Local orbital occupation and energy levels of Co in Na x CoO 2 : A soft x-ray absorption study

H.-J. Lin,1 Y. Y. Chin,1,2,3 Z. Hu,3 G. J. Shu,4 F. C. Chou,1,4 H. Ohta,5 K. Yoshimura,5 S. Hébert,6 A. Maignan,6 A. Tanaka,7 L. H. Tjeng,3 and C. T. Chen1 1National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan 2II. Physikalisches Institut, Universität zu Köln, Zülpicher Str.77, 50937 Köln, Germany 3Max-Planck-Institut für Chemische Physik fester Stoffe, Nöthnitzer Str. 40, 01187 Germany 4Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan 5Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan 6Laboratoire CRISMAT, UMR 6508, CNRS-ENSICAEN, 6 Bd. Maréchal Juin, 14050 Caen Cedex, France 7Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan Received 23 September 2009; revised manuscript received 5 March 2010; published 31 March 2010

The electronic properties of sodium cobalt oxides have attracted considerable experimental and theoretical attention in recent years since the discovery of the exceptionally high thermopower in NaCo 2 O 4 ͑Refs. 1 and 2͒ and superconductivity in the hydrated cobaltate Na 0.35 CoO 2 • 1.3H 2 O. 3 This class of compounds is particularly interesting because there are parallels to be made with the high-T c copper oxide superconductors: for both type of compounds, the electrical conductivity is governed by charge carriers which move mainly in two-dimensional planes separated by insulating layers. 4Yet there are also differences.The cuprates have a planar square lattice of the transition metal ions, while the sodium cobalt oxides exhibit a triangular lattice.The local coordination is also different.The Na x CoO 2 consists of alternating layers of CoO 2 and Na along the c axis, in which the edge-sharing CoO 6 octahedra in the CoO 2 layers have their threefold ͑111͒ axis oriented parallel to the c axis and show a noticeable compression along this direction.The local point group symmetry of the Co ion is therefore close to D 3d .
There is a considerable debate in the literature concerning the local electronic structure of the Co ions in Na x CoO 2 , in particular, about their spin state, orbital polarization and crystal field levels.  Fig1 illustrates the relevant oneelectron energy levels.28 In O h symmetry, the 3d atomic states are split into the well-known e g and t 2g orbitals.In D 3d symmetry, the t 2g states are further split into a two-fold degenerate e g and a nondegenerate a 1g , while the e g states remain doubly degenerated, now called e g . Ealier studies suggested a low-spin ͑LS͒ state for both the Co 3+ 3d 6 ion ͑S =0͒ and the Co 4+ 3d 5 ͑S =1/ 2͒ in the Na x CoO 2 system.5,7,8 Based on optical conductivity data of Na 0.82 CoO 2 , however, Bernhard et al. proposed that, following the formation of the Co 3+ / Co 4+ charge ordering state, the Co 4+ ion would drive a displacement of the neighboring oxygens towards itself and thus cause a local symmetry lower than D 3d for its adjacent Co 3+ ions.6 They concluded that this mechanism would lead to a further splitting of the e g doublet and stabilizes the intermediate-spin ͑IS͒ state with S = 1.Moreover, a subsequent theoretical spin-orbital-polaron model supported the IS state for Co 3+ in the Na x CoO 2 at large x, i.e., small hole concentrations, 9 with a local symmetry lower than D 3d .A recent polarization dependent soft x-ray absorption ͑XAS͒ experiment and a theoretical analysis thereof 10,11 claimed also to have observed the presence of appreciable crystal fields with a symmetry lower than D 3d , yet they supported the LS and not the IS scenario.It was also concluded, however, that the samples used in these XAS studies showed Co 3+ / Co 4+ spectral ratios which were inconsistent with their nominal Na contents.
Here we report our investigation on the local electronic structure of Na x CoO 2 using polarization dependent XAS experiments at the Co L 2,3 edges in combination with detailed full-multiplet cluster model calculations.We found that the Co 3+ / Co 4+ spectral contents do match the nominal Na concentration and that the spectral features can be excellently described using the D 3d local symmetry for the Co ions.We are also able to obtain quantitative estimates for the orbital polarization of the charge carriers.
