An ab initio study of 3s core-level x-ray photoemission spectra in transition metals
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n An ab initio study of s core-level x-ray photoemission spectra in transition metals Manabu Takahashi and Jun-ichi Igarashi Faculty of Engineering, Gunma University, Kiryu, Gunma 376-8515, Japan Faculty of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan
We calculate the s - and s -core-level x-ray photoemission spectroscopy (XPS) spectra in theferromagnetic and nonmagnetic transition metals by developing an ab initio method. We obtainthe spectra exhibiting the characteristic shapes as a function of binding energy in good agreementwith experimental observations. The spectral shapes are strikingly different between the majorityspin channel and the minority spin channel for ferromagnetic metals Ni, Co, and Fe, that is, largeintensities appear in the higher binding energy side of the main peak (satellite) in the majority spinchannel. Such satellite or shoulder intensities are also obtained for nonmagnetic metals V and Ru.These behaviors are elucidated in terms of the change of the one-electron states induced by thecore-hole potential. PACS numbers: 79.60.-i 71.15.Qe 71.20.Be
I. INTRODUCTION
X-ray spectroscopy has been extensively used forstudying electronic properties in solids. Core-level spec-troscopy is particularly useful for investigating the elec-tronic states through the dynamical response to thephoto-created core-hole. It is well known that the re-sponse function in metallic systems exhibits the singu-lar behavior near the Fermi edge.
In the case of thecore-level x-ray photoemission spectroscopy (XPS), thespectra display the asymmetric shapes as a function ofbinding energy in the vicinity of the threshold. Apartfrom the edge singularity, some structures have been ob-served in the high binding-energy region in some ferro-magnetic transition metals and their compounds. A no-table example is a satellite peak on the p XPS in fer-romagnetic metal Ni, which is located around the region eV higher from the threshold. Feldkamp and Davis analyzed these XPS spectra by evaluating the overlapbetween the excited states and the ground state, usinga numerical method on the linear-combination-atomic-orbital model. They clarified the origin of satellite as acombined effect of the core-hole screening and the inter-action between electrons.As regards the s XPS, many experiments have al-ready been carried out on ferromagnetic transition metalsand their compounds.
Several materials show satel-lite or shoulder structures as a function of binding energy,which are interpreted as a result of the s level splittingdue to the exchange interaction between the s electronsand the valence electrons in the polarized d states. Insome cases, they have been related to the local magneticmoment. On the other hand, having intensively inves-tigated Fe s XPS in various iron compounds, Acker etal. revealed that only poor correlation exists betweenthe satellite structures and the magnetic moments. Theyalso found that the Fe s XPS spectra show the satellitestructure even in some Pauli paramagnetic compounds.Furthermore, having investigated the Mn and Fe s XPSspectra in the insulating compounds, Oh et al. con-cluded that the splitting between and main and satellite peaks dose not reflect the d moment when the effect ofthe charge-transfer becomes important.We have developed an ab initio method to calculatethe XPS spectra by extending the theory of Feldkampand Davis. Here we briefly summarize the procedureof calculation. First, we carry out the band structurecalculation within the local density functional approxi-mation (LDA) to obtain the one-electron states in theground state. Next, instead of considering the systemwith only one core hole in crystal, we consider a sys-tem of super-cells with one core-hole per cell. Strictlyspeaking, we should consider the former system for theXPS event, but the latter system is expected to work bet-ter as increasing the cell size. These systems correspondto a kind of impurity problem, where the local chargeneutrality has to be satisfied according to the Friedelsum rule. We carry out the band structure calculationbased on the LDA, in which the exchange and Coulombinteractions between the core electrons and the conduc-tion electrons and between the conduction electrons aretaken into account through the exchange-correlation po-tential. The charge variation due to screening the corehole in the final states is also taken into account withinthe super-cell approximation. To guarantee the chargeneutrality, we add one extra conduction electron in eachsuper-cell, and seek the self-consistent solution. Then,with the calculated one-electron states, we discretize themomentum space into finite number of points, and con-struct final states by distributing electrons on these one-electron states. The final state with the lowest excitedenergy is given by piling the same number of electronsinto low energy one-electron levels at each k -point as thatin the ground state. We prepare the other final states bycreating one electron-hole (e-h) pair, two e-h pairs, andso on. Finally, we calculate the XPS spectra by evaluat-ing the overlaps between thus obtained final states andthe ground state with the help of the one-electron wavefunctions.The purpose of this paper is to systematically clarifythe relation between the spectral shapes and the screen-ing process by calculating the spin resolved s XPS spec-tra in a series of ferromagnetic metals Ni, Co, and Fe.The usefulness of our ab initio method is demonstrated.We have already reported the spectra in ferromagnetic Fein Ref. 19. In these metals, spectral shapes have char-acteristic dependence on elements and spin channels; thespectra have satellite or shoulder in the majority spinchannel, while the spectra show single peak structures inthe minority spin channel. Here we define the majority(minority) spin channel by the process that the