Table Of ContentIncident-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 inci-
dent 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 absorp-
tion: 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 geom-
etry, 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 reso-
lution 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 cho-
sen 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 N´eel 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 sub-
sequent measurements.23,28 Our primary focus here is to
investigate in greater detail the incident-energy and po-
larization 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 pho-
ton energy.
Figure3(a)showsrepresentativelinescansatthezone
center, taken at three different incident energies with
out-of-plane polarization. These data resemble previ-
ous work10, yet closer inspection reveals additional fea-
tures. 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 inci-
dent 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 po-
larization. Previous results10, obtained with out-of-plane po-
larization, 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 exci-
tation has no discernible dispersion, the latter appears
to disperse by 100 150 meV toward the zone bound-
ary. 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 polar-
ization. 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 differ-
ent in each case. It is possible that the suppression of
V. DISCUSSION the higher energy feature for the plane-polarized experi-
ment 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 dis-
tinct physical origin, consistent with the high-resolution
boring plaquette. It has been suggested that the non-
local 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 fea-
tures ought to be described in terms of a multiband pic-
ture. 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 polariza-
tion 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 disper-
sion 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 ener-
gies. 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 fol-
low the Raman-Stokes behavior, but the spectral weight
may be suppresseddue to the small density of states be-
low 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. Rev.B 44, 5176 (1991).
(2002). 22 A. Kotani and S.Shin, Rev.Mod. Phys. 73, 203 (2001).
3 P. Wernet, D. Nordlund, U. Bergmann, M. Cavalleri, M. 23 E. Collart, A. Shukla, J. P. Rueff, P. Leininger, H. Ishii,
Odelius, H. Ogasawara, L. A. Naslund, T. K. Hirsch, L. I. Jarrige, Y. Q. Cai, S. W. Cheong, and G. Dhalenne,
Ojamae, P. Glatzel, L. G. M. Pettersson, and A. Nilsson, Phys. Rev.Lett. 96, 157004 (2006).
Science 304, 995 (2004). 24 R. J. Birgeneau, M. Greven, M. A. Kastner, Y. S. Lee,
4 F. Sette, G. Ruocco, M. Krisch, U. Bergmann, C. Mas- B.O.Wells,Y.Endoh,K.Yamada,andG.Shirane,Phys.
ciovecchio, V. Mazzacurati, G. Signorelli, and R. Verbeni, Rev.B 59, 13788 (1999).
Phys. Rev.Lett. 75, 850 (1995). 25 O. P. Vajk, P. K. Mang, M. Greven, P. M. Gehring, and
5 M.d’Astuto,P.K.Mang,P.Giura,A.Shukla,P.Ghigna, J. W. Lynn,Science 295, 1691 (2002).
A.Mirone,M.Braden,M.Greven,M.Krisch,andF.Sette, 26 S. M. Heald, J. M. Tranquada, A. R. Moodenbaugh, and
Phys. Rev.Lett. 88, 167002 (2002). Y. W.Xu, Phys.Rev.B 38, 761 (1988).
6 J. P. Hill, C. C. Kao, W. A. L. Caliebe, M. Matsubara, 27 K.-W. Lee and W. E. Pickett, Private Communication
A.Kotani, J.L.Peng,andR.L.Greene,Phys.Rev.Lett. (2006).
80, 4967 (1998). 28 Y. J. Kim, privatecommunication (2006).
7 P.Abbamonte,C.A.Burns,E.D.Isaacs,P.M.Platzman, 29 P. Glatzel and U. Bergmann, Coord. Chem. Rev. 249, 65
L. L. Miller, S. W. Cheong, and M. V. Klein, Phys. Rev. (2005).
Lett. 83, 860 (1999). 30 G. D. Mahan, Many-Particle Physics (Springer Verlag,
8 K. H¨am¨al¨ainen, J. P. Hill, S. Huotari, C. C. Kao, L. E. New York,2000).
Berman, A.Kotani, T. Ide,J. L. Peng, and R. L. Greene, 31 F. deGroot, Chem. Rev.101, 1779 (2001).
Phys. Rev.B 61, 1836 (2000). 32 T. Nomura and J. Igarashi, Phys. Rev. B 71, 035110
9 M. Z. Hasan, E. D. Isaacs, Z. X. Shen, L. L. Miller, (2005).
K. Tsutsui, T. Tohyama, and S. Maekawa, Science 288, 33 K.Tsutsui,T.Tohyama,andS.Maekawa,Phys.Rev.Lett.
1811 (2000). 91, 117001 (2003).