Single crystals of various Na concentrations were prepared from Na 0.82 CoO 2 through subsequent electrochemical deintercalation procedures.Details of the crystal growth are described elsewhere. 29,30The XAS experiments were carried out at the dragon beamline of the National Synchrotron Radiation Research Center ͑NSRRC͒ in Taiwan.The photon energy resolution at the Co L 2,3 edges ͑h Ϸ 770-800 eV͒ was set at 0.3 eV.The degree of linear polarization of the incident light was ϳ99%.The spectra were recorded using the total electron yield method and a CoO single crystal was measured simultaneously in a separate chamber to calibrate the photon energy with an accuracy better than 0.01 eV for all polarization dependent measurements.Clean sample surfaces were obtained by cleaving the single crystals in situ at pressures in the low 10 −10 mbar range.
The samples were tilted with respect to the incoming beam, so that the Poynting vector of the light makes an angle of ␣ = 70°with respect to the c-axis surface normal.To change the polarization, the sample was rotated around the Poynting vector axis as depicted in Fig. 2, and , the angle between the electric-field vector E and the c axis, can be varied between 20°and 90°.This measurement geometry allows for an optical path of the incoming beam which is independent of , which guarantees a reliable comparison of the spectral line shapes as a function of polarization.The spectra for the electric field vector E parallel to the c axis can be obtained from the measured data by using the formula, I ʈ = ͓I ip − I Ќ cos 2 ͑␣͔͒ / sin 2 ͑␣͒, where I ip and I Ќ are the measured intensities with = 20°and = 90°, respectively, and sin 2 ͑␣͒ = 0.883 and cos 2 ͑␣͒ = 0.117.The isotropic spectra can be obtained from I ʈ +2 I Ќ .
Figure 3 shows the isotropic Co L 2,3 XAS spectra of Na 0.5 CoO 2 and Na 0.75 CoO 2 taken at room temperature.We have also included the spectra of LaCoO 3 ͑reproduced from Ref. 31͒ and BaCoO 3 .The displayed LaCoO 3 spectrum is the one measured in the low-temperature phase and serves as a reference for a LS Co 3+ system. 31Analogously, the BaCoO 3 is taken here as a reference for a LS Co 4+ system. 32For the BaCoO 3 spectrum we have removed the Ba M 4,5 white lines located at 784 and 798 eV energies using the Ba M 4,5 spectrum of BaFeO 3 . 32To facilitate the following discussion, the LaCoO 3 and BaCoO 3 spectra shown in Fig. 3 have been shifted by 0.6 eV and −0.4 eV, respectively, in energy.
The spectra are dominated by the Co 2p core-hole spinorbit coupling, which split the spectrum roughly into two parts, namely, the L 3 ͑h Ϸ 782 eV͒ and L 2 ͑h Ϸ 797 eV͒ white line regions.The lineshape of the spectrum depends strongly on the multiplet structure given by the atomiclike Co 2p −3d and 3d −3d Coulomb and exchange interactions, as well as by the solid including local crystal fields and hybridization with the oxygen ligands.In the Na 0.5 CoO 2 and Na 0.75 CoO 2 spectra we can observe distinct features which we label as A-B-C in the L 3 region, and D-E-F in the L 2 .
To help identify the origin of these features, we now turn our attention to the spectra of the reference materials LaCoO 3 and BaCoO 3 .There we can see that a LS Co 3+ has structures b-c and e-f resembling the Na x CoO 2 B-C and E-F, respectively, while a LS Co 4+ has peaks ␣ − ␤ and ␦ − ⑀ very similar to the Na x CoO 2 A-B and D-E, respectively.This strongly suggests that features A and D in the sodium cobaltates are pure signatures of the Co 4+ component, while C and F are due to the Co 3+ .Peaks B and E, on the other hand, have contributions from both the Co 3+ and Co 4+ .The assignment of peak A to Co 4+ is consistent with the observation that its intensity decreases in going from Na 0.5 CoO 2 to Na 0.75 CoO 2 .The same is also the case for peak D. It becomes even too weak to be visible in the x = 0.75 spectrum.