s -coreelectron is photo-excited to the vacuum state with thesame spin as the majority (minority) spin in the con-duction band states. As far as we know, such spectrahave been analyzed only by using a single band Hub-bard model, and has been related to the s - d exchangeinteraction. However, the model is, we think, too sim-ple to compare the calculated results quantitatively withthe experimental data and to draw definite conclusion.Applying the ab initio method, we calculate the spec-tra in good agreement with the experimental observa-tions. The screening effects are quite different betweenthe spin channels due to the exchange interaction be-tween the d electrons and the core hole. The d statesare modified by the core-hole potential at the core-holesite, and sometimes quasi-bound states are created nearthe bottom of the d band. The e-h pair excitationsfrom such quasi-bound states to the empty states cor-respond to the satellite or shoulder intensities. We findthat the presence of the quasi-bound states is not suffi-cient and the d bands have to be partially occupied inthe ground state, in order that the satellite or shoulderstructure appears. These considerations well explain thecharacteristics of the XPS spectra.Furthermore, to clearly show that the presence of satel-lite or shoulder has no direct relation to magnetic states,we calculate the s XPS spectra in nonmagnetic metalsV and Cu, and the s XPS spectra in nonmagnetic metalRu. We obtain shoulders in the high binding-energy re-gion in V and Ru. On the other hand, we have no suchstructure in Cu, although the localized bound states areclearly created below the bottom of the d band. Wecould explain these behaviors in the same way as in theferromagnetic metals.The present paper is organized as follows. In Sec. II,we formulate the XPS spectra with the ab initio method.In Sec. III, we present the calculated XPS spectra anddiscuss the behavior. The last section is devoted to theconcluding remarks. II. PROCEDURE OF CALCULATIONA. Formula for XPS spectra
We consider the situation that a core electron is excitedto a high energy state with energy ǫ by absorbing an x-rayphoton with energy ω q and that the interaction betweenthe escaping photo-electron and the other electrons couldbe neglected. The probability of finding a photo-electron with energy ǫ and spin σ could be proportional to I XPS σ ( ω q − ǫ ) =2 π | w | X f |h f | s σ | g i| δ ( ω q + E g − ǫ − E f ) , (1)where w represents the transition matrix element fromthe core state localized at a particular site to the stateof photo-electron, and is assumed to be independent ofenergy ǫ and spin σ . The s σ is the annihilation opera-tor of a relevant core electron, which is assumed to haveonly spin σ as the internal degrees of freedom. The kets | g i and | f i represent the ground state with energy E g and the final state with energy E f , respectively. We de-fine | f i by excluding the photo-electron. In the followingcalculation, we replace the δ -function by the Lorentzianfunction with the full width of half maximum (FWHM) s with Γ s = 1 . eV in order to take account of thelife-time broadening of the core level. B. Construction of final and initial states
In order to simulate the photo-excited states, we con-sider a periodic array of super-cells with one core-holeper cell, and calculate the one-electron states by meansof the band structure calculation based on the full po-tential linear augmented plane wave (FLAPW) method.We use the × × bcc super-cell for Fe and V as shownin Fig. 1 in Ref. 19, and the × × fcc super-cellfor Co, Ni, Cu and Ru as shown in Fig. 1, where thecore-hole sites form a bcc lattice and an fcc lattice, re-spectively. The larger the unit cell size is, the betterresults are expected to come out. The s - or s -corestates in transition metals are treated as localized stateswithin a muffin-tin sphere, so that we could specify thecore-hole site. To ensure the charge neutrality, we as-sume n e + 1 band electrons per unit cell instead of n e band electrons, where n e is the number of band electronsper cell in the ground state. One additional electron perunit cell would not cause large errors in evaluating one-electron states in the limit of large unit-cell size. Theself-consistent potential is obtained as the potential forthe fully relaxed (screened) state. We write the resultingone-electron state with energy eigenvalue ǫ σn ( k ) as ψ σn k ( r ) = 1 √ N c X j φ σn k ( r − R j ) exp( i k · R j ) , (2)with φ σn k ( r ) = u σn k ( r ) e i k · r , where u σn k ( r ) has the pe-riod of the super-cell, and j runs over N c super-cells.Needless to say, wave vector k ’s have N c discrete val-ues in the irreducible Brillouin zone. We use these one-electron states as substitutes of the states under a singlecore-hole. We distribute n e band electrons per super-cell on these states to construct the excited states. Inaddition, we carry out the band calculation in the ab-sence of the core-hole with assuming n e band electronsper super-cell. The wave function and energy eigenvalueare denoted as ψ (0) σn k ( r ) and ǫ (0) σn ( k ) , respectively. All thelowest N e = N c × n e levels are occupied in the groundstate.The final states | f i ’s are constructed by using the one-electron states calculated in the presence of the core-holein accordance with the following procedure. Defining n (0) σ ( k ) by the number of levels with spin σ and wavevector k below the Fermi level in the ground state, wedistribute n (0) σ ( k ) electrons with spin σ and wave vec-tor k in the states given in the presence of core hole.The final state | f i containing no e-h pair is constructedby distributing electrons from the lowest energy level upto the n (0) σ ( k ) ’th level with spin σ for each wave vec-tor k . The final states | f ν i ’s containing ν e-h pairs areconstructed by annihilating ν electrons in the occupiedconduction states and creating ν electrons in the unoc-cupied conduction states from | f i . Core Hole Site
