Shake-up valence excitations in CuO by resonant inelastic x-ray scattering

Transcription

1 PHYSICAL REVIEW B 70, 0855 (2004) Shake-up valence excitations in CuO by resonant inelastic x-ray scattering G. Döring, C. Sternemann, A. Kaprolat, 2 A. Mattila, 3 K. Hämäläinen, 3 and W. Schülke Institute of Physics, University of Dortmund, D-4422 Dortmund, Germany 2 European Synchrotron Radiation Facility, B.P. 220, F Grenoble Cedex, France 3 Department of Physical Sciences, P.O. Box 64, FIN-0004 University of Helsinki, Finland (Received 6 February 2004; published 30 August 2004) Shake-up satellites are found in the resonant inelastic x-ray scattering spectra of CuO single crystals, excited by means of monochromatized synchrotron radiation with energies near the Cu K-edge. Within the limits of third-order perturbation treatment, the satellites are attributed to charge transfer excitations between a bonding b g ground state and antibonding states of a g, b 2g, b g, and e g symmetry. These transitions are induced by Coulomb interaction of the -hole, s which constitutes the intermediate state of the resonant inelastic x-ray scattering process, with the Cu 3d system. We have modeled the dependence of the satellite s energy loss position on the incident photon energy by means of a third-order perturbation treatment, where two different Cu 3d s 9 4p excitations constituting the resonant inelastic x-ray scattering process must be taken into account. Together with a dipole forbidden 3d s 0 and a dipole allowed 3d s 0 L 4p transition, another dipole allowed Cu 3d s 9 4p excitation was investigated in detail by means of resonantly excited 2p s K emission, especially with respect to its dependence on incident polarization. Since this excitation does not leave behind a trace in conventional x-ray absorption spectra, symmetry assignment can be obtained only this way. An investigation of the dependence of the satellite s intensity on the scattering angle in vertical and horizontal scattering geometry has stressed the necessity to take into account the polarization and scattering angle dependence of the resonant elastic scattering process, being composed of the production and recombination of the virtual 4p s exciton, when it is regarded to be decoupled from the Cu 3d system. DOI: 0.03/PhysRevB PACS number(s): Ck, Ga, a I. INTRODUCTION Detailed information about the electronic structure of highly correlated transition metal oxides is of great importance for understanding the mechanism of high-t c superconductivity. Among these transition metal oxides, divalent CuO is of particular importance since it shares an integral component, namely the CuO 4 -plaquettes, with the high-t c cuprates. Therefore, experimental knowledge about details of its electronic structure and electronic excitations is crucial, especially since band structure calculations are precluded so that numerical techniques must be applied, utilizing small clusters of ions taking into account on-site and cluster interactions. These theoretical calculations need thorough confirmation by experiment. Thus a lot of experimental techniques have been brought into action. Those which study the occupied states near the Fermi level, like nonresonant and resonant 2 photoemission spectroscopy, while different methods probe the unoccupied states like x-ray absorption near edge structure (XANES) spectroscopy at the Cu K-, the Cu L-, and the O K-edge. 3 7 An angular and polarization dependent Cu K-edge XANES study by Bocharov et al. 8 must receive special attention since both the experimental results and the real-space multiple scattering calculations were decomposed into symmetry-resolved components. Nevertheless, the calculation is strictly of one-electron nature, so that, for instance, many-body shake-down processes as claimed by Bair and Goddard 9 were not taken into account. Lately, also resonant inelastic x-ray scattering (RIXS) has been utilized as far as investigations of unoccupied states and excitations of the valence electron system of CuO and the cuprates are concerned. This technique as applied by Krisch et al. 0 is able to reveal fine details of unoccupied states, which remain unresolved with conventional x-ray absorption spectroscopy (XAS). RIXS spectra in the vicinity of 2p s emission taken at the Cu K-edge of CuO (Refs. 3) have revealed a transition from the s core level into an unoccupied (most probably) 4p state in the pre-edge range, undistinguishable in the XAS spectrum. A symmetry assignment of this newly revealed transition on the basis of polarization resolved RIXS is highly desired and will be given in this paper. Furthermore, the RIXS process can include excitations in the complicated