As an ansatz for further confirmation, we have made composite spectra consisting of 50% LaCoO 3 plus 50% BaCoO 3 and 75% LaCoO 3 plus 25% BaCoO 3 .In doing this we have shifted the LaCoO 3 and BaCoO 3 slightly as already mentioned above.We would like to note that it is an experimental observation to have compounds showing slight energy differ- ences despite the fact that they have the same valence and we ascribe this to differences in the screening during the XAS process associated with differences in the local and extended environment of the ions in the compounds.We can see from curves ͑aЈ͒ and ͑bЈ͒ in Fig. 3 that these composite spectra reproduce reasonable well those of Na 0.5 CoO 2 and Na 0.75 CoO 2 , suggesting that the Co 3+ / Co 4+ spectral contents are quite consistent with the nominal Na concentrations.In fact, these experimental findings falsify the peak assignments made in the recent XAS study 10,11 mentioned above and thus also their claim concerning the improper Na concentration in their samples.
In our attempt to interpret the isotropic spectra of the sodium cobaltates, we note first of all that the match with the composite spectra of the LaCoO 3 and BaCoO 3 as described above is very reasonable but not perfect.This is not surprising since it simply indicates that the local and extended coordination of the Co ions in the three types of compounds is sufficiently different to cause observable differences in, e.g., the crystal field and Madelung parameters.To obtain more detailed information about the local electronic structure of the sodium cobaltates we now will make use of the polarization dependence of the spectra.Figure 4 shows the Co L 2,3 XAS spectra of Na 0.5 CoO 2 ͑curve a͒ and Na 0.75 CoO 2 ͑curve e͒ for E Ќ c and E ʈ c.We can clearly observe that there is a pronounced polarization dependence and that this so-called linear dichroic effect is stronger for the x = 0.5 composition than for the x = 0.75.This directly suggests that the hole-like charge carriers are orbitally very anisotropic.
To extract the information concerning the orbital occupation and level splittings, we will analyze the L 2,3 XAS spectra and the linear dichroism therein using the configurationinteraction cluster model that includes the full-atomic multiplet theory and the hybridization of the central transition metal ion with the O 2p ligands. 33,346][37][38][39][40] Crystal structure data 41 reveal that the local coordination of the Co ions can be well approximated by a CoO 6 cluster in D 3d point group.We will use this cluster in our calculations with the XTLS 8.3 code. 34n the simulations for the Co 3+ site we started with the parameter values found for LiCoO 2 , 42,43 a compound having the Co 3+ ions arranged also in a triangular lattice.These parameter values are very close to those of LaCoO 3 , 31,[44][45][46][47][48][49][50] except that here the O 2p to Co 3d charge transfer energy ⌬ 3+ is about 4 eV instead of about 2 eV, a finding which is probably related to a difference in the Madelung potential associated with the different arrangement of the Co ions in the crystal.5][46] A summary of the parameter values is listed in Ref. 51.In the calculations we have tuned only the trigonal crystal-field parameters 28 D and D as to obtain the best simulation of the experimental spectra.In the calculations, we have used slightly different optimal parameter values for the two different compositions as we will describe below in more detail.
The results of the simulations are depicted in Fig. 4. We can observe that we are able to obtain excellent simulations of the experimental spectra.Also here we have made composite spectra consisting of incoherent sums of 50% Co 3+ plus 50% Co 4+ and 75% Co 3+ plus 25% Co 4+ to reproduce the Na 0.5 CoO 2 and Na 0.75 CoO 2 spectra, respectively.The excellence of the fits indicates first of all that we have obtained a reasonable detailed understanding of the local electronic structure.It also demonstrates that a strict D 3d local symmetry for both Co 3+ and Co 4+ ions is sufficient to explain the observed spectral features and polarization dependence, refuting the need for a local symmetry lower than D 3d for the Co 3+ ion as claimed previously. 11Moreover, the fact that the Co 3+ / Co 4+ spectral fractions used to reproduce the spectra matches the nominal Na concentrations also indicates that the samples are of proper stoichiometry.We note that the separate Co 3+ and Co 4+ simulations also match reasonably well those of LaCoO 3 and BaCoO 3 , respectively, justifying our approach in Fig. 3 to explain the sodium cobaltate spectra using these two reference compounds.