Figure 1: Sketch of a super-cell containing core-holes in fccstructure. Core-hole sites indicated by solid circles are as-sumed forming a × × fcc lattice. The super-cell for thebcc structure is shown in fig. 1 in Ref. 19. C. Overlap integrals
We assume that the transition matrix elements be-tween the core state and the photo-excited states areconstant. The remaining matrix elements connecting theground and final states are expressed by h f ν | s σ | g i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S , S , ..... S N e , S , S , ..... S N e , .... .... .... ....S ,N e S ,N e ...... S N e ,N e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3)with S i,i ′ = Z φ ∗ i ( r ) φ (0) i ′ ( r )d r, (4)where the integral is carried out within a unit cell. Sub-scripts i = ( σ, n, k ) and i ′ = ( σ ′ , n ′ , k ′ ) are running over occupied conduction states in the presence of core holeand in the ground state, respectively. We eliminate theoverlaps between the wave functions for the core levels.The corresponding energies difference is given by ∆ E = E f ν − E g = E f ν − E f + E f − E g , (5)where E f − E g includes the energy of core hole, and istreated as an adjustable parameter in the present studysuch that the threshold of XPS spectra coincides withthe experimental value. The excitation energy E f ν − E f with ν e-h pairs is given by E f ν − E f = X ( i,j ) ( ǫ j − ǫ i ) , (6)where ǫ i ’s are the Kohn-Sham eigenvalues, and ǫ j − ǫ i stands for the energy of e-h pair of an electron at level j and a hole at level i . Although the Kohn-Sham eigenval-ues may not be proper quasi-particle energies, they prac-tically give a good approximation to quasi-particle ener-gies, except for the fundamental energy gap. Substi-tuting Eqs. (3) and (5) into Eq. (1), we obtain the XPSspectra.In the actual calculation, instead of N c k -points, wepick up only the Γ point as the sample states for cal-culating XPS spectra. For Ni, we pick up the X point(and the equivalent Y and Z points) in addition to the Γ point, since the d band states at the Γ point are fullyoccupied by both up-spin and down-spin electrons eventhough the × × fcc super-cell is used.Before closing this section, we briefly mention the XPSintensity at the energy of threshold. The final state | f i with the lowest energy (no e-h pair) has a finite overlapwith the ground state | g i , giving rise to intensities at thethreshold. In principle, such overlap converges at zerowith N e → ∞ , according to the Anderson orthogonal-ity theorem. In such infinite systems, energy levels be-come continuous near the Fermi level and thereby infinitenumbers of e-h pairs could be created with infinitesimalexcitation energies, leading to the so called Fermi edgesingularity in the XPS spectra. The finite contributionobtained above arises from the discreteness of energy lev-els and could be interpreted as the integrated intensityof singular spectra near the threshold, in consistent withthe model calculations for other systems.
III. RESULTS AND DISCUSSIONA. Ferromagnetic Transition Metals
In this subsections, we refer to majority(minority) spinas up(down)-spin. The Ni metal takes an fcc structure.For simplicity, the Co metal is assumed to take an fccstructure, although it actually takes an hcp structure.
Table I: Screening electron number in the d -symmetric statesinside the muffin-tin sphere at the s core-hole site. The radiiof the muffin-tin spheres are . Bohr.3s hole spin ∆ n d ↑ ∆ n d ↓ ∆ n d ↑ + ∆ n d ↓ Fe up -1.44 2.38 0.94dn 0.47 0.48 0.95Co up -0.52 1.55 1.03dn 0.24 0.75 0.99Ni up 0.07 0.87 0.94dn 0.26 0.65 0.91
The Fe metal takes a bcc structure. Figures 2 and 3show DOS’s projected onto the states with d symmetry( d -DOS) at the core-hole site for Ni and Co, respectively.The corresponding DOS’s for Fe are shown in Fig. 3 inRef. 19. In these calculations, six k -points are pickedup in the irreducible Brillouin zone for super-cell sys-tems. The DOS’s calculated with no core-hole are essen-tially the same as those reported by Moruzzi, Janak andWilliams. Table I lists the screening electron number ∆ n dσ in the d -symmetric states with spin σ , that is, thedifference of the occupied electron number between in thepresence and in the absence of the core hole inside themuffin-tin sphere.On the basis of these one-electron states, we calcu-late the s XPS spectra, by following the procedure de-scribe in Sec. II. Figures 4, 5 and 6 are the spectra thuscalculated as a function of the binding energy ω q − ǫ for Ni, Co, and Fe, respectively, in comparison with theexperiments. The spectral shape in Fig. 6 for Fe isslightly different from our previous result (Fig. 4 in Ref.19), since the present calculation takes full account of ex-citations up to three e-h pairs in comparison with onlyup to two e-h pairs in Ref. 19. The spectra are strikinglydifferent between the up-spin channel and the down-spinchannel and strongly depend on elements, in good agree-ment with the experiments. In the following, we explainthe origin of these behaviors in relation to one-electronstates screening the core hole. -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spin(a) fcc Ni -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spinup spindn spin(b) -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spinup spindn spin(c) Figure 2: Calculated d -DOS at the core-hole site in the super-cell system in ferromagnetic nickel; (a) d -DOS with no core-hole, (b) d -DOS when the s up-spin electron is removed, (c) d -DOS when the s down-spin electron is removed. -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spin(a) fcc Co -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spinup spindn spin(b) -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spinup spindn spin(c) Figure 3: Calculated d -DOS at the core-hole site in the super-cell system in ferromagnetic cobalt; (a) d -DOS with no core-hole, (b) d -DOS when the s up-spin electron is removed, (c) d -DOS when the s down-spin electron is removed.
