1 2 s 9 at the well-screened state (1s3d10L4p). This is different terpreted through the following complex dynamical pro- from findings for CuO and from the argument that the cess illustrated in Fig. 8: La CuO absorbs a photon of 2 4 observed shake-up excitation requires the 3d9 open-shell energyE ∼ 8995eV with polarizationE⊥c,creatingon i configuration and should be absent at the well-screened a Cu site a well-screened 1s core hole and an electron in state19. However, we note that the molecular-orbital a 4p orbital [Fig. 8 (b)]. The 4p electron then evolves σ σ excitation was observed to be resonant at both well- into a 4p state [Fig. 8 (c)] via a subsequent relaxation z screened and poorly-screened states in superconducting process. Finally, the 4pz electron recombines with the HgBa CuO 12. 1s core hole, emitting a photon with energy ∼ 8991 eV. 2 4+δ We will now discuss a different interpretation for the However, for out-of-plane polarization, electrons are al- slope-1componentofthestreakofintensitythatrelatesit ready excited into a 4p state, and this relaxation will z toaresonantexcitationintothecontinuumofunoccupied not occur. Nevertheless, the envelope for out-of-plane states.45,46 Figure 2 shows the zone-center RIXS spectra polarization lies also 4 eV below that of the main edge. together with the respective absorption spectra for both This interpretation therefore requires the existence of a polarization condtions. The RIXS intensity is plotted discrete lower-energy state with energy 8987 eV for 4pz versusfinalphotonenergy(E )insteadofenergytransfer electron to relax into. Interestingly, in Ref. 7, a res- f (ω). Wefindthatthe spectracollapsetoapeak,withan onance of approximately this energy was observed with envelope of approximatelyLorentzianshape, centered at in-planepolarization. Furtherexperimentationcombined about E = 8991 eV for in-plane polarization and E = with theoreticalmodeling canbe expected to resolvethe f f 8987eVforout-of-planepolarization. The peak position origin of the slope-1 streak of intensity. lies∼4eVbelowthephotoabsorptionthresholdforboth polarization conditions. This observation is consistent with the presence of slope-1 streaks in the contour plots VI. SUMMARY forbothpolarizationconditions: forin-planepolarization this streak starts at E ∼ 8995 eV and for out-of-plane i In summary, we have presented a detailed Cu K- polarization it starts at E ∼ 8991 eV. i edge RIXS study of La CuO in which we resolve a 2 4 The photon-absorption process near the Cu K edge multiplet of charge-transfer excitations in the 1-7 eV is comprised of transitions to either narrow molecular range. We suggest several interpretations to explain the orbitals or continuum unoccupied states. For discrete polarization-dependent spectra. A calculation applying levels, such as the well-screened and poorly screened third-orderperturbationtheoryintroducesafinal-energy states, induced valence excitations resonate and follow resonanceandsuccessfullysimulatesthe maincharacter- the Raman-Stokes law as E crosses the discrete levels, i istics of the spectra for both polarizationconditions. On i.e., the excitation energies ∆ do not vary with incident the other hand, transitions to continuum 4p bands that photon energy. On the other hand, a resonance due to begin at the main absorption edge as well as 4p → 4p σ z transitionstocontinuumstatesbehavesdifferently. Here, relaxation are offered as alternative explanations to the the resonance condition is fulfilled for every incident en- fluorescence-like component in the contours. These pro- ergy tuned to the continuum, and the subsequent emis- posalsallemphasizetheimportantroleofthe4pelectrons sion is independent of E once the incident energy in- i in the RIXS cross section. creasesabovetheloweredgeofthecontinuum. WhenE i is below the edge, the resonant emission should also follow the Raman-Stokes behavior, but the spectral weight may be suppresseddue to the small density of states below the edge. Our observation is consistent with the ex- VII. ACKNOWLEDGEMENTS istenceofacontinuousunoccupiedbandof4psymmetry, with an edge at ∼ 8991 eV and ∼ 8995 eV for the re- The authors gratefully acknowledge valuable discus- spective polarizations,and a width of about 34 eV. This sions with P. Abbamonte, U. Bergmann, J. van den would support the view that a combination of an ex- Brink, T. P. Devereaux, M. V. Klein, Y. J. Kim, K.-W. tended picture of itinerant 4p electrons and of localized Lee, R. S. Markiewicz, W. E. Pickett, K. M. Shen, M. molecular orbitals is necessary to interpret the K-edge vanVeenendaal,andF.C.Zhang. The workatStanford absorption spectra21 and the nature of the intermediate University was supported by the DOE under Contract states in RIXS. No. DE-AC02-76SF00515 and by the NSF under Grant Wenotethattheenvelopeofthein-planedatacollapse No. 0405655. Work at the CMC-XOR Beamlines is sup- showninFig. 2(a)isverysimilarinshapeandpositionto ported in part by the Office of Basic Energy Sciences of theXAScorrespondingtothewell-screenedstateforout- the U.S. Dept. of Energy and by the National Science of-planepolarization[Fig. 2(b)]. Thisleadsustoathird Foundation Division of Materials Research. Use of the interpretation of the slope-one contribution to the RIXS Advanced Photon Source is supported by the Office of cross section shown in Fig. 9(a). Specifically, it suggests BasicEnergySciences ofthe U.S. DepartmentofEnergy thatpartofthein-plane-polarizedRIXSsignalcanbein- under Contract No. W-31-109-Eng-38 10 1 C.A.Burns,P.Abbamonte,E.D.Isaacs,andP.M.Platz- (2005). man, Phys. Rev.Lett. 83, 2390 (1999). 21 J. M. Tranquada, S. M. Heald, W. Kunnmann, A. R. 2 C. Dallera, M. Grioni, A. Shukla, G. Vanko, J. L. Sarrao, Moodenbaugh, S. L. Qiu, Y. W. Xu, and P. K. Davies, J. P. Rueff, and D. L. Cox, Phys. Rev. Lett. 88, 196403 Phys. 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