Let us analyze the origin of the polarization dependence in the XAS spectra of Na x CoO 2 .In the absence of the trigonal crystal field, the t 2g eigenstates of the O h point group can be rewritten as

the t 2g
Ϯ and e g Ϯ , i.e., V mix = ͗t 2g Ϯ ͉H͉e g Ϯ ͘, leading to the formation of the e g and the e g orbitals.We will call D trig 0 the bare trigonal crystal-field splitting between the a 1g and e g orbitals: bare because the splitting is taken without the energy stabilization of the e g orbital due to the mixing of the t 2g Ϯ with the higher lying e g Ϯ orbitals.We will further define one more quantity, namely, D trig ef f , which represents the effective a 1g − e g splitting for the Co 4+ cluster after including the t 2g Ϯ − e g Ϯ mixing and the hybridization with the oxygen ligands.This quantity can be read directly from the Co 4+ total energy diagram calculated with the Co 3d spin-orbit interaction switched off.
Figure 5 panel ͑a͒ shows the polarization dependence for the Co 3+ site ͑left͒ and the Co 4+ ͑right͒ as a function of the bare trigonal crystal field D trig 0 .Starting with the spectrum for the Co 3+ , it is evident that there is no polarization dependence when the trigonal crystal field is absent since in this effectively O h symmetry the electronic configuration for this low spin Co 3+ is a highly symmetric closed shell 3dt 2g 6 .The peaks BЈ, CЈ, EЈ, and FЈ can be assigned to transitions from the 2p 3/2 and 2p 1/2 core to the empty 3d e g orbitals.The splitting between BЈ and CЈ, as well as between EЈ and FЈ, is a multiplet effect as explained in a previous theoretical work. 52pon switching on D trig 0 , one can observe the occurrence of some linear dichroism for the Co 3+ : the intensities of peaks BЈ and EЈ become a function of the polarization and also the positions of peaks BЈ and EЈ are somewhat polarization dependent.These tiny energy shifts in the peak positions reflect directly the splitting within the e g orbitals.Nevertheless, the main conclusion which one can draw from this figure is that D trig 0 induces only a very small linear dichroic effect in the Co 3+ spectrum.All this is consistent with the fact that the 1 A 1 ground-state symmetry of the LS Co 3+ is not split by these bare trigonal crystal fields.
We now focus on the polarization dependence of the Co 4+ due to the bare trigonal crystal field.We start first with the D trig 0 = 0 case, for which there is no dichroism to be observed.In comparison to a Co 3+ 3d 6 ion, the Co 4+ 3d 5 has the additional low lying peaks labelled AЉ and DЉ.These reflect the presence of a hole in the t 2g subshell, giving an extra channel into which a 2p 3/2 or a 2p 1/2 core electron, respectively, can be excited during the XAS transition.This hole is isotropically distributed over the a 1g and t 2g Ϯ orbitals in this effectively O h surrounding.Upon switching on D trig 0 , one can make the striking observation from the curves on the right of panel ͑a͒ in Fig. 5 that already a small value of D trig 0 is sufficient to generate a pronounced linear dichroism in peaks AЉ and DЉ.The dichroic effect here, defined as the intensity difference for the two polarizations, is very much larger than for the Co 3+ .
A detailed look given in panel ͑a͒ of Fig. 6 reveals in fact, that after an initially rapid increase, the dichroic effect reaches a high but rather constant value as a function of D trig 0 or D trig ef f ͑these two quantities are practically equal, i.e., they differ by less than 10%-see the legends on the right in panel ͑a͒ of Fig. 5.This can be understood as follows.The introduction of a positive D trig 0 and consequently, a positive D trig ef f , leads to the unoccupation of the minority-spin a 1g orbital.Since the a 1g wave function is also highly directional, this leads naturally to the very strong polarization dependence of the probability with which the Co 2p → 3d transition occurs. 53Here we note that in getting the a 1g orbital unoccupied, D trig ef f has to compete with the Co 3d spin-orbit interaction, which mixes the a 1g with the e g orbitals.In other words, D trig ef f has to be large enough as to lift the mixing of these orbitals.This explains why the linear dichroism reaches its largest asymptotic values only for D trig ef f exceeding about 80 meV, which is the strength of the 3d spin-orbit interaction. 54ext, we investigate the influence of V mix on the spectra.The results are depicted in panel ͑b͒ of Fig. 5: the curves on the left are for the Co 3+ and on the right for the Co 4+ .The calculations were done for V mix ranging from 0 to 0.5 eV, with D trig 0 set to zero.For the Co 3+ we can clearly observe a strong polarization dependence for peaks BЈ, CЈ, EЈ, and FЈ.The dichroic effect scales rather linearly with V mix as depicted more clearly for peak BЈ in Fig. 6.The origin of this lies in the fact that the mixing between the cubic t 2g Ϯ and e g Ϯ wave functions results in the formation of the noncubic e g and the e g orbitals. 28This changes the ground state of the low spin Co 3+ system from the highly symmetric formally closed shell t 2g 6 configuration into a state in which the electron distribution is anisotropic: a positive V mix makes the occupied e g to be directed more along the c axis and unoccupied the e g more into the a − b plane, which is reflected by the higher absorption intensity for E Ќ c as compared to E ʈ c as shown by the left curves in panel ͑b͒ of Fig. 5.