1. Up-spin channel
First we consider the situation that a up-spin s elec-tron is removed in a unit cell. As shown in Figs. 2 (b)and 3(b), the d -DOS at the core-hole site are stronglymodified by the core-hole potential for Ni and Co. Thesituation is similar to Fe, as shown in Fig. 3 in Ref. 19.The d -DOS’s of the down-spin states are strongly pulleddown, forming quasi-bound states around the bottom of Table II: Absolute squares | A ↑ | and | A ↓ | , where A ↑ and A ↓ represent the up- and down-spin parts of the overlap integral,respectively, between | f i and s σ | g i in the σ -spin channel,that is, h f | s σ | g i = A ↑ A ↓ .up-spin core-hole dn-spin core-hole | A ↑ | | A ↓ | | A ↑ | | A ↓ | Ni .
940 0 .
234 0 .
770 0 . Co .
983 0 .
049 0 .
980 0 . Fe .
967 0 .
127 0 .
969 0 . the d band. On the other hand, the d -DOS’s of the up-spin states are slightly pulled down for Ni, and pushedupward to the higher energy region for Co and Fe. The d -DOS’s at the site without a core-hole are essentiallythe same as those in the ground state with no core-holeSince up-spin electrons are prevented from comingclose to the core-hole site due to the exchange interac-tion with the up-spin core hole, the screening is almostdone by down-spin electrons. This tendency is clear inNi; ∆ n d ↓ = 0 . , while ∆ n d ↑ = 0 . , as shown in Table I.For Co and Fe, the core-hole potential is overscreened bydown-spin electrons; ∆ n d ↓ = 1 . (Co), ∆ n d ↓ = 2 . (Fe).This overscreening is compensated by up-spin electrons; ∆ n d ↑ = − . (Co), ∆ n d ↑ = − . (Fe). As a result, thescreening electron numbers become almost unity: . , . , and . for Ni, Co, and Fe, respectively, indi-cating that the screening is nearly completed by the d -symmetric states. The magnetic moments at the core-hole site are − . µ B , − . µ B , and − . µ B for Ni, Co,and Fe, respectively, which are opposite to those withoutthe core-hole, . µ B , . µ B , . µ B , for Ni, Co, and Fe,respectively.We note that the magnitude of the overscreening by thedown-spin electrons becomes weaker Fe, Co, and Ni inthat order. In Ni, the number of the down-spin electronsscreening the core-hole is just about unity. Consequently,the up-spin d states become to be less pushed upward tothe high energy region in that order. In Ni, the effects ofthe core-hole potential and the Coulomb repulsion fromthe screening down electrons subtly cancel out each otherand the d -DOS of up-spin states is hardly modified.It is inferred from these changes in the d -DOS’s thatone-electron wave functions are largely modified by thecore-hole potential particularly for down-spin states. It isnecessary to use both the occupied and unoccupied statesof the ground state in order to expand those modifiedone-electron wave functions for the down-spin electrons,since the down-spin d bands are partially occupied inthe ground state. For this reason, the absolute square | A ↓ | becomes rather small, as shown in table II. Here A ↑ ( A ↓ ) represents the the up(down)-spin part of the overlapintegral between the lowest-energy final state | f i (no e-hpair) and s ↑ | g i , and thereby h f | s ↑ | g i = A ↑ A ↓ .The one-electron wave functions for up-spin electronsare also modified from those in the ground state. In spiteof such modification, | A ↑ | ’s are nearly unity as shown inTable II. This could be understood as follows. Since theup-spin d bands are almost fully occupied in the groundstate, up-spin one-electron states constituting the finalstate | f i could be represented by a unitary transform ofthose constituting the ground state | g i . Therefore, sincethe determinant is invariant under unitary transforma-tion, A ↑ ’s are close to unity.Final states | f ν i ’s containing up -spin e-h pairs couldgive rise to only small intensities, since the states of theexcited electrons with up-spin are almost orthogonal tooccupied states with up-spin in the ground state | g i , andthereby the overlap determinants would vanish. On theother hand, the final states | f i ’s containing one down -spin e-h pair could give rise to considerable intensities,since the corresponding one-electron wave functions con-tain the amplitudes of the unoccupied one-electron statesin the ground state | g i , and thereby the overlap deter-minants would not vanish. Considering various combina-tions of one e-h pair, we obtain intensities distributed ina wide range of binding energy. I n t e n s it y ( a r b . un it s ) expt.total|f >dn |f >|f >|f >95100105110115120125 Binding Energy (eV) I n t e n s it y ( a r b . un it s ) expt.total|f >up |f >dn |f >|f >a) up spin channelb) dn spin channel Ni 3s XPS
Figure 4: s XPS spectra in ferromagnetic nickel as a functionof binding energy. (a) and (b) are for the up-spin and down-spin channels, respectively. The experimental data are takenfrom Ref. [13].