valence electron system of Cu oxides, where these excitations do not need to follow the dipole selection rules and their final states do not suffer from core hole effects as do final states in a conventional XAS process. Kuiper et al. 4 have applied RIXS to cuprates at the Cu 3p edge in the soft x-ray regime. These investigations were extended to polarization resolved RIXS at the Cu 2p edge by Duda et al., 5,6 and the results were compared to theoretical predictions by Tanaka and Kotani, 7 performed within second-order perturbation theory utilizing the Anderson impurity model. It must be stressed that RIXS studies at the Cu 2p and 3p edge emphasize excitation processes due to d-d transitions in the valence electron system compared to charge transfer transitions, which dominate the excitation processes /2004/70(8)/0855(5)/$ The American Physical Society

2 DÖRING et al. PHYSICAL REVIEW B 70, 0855 (2004) of RIXS in the cuprates at the Cu K-edge. Investigations to detect charge transfer excitations were initiated by Kao et al. 8 on NiO, and followed by an investigation of Hill et al. 9 on Ni 2 CuO 4, further completed by Hämäläinen et al. 20 stressing the polarization and momentum transfer dependence of these RIXS spectra. They found that the polarization of the incident radiation selects the symmetry of the intermediate 4p states and that under vertical scattering geometry no momentum transfer dependence of the peaks caused by charge transfer excitation could be observed neither with respect to energy position nor intensity. Both experimental RIXS studies were accompanied by theoretical calculations on the basis of second-order perturbation theory and the Anderson impurity model. By extending these calculations from a relatively simple cluster, where only one Cu atom was surrounded by many oxygen atoms to more complicated clusters containing also neighboring Cu atoms, the authors were able to give reasons for the fact that charge transfer scattering could not be observed when the intermediate state should give rise to a delocalized Zhang-Rice-singlet. 2 In another RIXS study on La 2 CuO 4 Abbamonte et al. 22 have treated the occurrence of excitations in the valence electron system in third-order perturbation theory so that the peak-structure due to these excitations must be considered as shake-up satellite caused by Coulomb interaction of the 4p s exciton with the valence electron system. They assigned this satellite, according to calculations by Simon et al. 23 to a strong local transitition from the b g ground state to a high energy a g state composed of symmetric combinations of a Cu 3d x 2 y2 orbital with the surrounding O 2p orbitals. The authors were able to explain by means of the perturbation theory at least for one of the detected shake-up satellite features, the dependence of its energy position on the incident photon energy, whereas the former RIXS studies 9,20 have not revealed any variation with energy. Moreover, by taking into account the incident energy dependence as being caused by the double resonance denominator of the third-order perturbation treatment, Abbamonte et al. 22 could reduce all shake-up features observed for different incident photon energies to a single peak, which they brought in connection with the dynamic structure factor of the valence electron system. These considerations drew their attention to the matrix element of the Coulomb interaction, which gives rise to the shake-up excitations of the 3d system. They believed that it could be a possible source of a dependence of the satellite s intensity and shape on the momentum transfer. Indeed, Abbamonte et al. 22 found such a dependence on momentum transfer in the shake-up satellites of Sr 2 CuO 2 Cl 2, at least with respect to their shape, intensity and possibly also their energy position so that they concluded that dispersion information could be measured with RIXS. It should be stressed that Hämäläinen et al. 20 did not find such a dependence on momentum transfer neither with respect to energy position nor with respect to intensity of the satellites, when using a geometry where both the polarization ˆ of the incident beam and the direction of the momenum transfer q relative to the crystal coordinates remains unaltered when the magnitude of q is varied. In this paper we aim to solve this puzzle by looking into another source of momentum transfer dependence of the CuO satellite structures, which can be traced back on the dependence of the underlying production and recombination of a virtual 4p s exciton on the scattering angle. In agreement with theoretical predictions of Tsutsui et al. 24 and more general considerations of Platzman and Isaacs, 25 it