10 Y. J. Kim, J. P. Hill, C. A. Burns, S. Wakimoto, R. J. 34 F.C.ZhangandT.M.Rice,Phys.Rev.B37,3759(1988).
Birgeneau, D. Casa, T. Gog, and C. T. Venkataraman, 35 F.C.ZhangandK.K.Ng,Phys.Rev.B58,13520(1998).
Phys. Rev.Lett. 89, 177003 (2002). 36 A. S. Moskvin, R. Neudert, M. Knupfer, J. Fink, and
11 Y.J.Kim,J.P.Hill,S.Komiya,Y.Ando,D.Casa,T.Gog, R.Hayn,Phy.Rev.B 65, 180512 (2002).
andC.T.Venkataraman,Phys.Rev.B70,094524(2004). 37 H. Tolentino, M. Medarde, A. Fontaine, F. Baudelet,
12 L. Lu, X. Zhao, G. Chabot-Couture, J. N. Hancock, N. E. Dartyge, D. Guay, and G. Tourillon, Phys. Rev. B 45,
Kaneko, O. P. Vajk, G. Yu, S. Grenier, Y. J. Kim, D. 008091 (1992).
Casa,T.Gog,andM.Greven,Phys.Rev.Lett.95,217003 38 B. O. Wells, Z.X. Shen,A.Matsuura, D. M. King, M. A.
(2005). Kastner,M.Greven,andR.J.Birgeneau,Phys.Rev.Lett.
13 K.Ishii, K. Tsutsui, Y.Endoh,T. Tohyama, S.Maekawa, 74, 964 (1995).
M. Hoesch, K. Kuzushita, M. Tsubota, T. Inami, J. 39 C. Du¨rr, S. Legner, R. Hayn, S. V. Borisenko, Z. Hu, A.
Mizuki, Y. Murakami, and K. Yamada, Phys. Rev. Lett. Theresiak,M.Knupfer,M.S.Golden,J.Fink,F.Ronning,
94, 207003 (2005). Z.-X. Shen, H. Eisaki, S. Uchida, C. Janowitz, R. Mu¨ller,
14 K.Ishii,K.Tsutsui,Y.Endoh,T.Tohyama,K.Kuzushita, R. L. Johnson, K. Rossnagel, L. Kipp, and G. Reichardt,
T. Inami, K. Ohwada, S. Maekawa, T. Masui, S. Tajima, Phys. Rev.B 63, 014505 (2000).
Y. Murakami, and J. Mizuki, Phys. Rev. Lett. 94, 919 40 F.Ronning,C.Kim,K.M.Shen,N.P.Armitage,A.Dam-
(2005). ascelli, D. H. Lu, D. L. Feng, Z.-X. Shen, L. L. Miller, Y.
15 K.Ishii,T.Inami,K.Ohwada,K.Kuzushita,J.Mizuki,Y. J.Kim,F.Chou,andI.Terasaki,Phys.Rev.B67,035113
Murakami, S.Ishihara,Y.Endoh,S.Maekawa, K.Hirota, (2003).
and Y.Moritomo, Phys.Rev.B 8, 224437 (2004). 41 K. H. Ahn, A. J. Fedro, and M. van Veenendaal, cond-
16 S. Grenier, J. P. Hill, V. Kiryukhin, W. Ku, Y. J. Kim, mat/0412635 (2004).
K. J. Thomas, S. W. Cheong, Y. Tokura, Y. Tomioka, D. 42 R. S. Markiewicz and A. Bansil, Phys. Rev. Lett. 96,
Casa, and T. Gog, Phys. Rev.Lett. 94, 047203 (2005). 107005 (2006).
17 C. C. Kao, W. A. L. Caliebe, H. J. B, and J. M. Gillet, 43 H.Eskes and G. Sawatzky,Phys. Rev.B 44, 9656 (1991).
Phys. Rev.B 54, 16361 (1996). 44 P.M.Platzman and E.D.Isaacs, Phys.Rev.B 57, 11107
18 M. H. Krisch, C. C. Kao, F. Sette, W. A. Caliebe, (1998).
K. H¨am¨al¨ainen, and J. B. Hastings, Phys. Rev. Lett. 74, 45 F. Gel’Mukhanov and H. ˚Agren, Phys. Rev. B 57, 2780
4931 (1995). (1998).
19 G. D¨oring, C. Sternemann, A. Kaprolat, A. Mattila, 46 F. Gel’Mukhanov and H. ˚Agren, Phys. Rep. 312, 87
K.H¨am¨al¨ainen,andW.Schulke,Phys.Rev.B70,085115 (1999).
(2004).
20 P. Glatzel and U. Bergmann, Coord. Chem. Rev. 249, 65