Figure 4(a) shows the calculated spectra for Ni in com-parison with experimental observations in the up-spinchannel. The calculated spectra have maximum intensityat the threshold around ω q − ǫ = 110 eV and the signifi-cant satellite intensity around ω q − ǫ = 115 eV. Note thatthat the final states containing one down-spin e-h pairgive rise to considerable intensities, extending over mainand satellite regions. The satellite intensity correspondsto excitations of one e-h pair from the quasi-bound states to the unoccupied states with down-spin. This excitationmay be considered as a core-hole plus d , since one ofthe quasi bound states, which are almost localized at thecore-hole site, is empty. In this calculation, however, thestates are not split off from the band bottom edge, indi-cating that the core-hole-dn-hole states are only weaklybound. The satellite binding energy of the calculatedspectra is smaller than that of the observed spec-tra. This discrepancy might be owing to the LDA. TheExcitations of two e-h pairs would give rise to consider-able intensities in the energy range ω q − ǫ = 110 ∼ eV. Excitations of three e-h pairs give rise to only smallintensities. I n t e n s it y ( a r b . un it s ) expt.total|f >dn |f >|f >|f > 80859095100105110115 Binding Energy (eV) I n t e n s it y ( a r b . un it s ) expt.total|f >dn |f >|f >a) up spin channelb) dn spin channel Co 3s XPS
Figure 5: s XPS spectra in ferromagnetic cobalt as a functionof binding energy. (a) and (b) are for the up-spin and down-spin channels, respectively. The experimental data are takenfrom Ref. [10].
Figure 5(a) shows the calculated spectra for Co. Thecalculated spectra have a broad peak structure with themaximum intensity at the threshold around ω q − ǫ = 101 eV. They also have large shoulder intensities around ω q − ǫ = 104 eV. The former peak originates from theexcitations with one and two e-h pairs. The contributionof the lowest-energy final state | f i (no e-h pair) is quitesmall due to small | A ↓ | . The latter shoulder originatesfrom excitations of two and three e-h pairs, probably in-cluding the excitations from the quasi-bound states tothe unoccupied states with down-spin. I n t e n s it y ( a r b . un it s ) expt.total|f >dn |f >|f >|f > 7580859095100105 Binding Energy (eV) I n t e n s it y ( a r b . un it s ) expt.total|f >dn |f >a) up spin channelb) dn spin channel Fe 3s XPS
Figure 6: s XPS spectra in ferromagnetic iron as a functionof binding energy. (a) and (b) are for the up-spin and down-spin channels, respectively. The experimental data are takenfrom Ref. [11].
Figure 6(a) shows the calculated spectra for Fe. Thecalculated spectra consist of a peak around ω q − ǫ = 92 eV and a satellite peak around eV. The satellite peakis larger than the peak around the threshold. The finalstates | f i ’s containing one down-spin e-h pair give rise tothe satellite intensity. The final states | f i ’s containingtwo e-h pairs give rise to a shoulder to the satellite around ω q = ǫ = 97 ∼ eV, as shown in the figure. Theexcitations of three e-h pairs gives rise to finite but smallintensities in the wide energy range around eV.