The effect of V mix for the Co 4+ is also very large.The spectral changes are more complicated: here the polarization dependence of peaks AЉ and DЉ is the same as that of peaks BЉ and EЉ for small V mix values, but becomes opposite for large V mix numbers.To understand this result we also have calculated D trig ef f , the effective a 1g − e g splitting, as indicated by the legends next to the curves on the right in panel ͑b͒ of Fig. 5. Please note that this effective splitting here is caused by V mix only, since D trig 0 is set to zero; in other words, we see here the stabilization of the e g orbitals with respect to the a 1g due to the mixing of these e g orbitals with the higher lying e g .We then plot the linear dichroic effect for peak AЉ as a function of D trig ef f in panel ͑a͒ of Fig. 6, and observe a very similar behavior as when the linear dichroism was induced by D trig 0 as discussed above.This is enlightening since D trig ef f reflects how much the a 1g is more unstable than the e g orbital, the curves in Fig. 6 basically show that the linear dichroism in peak AЉ can be rather directly linked to the hole occupation in the a 1g orbital, no matter whether the dichroism is caused by D trig 0 or V mix or a combination of these.Concerning the linear dichroism in peak BЉ of the Co 4+ we now can infer the following picture: the presence of V mix induces two effects, namely the unoccupation of the a 1g orbital giving the dichroism as can be seen for peak BЉ ͑and AЉ͒ due to D trig 0 ͓see curves on the right of panel ͑a͒ in Fig. 5͔, and the redistribution of the electron density between the cubic t 2g Ϯ and e g Ϯ wave functions leading to the formation of the noncubic e g and the e g like in the case of peak BЈ of the Co 3+ ͓see curves on the left of panel ͑b͒ in Fig. 5͔.These two dichroic signals have opposite signs, and for large values of V mix the latter wins, explaining the nonmonotonic behavior of the net dichroic signal in the Co 4+ BЉ peak as depicted in panel ͑b͒ of Fig. 6.
The results of Figs. 5 and 6 give us a clear strategy for how to determine the trigonal crystal-field parameters.By fitting the dichroism in the calculated peak AЉ to the experimental one in peak A ͑see Fig. 4͒, we can obtain a reliable estimate for D trig ef f .The value for V mix can be fixed from fitting the dichroism in the calculated peaks BЈ and BЉ to the experimental signal in peak B ͑see Fig. 4͒.Since D trig ef f is determined by a combination of D trig 0 and V mix , we then also have pinpointed the value for D trig 0 .We found for the Na 0.75 CoO 2 composition that D trig ef f = 0.027 eV, V mix = 0.1 eV and D trig 0 = −0.065eV ͑D = 0.050 eV and D = −0.013eV͒.Here we have used the same crystal-field parameters for the Co 3+ and Co 4+ sites since there is no experimental indication available to suggest that the two sites are different, i.e., the system is metallic and there is no charge ordering observed.For the Na 0.5 CoO 2 we have obtained essentially the same parameter values, except for V mix = 0.3 eV for the Co 3+ ͑D = 0.13 eV and D = −0.050eV͒.Here we found the 3+ site to be slightly different from the 4+, not inconsistent with the experimentally observed presence of charge ordering in this insulating compound. 41t was claimed in a recent polarization dependent XAS study that there is a need for a local symmetry lower than D 3d for the Co 3+ ion to explain the spectra. 10,11However, the authors were still using cubic wave functions in their Co 3+ simulations in D 3d symmetry, thus essentially neglecting the anisotropic mixing between t 2g with the e g states due to the presence of the trigonal crystal field. 28It is this mixing which is responsible for the polarization dependence in peak BЈ.Using the proper wave functions and full multiplet theory we are able to reproduce the polarization dependent XAS spectra in detail as we have shown above.