was recently demonstrated by Hasan et al. 26 that by varying the direction and the magnitude of q in a RIXS study on Ca 2 CuO 2 Cl 2, dispersion information of an excitation could be obtained. This excitation was assigned to a transition between a Zhang-Rice singletlike lower Hubbard band and the upper Hubbard band, both sufficiently delocalized states. Even the satellite structure, which was attributed to a strongly localized charge transfer transition b g a g (like in the study of Abbamonte et al. 22 ), exhibited a distinct change of intensity and shape but not of the energy position, when varying the direction and the magnitude of q. Therefore, also this behavior needs further clarification. Hasan et al. 27 have extended their investigation of dispersion information by RIXS to D Mott insulators (SrCuO 2 and Sr 2 CuO 3 ). It must not be forgotten that the above described low-lying dispersing excitations across the Mott gap of Sr 2 CuO 2 Cl 2 and their anisotropic dispersion were already earlier measured using electron energy loss spectroscopy (EELS) by Wang et al. 28 and by Neudert et al. 29 All the previously mentioned RIXS shake-up studies have been performed on CuO 4 -plaquettes more or less surrounded and influenced by other constituents of the related cuprates. Therefore, it seemed to us timely to investigate the excitations behind the shake-up process as well as the process itself on a Cu compound, where these plaquettes are present in a pure form, that is, on single crystalline CuO. Along with the measurements of the charge transfer excitation induced shake-up satellites the following problems arisen in the course of the above summarized RIXS investigations should be tackled: (i) Symmetry and energy assignment of the excited electron with 4p symmetry, which together with the s hole builds a virtual exciton and scatters off the valence electron system. (ii) Search for criteria whether or not a third-order perturbation treatment must be invoked in order to explain the energy position, shape and the resonance behavior of the shake-up satellites as a function of the incident photon energy. (iii) Find out reasons for the q-dependence of the shake-up spectra of strongly localized excitations by fixing all relevant parameters of the scattering process: incident and scattered polarization as well as the amount and direction of q relative to the crystal coordinates. In order to reach this goal, the paper is organized as follows: In Sec. II we shortly present the basic relations of a third-order treatment of the RIXS process connected to shake-up processes. Section III describes the experiment and specifies the main ingredients of the CuO crystal structure. Section IV contains the results of a high-resolution measurement of the Cu K fluorescence by tuning across the K threshold and varying the incident polarization in order to separate out resonances in the unoccupied states and to find out their symmetry. Section V provides the measurements of

3 SHAKE-UP VALENCE EXCITATIONS IN CuO BY PHYSICAL REVIEW B 70, 0855 (2004) the shake-up satellites as a function of the incident energy, discusses the corresponding excitations of the valence electron system, and describes the results in terms of third-order perturbation theory. In Sec. VI an eventual q-dependence of the shake-up satellite is explained. Finally, Sec. VII offers the conclusions. II. BASIC RELATIONS The leading contribution to the double differential scattering cross section of an inelastic scattering experiment near the K-resonance, that is for incident photon energies near the binding energy of the s electron, is given within the secondorder perturbation theory by d 2 d d 2 2 f m f ˆ 2* j p j e ik 2 r j m m ˆ j p j e ik r j i E m E i i m 2 E f E i, where i, f, and m are the initial, final, and an intermediate state of the scattering electron system with their corresponding energies E i, E f, and E m, respectively. The intermediate state m is assumed to be connected with a virtual s -core hole at the copper site and, in our special case with an electron excited into a state with Cu 3d and/or 4p symmetry contributions which, in the latter case, can be considered as a virtual s 4p exciton. m denotes its lifetime. K,, ˆ and K 2, 2, ˆ 2 are the wave vectors, frequencies and polarization vectors of the incident and the scattered photons, respectively, and = 2. The summation j is carried out over all electrons of the system. In order to discuss the dependence of the RIXS spectra on the polarization of the incident photons relative to the crystal coordinates, we have to discuss the matrix element M i,m = m ˆ p j e ik r j i 2 j which describes in Eq. () the absorption of the incident