2. Down-spin channel
When a down-spin s electron is removed in a unitcell, the screening behavior is quite different from thesituation where a up-spin s electron is removed. Asshown in Figs. 2(c) and 3(c), and the bottom panel inFig. 3 in Ref. 19, the d -DOS’s at the core-hole site forNi, Co, and Fe are strongly modified by the core-holepotential. Different from the up-spin channel, the effectis larger for up-spin conduction states than for down-spinconduction states; large weights are transferred to thebottom of the conduction band in the up-spin d -DOS’s,while the weights are slightly shifted downward to thelower energy region in the down-spin d -DOS’s.The screening electron numbers are rather smaller inthe up-spin state than in the down-spin state, as listed inTable I. This difference arises from the fact that up-spin d states are almost occupied in the ground state. The total screening electron numbers are almost unity, . , . , and . for Ni, Co, and Fe, respectively. The localmagnetic moments at the core-hole site are not signifi-cantly changed from those in the ground states.The | A ↑ | ’s are not far from unity as listed in Table II,by the same reason as in the up-spin channel. Note that | A ↓ | ’s are larger than in the up-spin channel, indicatingthat one-electron wave functions for down-spin conduc-tion electrons are less modified in the down-spin channelthan in the up spin-channel.Figure 4(b) shows the calculated spectra for Ni, incomparison with the experiment. The final states | f i ’scontaining one e-h pair with down-spin give rise to con-siderable intensities in a wide energy region ∼ eV, while the final states | f i ’s containing two e-h pairsgive rise to small intensities in a wide energy region withthe maximum around ω q − ǫ = 115 eV. We obtain aasymmetric shape with a tail in the high-energy regionin agreement with the experiment.Figure 5(b) shows the calculated spectra for Co. Thelowest-energy final states | f i gives rise to a peak at thethreshold around ω q − ǫ = 100 eV. Different from Ni,final states | f i ’s and | f i ’s give rise to intensities on alimited region near the threshold. This suggests that one-electron wave functions are modified only for levels in thevicinity of the Fermi level. Contributions of final states | f i ’s (three e-h pairs) are found negligible.Figure 6(b) shows the calculated spectra for Fe. Thefinal state | f i gives rise to a main peak at the thresh-old ω q − ǫ = 91 eV with the largest contribution in thethree metals. Final states | f i ’s with one down-spin e-hpair give rise to a broad peak around eV, which isquite small in comparison with the main peak. Althoughthe observed spectra show small satellite intensity around ω q − ǫ = 97 . eV, the present calculation gives no inten-sity there. It is unclear on the mechanism giving theintensity to our knowledge. B. Nonmagnetic metals
In ferromagnetic metals Fe, Co, and Ni, the XPS spec-tral shapes remarkably depend on the spin channel, ow-ing to the exchange interaction between the core hole andthe conduction electrons. Although the spectral shape isfound closely related to the filling of band states, the pres-ence of the satellite in the XPS spectra seems to have nodirect relation to ferromagnetic states. In this subsection,taking up typical nonmagnetic metals Cu, V and Ru, weclarify this issue. Since the spectra are independent ofthe spin channel in nonmagnetic metals, we consider thedown-spin channel in the following. We assume the fccstructure for Cu and Ru, and the bcc structure for V.Figure 7 show the d -DOS for Cu. The DOS calcu-lated with no core-hole are essentially the same as thosereported by Moruzzi, Janak and Williams. The d -DOSat the core-hole site is shifted downward to the deeper en-ergy region. The localize bound-states are clearly createdbelow the d bands in both up- and down-spin states.As listed in Table III, the screening electron numbers aregiven by ∆ n d ↑ = 0 . , ∆ n d ↓ = 0 . , and the total screen-ing electron number is given by ∆ n d ↑ +∆ n d ↓ = 0 . . Thisindicates that the core-hole potential is not sufficientlyscreened by the d electrons at the core-hole site. Here wenote that the screening mechanism seems to be somewhatdifferent from the other metals which have partially oc-cupied d -bands and do not show the bound states. Sincethe d band states are almost fully occupied in the groundstate, the change of the charge density is hardly achievedby a unitary transform of the d band states. Since thebound states are split off from the band bottom edge, theradial part of the local atomic wavefunctions constitutingthe bound states at the core-hole site is slightly shrunkcompared to that in the ground state with no core-hole,leading to the small change of the charge density. Thisshrink of radial part of the local atomic wavefunctionscannot be described by a unitary transform of the d band states in the ground state. The states with muchhigher energy in the ground state necessarily constitutethe bound states to some extent. -10 -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spin(a) fcc Cu -10 -8 -6 -4 -2 0 2 4Energy (eV)42024 DO S ( s t a t e s / e V ) total/27c site 3dup spindn spin(b) Figure 7: (a) DOS calculated in a system of super-cell withno core-hole in nonmagnetic copper. The solid line representsthe DOS projected onto the d -symmetry within the muffin-tinsphere. (b) DOS at the site of s down-spin core-hole. Thethin line represents the total DOS divided by the number ofatoms in a unit cell. Figure 8 shows the d -DOS for V. The d -DOS at thecore-hole site is shifted downward to the deeper energyregion for up-spin states, while the d -DOS is shifted up-ward for down-spin states. No bound state is formed.As listed in Table III, the screening electron numbers aregiven by ∆ n d ↑ = 1 . , ∆ n d ↓ = − . , and the total num-ber by ∆ n d ↑ + ∆ n d ↓ = 1 . . The core-hole potential isoverscreened by up-spin electrons, and the overscreeningis compensated by down-spin electrons. Note that thescreening is complete within the d electrons at the core-hole site. The screening is effective because the d bandsare partially occupied. -4 -2 0 2 4 6 8Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spin(a) bcc V -4 -2 0 2 4 6 8Energy (eV)42024 DO S ( s t a t e s / e V ) c site 3dup spindn spin(b) Figure 8: (a) DOS calculated in a system of super-cell with nocore-hole in nonmagnetic vanadium. The solid line representsthe DOS projected onto the d -symmetry within the muffin-tinsphere. (b) DOS at the site of s down-spin core-hole.The solid line represents the DOS projected onto the d -symmetry within the muffin-tin sphere. Figure 9 shows the d -DOS for Ru. The d -DOS’s atthe core-hole site are shifted downward to the deeperenergy region with both spin states, but the change isthe smallest in the three cases without any bound states.This is probably related to the fact that the d electronsare more itinerant than the d electrons. As listed inTable III, the screening electron numbers are ∆ n d ↑ =0 . , ∆ n d ↓ = 0 . , and ∆ n d ↑ + ∆ n d ↓ = 1 . , indicatingthat the screening is nearly completed by the d electronsat the core-hole site. -8 -6 -4 -2 0 2 4Energy (eV)202 DO S ( s t a t e s / e V ) c site 4dup spindn spin(a) fcc Ru -8 -6 -4 -2 0 2 4Energy (eV)202 DO S ( s t a t e s / e V ) c site 4dup spindn spin(b) Figure 9: (a) DOS calculated in a system of super-cell with nocore-hole in nonmagnetic ruthenium. The solid line representsthe DOS projected onto the d -symmetry within the muffin-tinsphere. (b) DOS at the site of s down-spin core-hole.Table III: Screening electron number with respect to the d symmetry within the muffin-tin sphere at the s ( s ) down-spin core-hole site. The radii of the muffin-tin spheres are . , . , and . Bohr for Cu, V, and Ru, respectively.3s hole spin ∆ n d ↑ ∆ n d ↓ ∆ n d ↑ + ∆ n d ↓ Cu dn .