Interestingly, the existence of lower than D 3d symmetry has been also proposed from optical spectroscopy, but then only for compositions and temperatures for which a chargeordered state appears. 6This proposal need not to be inconsistent with our results.It is conceivable that the optical features appearing only in the charged ordered state are indeed caused by low symmetry crystal fields but these then must have a very small strength in comparison to that of the trigonal crystal field, so that in XAS the spectral intensities associated with the trigonal crystal field overwhelm or dominate over the possible low-symmetry features.Yet, the claimed stabilization of the IS state of the Co 3+ ions in the chargeordered state 6 cannot be evidenced from our present XAS study.
1][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] The negative value we found for D trig 0 , the bare trigonal crystal-field splitting, seems to be consistent with the expectation that point charges for an octahedron compressed along the ͑111͒ axis will stabilize the a 1g orbital with respect to the e g . 7,13,21Our finding for the presence of V mix and corresponding destabilization of the a 1g orbital is an aspect which was also mentioned in some of the theoretical reports. 27Yet, the net result of these two opposing effects, namely, that D trig ef f in our study is positive, should not be taken as evidence that the centroid of the a 1g band is indeed higher in energy than that of the e g . The analysis we have made here is based on a single-site cluster method which does not take into account the band formation.A positively valued D trig ef f merely means that there is more holes in the a 1g than in the e g orbitals.This can be achieved even in the case that the centroid of the a 1g is lower than that of the e g , provided that the top of the partially emptied t 2g band has more a 1g than e g character. 12,17,22,27Yet, we can provide an explicit determination of the orbital polarization of the t 2g holes.From the cluster calculations which yield the excellent simulation of the experimental polarization dependent spectra ͑see Fig. 4͒ we find that the Co 4+ has about 0.58 hole in the a 1g orbital and 0.42 in the e g .These numbers apply for both the Na 0.5 CoO 2 and the Na 0.75 CoO 2 compositions.Band-structure calculations 12,17,27 predict that for low x contents, i.e., x Ͻ 0.67, the t 2g hole has indeed a mixed a 1g and e g character.However, for high x contents, i.e., 0.67Ͻ x Ͻ 1, the hole should reside only in the a 1g band.This is in contradiction with our findings.If the hole were of pure a 1g character, the linear dichroism of peak A would have been larger by a factor of 2-3 than experimentally observed.At the moment it is not clear how to reconcile this disagreement.One possibility is that the band-structure calculations reported so far in the literature did not take into account the Co 3d spin-orbit interaction, which otherwise would have resulted in a mixing between the a 1g and e g bands.
Another important aspect of our polarization dependent XAS work is that we found good agreement between the Co 3+ / Co 4+ spectral intensity ratios and the nominal Na concentration.In the earlier study, 10,11 however, it was claimed that the spectroscopic data are not consistent with the specified Na content.This is quite puzzling since the experimental method is the same.We therefore set out to look into the details of that analysis and we have found that this is mainly due to the neglect in that earlier study of the Co 2p −3d exchange interaction when carrying out the simulations.The consequence of such a neglect is that peaks CЈ and FЈ in the calculated Co 3+ spectrum disappear ͑while in our simulations they are clearly present, see Figs. 4 and 5͒.The authors of that earlier study then had to attribute the experimentally observed peak C to a feature of the Co 4+ ion.They used an unrealistic large value for the O 2p to Co 3d transfer integral, in order to position what is peak BЉ in our calculations ͑see Fig. 5͒ at the energy of the experimental peak C. The final result is that all of the L 3 intensity of their simulated Co 4+ ion is concentrated at the energies of the experimental peaks A and C. It is then of no surprise that peak C in their theoretical spectrum has too much intensity than that found experimentally: for Na 0.4 CoO 2 they found that the calculated feature C has twice the weight than the experimental one.They attributed this discrepancy erroneously to a proportionally lower Co 4+ fraction than expected on the basis of the nominal Na content.