31 0 .
24 0 . V dn . − .
44 1 . Ru dn .
77 0 .
24 1 . We calculate the up- and down-spin parts of overlapintegrals between the lowest-energy final state | f i andthe ground state | g i , which values are listed in Table IV. Table IV: Absolute squares | A ↑ | and | A ↓ | , where A ↑ and A ↓ represent the up- and down-spin parts of the overlap integralbetween | f i and s ↓ | g i , that is, h f | s ↓ | g i = A ↑ A ↓ .dn-spin core-hole | A ↑ | | A ↓ | Cu .
910 0 . V .
571 0 . Ru .
681 0 . For Cu, | A ↑ | and | A ↓ | are close to unity, althoughthe one-electron wave functions are strongly modified.Since the d bands are fully occupied in the groundstate, one-electron wave functions constituting | f i arenearly expressed by a unitary transform of those con-stituting | g i , except for the shrink of the radial part ofthe atomic wavefunctions at the core-hole site. There-fore, the squares of the overlap determinant | A ↑ | and | A ↓ | are nearly unity. Final states | f i (one e-h pair), | f i (two e-h pairs), and so on, could have merely verysmall overlaps with s ↓ | g i , since the one-electron state onwhich the excited electron sits is nearly orthogonal to theone-electron states constituting the ground state. Thuswe have a simple single peak structure coming from | f i without any noticeable intensities on the higher bindingenergy side, as shown in Fig. 10. This result is consis-tent with the experimental observation. Note that thechange of the wave functions is not directly related to thespectral shape. The effect due to forming strong boundstates is not seen. -15-10-5051015
Binding Energy (eV) I n t e n s it y ( a r b . un it s ) total|f >Cu 3s XPS Figure 10: s XPS spectra in nonmagnetic copper as a func-tion of binding energy in the down-spin channel.
For V, | A ↓ | is close to unity. This suggests that one-electron wave functions for down-spin electrons are littlemodified from those in the ground state. On the otherhand, | A ↑ | is rather smaller than unity. This suggeststhat up-spin one-electron wave functions constituting | f i include the amplitudes of the unoccupied one-electronstates in the ground state. In such a situation, the finalstates | f i ’s containing one e-h pair with up-spin couldhave finite overlaps with the ground state. Figure 110shows the calculated spectra. We have a main peak com-ing from the final state | f i at the threshold and thenoticeable shoulder coming from the final states | f i ’s. -15-10-5051015 Binding Energy (eV) I n t e n s it y ( a r b . un it s ) total|f >up |f >dn |f >|f >V 3s XPS Figure 11: s XPS spectra in nonmagnetic vanadium as afunction of binding energy in the down-spin channel.
For Ru, | A ↓ | is again close to unity by the same reasonas in V. The | A ↑ | is smaller than unity, although it islarger than that in V. This indicates that one-electronwave functions constituting | f i are less modified by thecore-hole potential in comparison with V. Final states | f i ’s containing one e-h pair with up-spin give rise to ashoulder structure with a little smaller intensity than inV. -15-10-5051015 Binding Energy (eV) I n t e n s it y ( a r b . un it s ) total|f >up |f >dn |f >|f >Ru 4s XPS Figure 12: s XPS spectra in nonmagnetic ruthenium as afunction of binding energy for the s down-spin core-hole. IV. CONCLUDING REMARKS
We have developed an ab initio method to calculate the s and s core-level XPS spectra in ferromagnetic metalsNi, Co, and Fe, and in nonmagnetic metals Cu, V, andRu. For the ferromagnetic metals, we have found thatthe spectral intensities are distributed in a wide rangeof binding energy with satellite or shoulder structures for the up-spin channel, while the intensities are con-centrated near the threshold with no satellite peak forthe down-spin channel. The origin of such behavior hasbeen explained in relation to the d band modified by thecore-hole potential and the overlap integral between thefinal states and the ground state. Bound or quasi-boundstates are formed by the core-hole potential, and the e-hexcitations from such quasi-bound states to the unoccu-pied levels would usually give rise to satellite intensities.However, the presence of the quasi-bound state is nota sufficient condition to the presence of satellite; the d -band should be partially occupied in the ground state,and thereby the one-electron wave functions constitutingthe final states include the amplitudes of the unoccupiedone-electron states in the ground state. If the d -band isfully occupied, the satellite intensity would not come outeven in the presence of the bound state. Note that thesatellite peak position has no direct relation to the s level exchange splitting; the LDA calculation gives suchsplittings as . , . , and . eV for Ni, Co, and Fe, re-spectively.These results indicate that the presence of satellite isnot directly related to the ferromagnetic ground state.We have clarified this point by calculating the spectra innonmagnetic metals Cu, V, and Ru. For V and Ru, wehave obtained shoulder structures in the XPS spectra,although the structure is rather small for Ru. The originof these behaviors is the same as in the ferromagneticmetals. For Cu, only a symmetric peak is found with nostructure, although the localized bound states are clearlyformed below the bottom of the conduction band. This isbecause the d band is completely occupied in the groundstate.We have calculated the XPS spectra in Ni, Co, Fe,and Cu in good agreement with the experiment, whilewe could not find experimental XPS data for V and Ru.Acker et al. observed the satellite structures even in somePauli paramagnetic Fe compounds, The present resultswould provide an interpretation of their findings. Fi-nally, as regards the L -edge spectra, experimental datafor XPS spectra and the x-ray absorption spectra are ac-cumulated, and ab initio approach has been tried. Theextension of the present method to the L -edge spectra isleft in future study. Acknowledgments
We used the FLAPW code developed by NoriakiHamada. We thank him for allowing us to use his codeand fruitful discussions. This work was partially sup-ported by a Grant-in-Aid for Scientific Research in Pri-ority Areas “Development of New Quantum Simulatorsand Quantum Design” (No.19019001) of The Ministryof Education, Culture, Sports, Science, and Technology,Japan.1 P. W. Anderson, Phys. Rev. Lett. , 1049 (1967). G. D. Mahan, Phys. Rev. , 612 (1967). P. Noziéres and C. T. de Dominicis, Phys. Rev. , 1097(1969). S. Doniach and M. Sunjic, J. Phys. C , 285 (1970). S. Hufner and G. W. Wetheim, Phys. Lett. , 301(1975). L. A. Feldkamp and L. C. Davis, Phys. Rev. B , 3644(1980). C. S. Fadley and D. A. Shirley, Phys. Rev. A , 1109(1970). J. F. van Acker, Z. M. Stadnik, J. C. Fuggle, H. J. W. M.Hoekstra, K. H. J. Buschow, and G. Stroink, Phys. Rev.B , 6827 (1988). F. U. Hillebrecht, R. Jungblut, and E. Kisker, Phys. Rev.Lett. , 2450 (1990). D. G. Van Campen and L. E. Klebanoff, Phys. Rev. B ,2040 (1994). Z. Xu, Y. Liu, and P. D. Johnson, Phys. Rev. B , 7912(1995). A. K. See and L. E. Klebanoff, Phys. Rev. B , 7901(1995). A. K. See and L. E. Klebanoff, Phys. Rev. B , 11002(1995). W. J. Lademan and L. E. Klebanoff, Phys. Rev. B , 6766(1997). I. N. Shabanova, N. V. Keller, V. A. Sosnov, and A. Z.Menshikov, J. Electron. Spectrosc. Relat. Phenom. , 581 (2001). N. Kamakura, A. Kimura, T. Saitoh, O. Rader, K. S. An,and A. Kakizaki, Phys. Rev. B , 094437 (2006). P. S. Bagus and J. V. Mallow, Chem. Phys. Lett. , 695(1994). S.-J. Oh, G.-H. Gweon, and J.-G. Park, Phys. Rev. Lett. , 2850 (1992). M. Takahashi, J. Igarashi, and N. Hamada, Phys. Rev. B , 155108 (2008). J. Friedel, Nuovo Cim. Suppl. , 287 (1958). Y. Kakehashi, K. Becker, and P. Fulde, Phys. Rev. B ,16 (1984). M. S. Hybertsen and S. G. Louie, Phys. Rev. B , 5390(1986). N. Hamada, M. Hwang, and A. J. Freeman, Phys. Rev. B , 3620 (1990). A. Kotani and Y. Toyozawa, J. Phys. Soc. Jpn. , 912(1974). V. L. Morruzi, J. F. Janak, and A. R. Williams,
Cal-culated Electronic Properties of Metals (Pergamon, NewYork, 1978). P. Kruger and C. R. Natoli, Phys. Rev. B70