Although satisfactory, our finding that the Co 3+ / Co 4+ spectral intensity ratio corresponds well with the nominal Na content is also surprising.The simulations in Fig. 3 were done by constructing the composite spectra as an incoherently weighted sum of the Co 3+ and Co 4+ contributions.In principle this summation has to be done coherently as also argued in the earlier XAS study. 11In a mixed valent situation such as we have for the Na x CoO 2 case, the spectral intensities of the Co 3+ and Co 4+ constituents could be appreciably redistributed in favor for the lowest lying spectral features.An example for such a redistribution process can be found for instance in the photoemission and inverse-photoemission spectra of strongly correlated insulators upon doping. 55,568][59][60] The ground state of Na x CoO 2 is therefore a coherent state of all Co 3+ and Co 4+ present in the lattice.Yet, the solution for this paradox may be found in the energetics of the different XAS final states which have to be compared to the one-electron nearest-neighbor hopping integrals.It is a well-known that XAS spectra are highly sensitive to the valence state: an increase in the valence state of the metal ion by one causes a shift of the XAS L 2,3 spectra by one or more eV towards higher energies. 38,61,62This shift is due to a final state effect in the x-ray absorption process.The energy difference between the XAS final states of a 3d n ͑3d 6 for Co 3+ ͒ and a 3d n−1 ͑3d 5 for Co 4+ ͒ configuration is ⌬E = E͑2p 6 3d n−1 → 2p 5 3d n ͒ − E͑2p 6 3d n → 2p 5 3d n+1 ͒ϷU pd − U dd , where U dd is the Coulomb energy between two 3d electrons and U pd the one between a 3d electron and the 2p core hole.Experimentally U pd − U dd is found to be Ϸ1 ϳ 2 eV.On the other hand, the one-electron nearest-neighbor hopping integrals in this triangular lattice system are much smaller, especially the ones for the a 1g hole.Tight-binding fits 21 to the band-structure results 12 yield numbers less than 0.1 eV.This implies that the Co 3+ and Co 4+ XAS final states hardly mix with each other.Their intensities are then simply given by their population numbers in the coherent ground state, which in turn is determined by the Na concentration, as we have observed in our work.
In conclusion, we have carried out a detailed experimental and theoretical study on the polarization dependence of the Co− L 2,3 soft x-ray absorption spectra of Na 0.5 CoO 2 and Na 0.75 CoO 2 .We found that the t 2g holes have mainly an a 1g character.Yet we also observed a non-negligible e g / e g signature, which is surprising especially for Na 0.75 CoO 2 in view of the presently available band-structure results.The data can be well explained using a local D 3d point group in which there is mixing between the cubic t 2g Ϯ and e g Ϯ orbitals.The Co 3+ / Co 4+ spectral ratios agree well with the nominal Na concentration.
This work was supported in part by the National Science Council of Taiwan and by the Deutsche Forschungsgemeinschaft through Grant No. SFB 608 and by the European Union through the ITN SOPRANO network.

FIG. 2 .
FIG.2.Experimental geometry with polarization of the light in the horizontal plane. is the angle between the electric field vector E ជ and the c-axis surface normal, and ␣ the tilt between the Poynting vector and the surface normal.

FIG. 5 .
FIG. 5. ͑Color online͒ Calculated polarization dependent Co L 2,3 XAS spectra of Co 3+ ͑left͒ and Co 4+ ͑right͒ as a function of the bare trigonal crystal field splitting D trig 0 ͑top panel a͒ and as a function of the mixing parameter V mix in D 3d symmetry.See text for definitions of D trig 0 and V mix .

FIG. 6 .
FIG.6.͑Color online͒ Panel ͑a͒: Intensity difference between the E Ќ c and E ʈ c alignments for peak AЉ in the calculated Co 4+ spectra as a function of D trig ef f .Panel ͑b͒: idem for peak BЉ of the Co 4+ and peak BЈ of the Co 3+ as a function of V mix .