photon. If we specialize on the one-electron case, restrict the exponential to terms of order zero (dipole terms) and of order one (quadrupolar terms) by writing and utilize the commutator we end up with e ik r +ik r, p = m i r,h 0 M i,m 2 M D i,m + M Q i,m 2 m ˆ r i m ˆ rk r i 2, 5 where we have made use of the orthogonality of i and m and of the fact that the dipole matrix element M D,m is imaginary and the quadrupolar matrix element M Q i,m is real. Choosing now the real representations of the spherical harmonics Y lm, as the basis for the states i and m by writing for the general case n = P l,m R l r Y l,m, l,m this leads to the dipole matrix element 6 M D,m x P, + y P, + z P,0 7 and to the quadrupolar matrix element M Q i,m = 4 5 xk y + y K x P 2, 2 + x K z + z K x P 2, + 3 z K z P 2, 0 + y K z + z K x P 2, + x K x y K y P 2,2. 8 The squares of the polarization and wave vector dependent prefactors of the expansion coefficients P l,m in Eqs. (7) and (8) are denoted as their partial spectral weights. They weight the different contributions P l,m to the RIXS spectra dependent on the scattering geometry. We will now consider the Coulomb interaction between the virtual s 4p exciton and the electron system. In principle, this interaction H C can be treated within the limits of an Anderson impurity model by adding H C to the model Hamiltonian of the system, as has been done, for example, by Tsutsui et al. 24 We will take H C into consideration in a more general form by following Platzman and Isaacs, 25 and Abbamonte et al., 22 respectively, on the basis of a third-order perturbation treatment. Therefore, we consider H C as a perturbing Hamiltonian by writing H e = H 0 + H C, where H 0 is the Hamiltonian of the unperturbed system. Using the relation H e z = m m m E m i m z we can write Eq. () the following way:

4 DÖRING et al. PHYSICAL REVIEW B 70, 0855 (2004) where d d d 2 2 f b 2 f H e E i + i b E f E i, b = ˆ p j e ik r j, j b 2 = ˆ 2* p j e ik2 r j. j With the definitions 2 G 0 z = H 0 z, G e z = H e z, 3 Eq. (9) can easily be transformed into the Dyson-type equation G e z = G 0 z G 0 z H C G e z 4 whose first order iteration can be written G e z = G 0 z G 0 z H C G 0 z 5 so that Eq. () becomes d d d 2 f b 2 f H 0 E i + b i 2 + f b 2 H 0 E i + H C H 0 E i + i b E f E i 6 and after inserting a complete set of eigenfunctions of H 0, d d d 2 f b 2 m m b i f m E m E i i m f b + 2 n n H C m m b i E m E i i m E n E i i n m,n E f E i. 7 The first term of Eq. (7) is equivalent to Eq. (). The second term describes a scattering process, where first a photon with an energy of is absorbed exciting the electron system from the ground state into the intermediate state m, which in our special case is the virtual exciton 4p, s together with the Cu 3d related valence electron system in its ground state, s 4p,v. This virtual exciton takes up the full momentum of the incident photon and Coulomb-interacts with the valence electrons, so that a so-called shake-up process can take place, leading to an excitation into a new intermediate state n, in our case s 4p,v, which should not be specified in the first instance. This intermediate state decays into the final state f emitting a photon with an energy of 2. The double resonance denominator of the second term of Eq. (7) deserves special attention. Assuming that the Cu 3d related valence electron system is decoupled from 4p electrons, we can separate the excitation energy of the 3d system from the excitation energy E ex of the remaining system by writing 2 FIG.. Resonant inelastic scattering scheme including a shake-up process. E m E i = E ex, ; E m E n = ; E n E f = E ex,2, 8 where we additionally assume that the excitation of the remaining system and also the de-excitation can be reperesented by a single energy, each with a single lifetime broadening. Along with the energy conserving -function, the second term of Eq. (7), the shake-up amplitude A s gets the form, A s = f b 2 n E ex,2 2 i 2 E ex, i m,n n H C m m b i. 9 It should be mentioned that the scattering amplitude of Eq. (9) has the same form as the one which describes the phonon contribution to the optical resonant Raman scattering. 30 Figure shows the corresponding inelastic scattering scheme, which includes a shake-up process. A s provides only an additional contribution to the total scattering intensity while the first term of Eq. (7) produces the main contribution, whose amplitude, A f, can be written as A f = E ex, i f b 2 m m b i. m 20 In what follows, we will designate contributions of A s to the scattering spectra by shake-up satellites, whereas A f gives rise to the main fluorescence line. The relevant shake-up excitations will be now discussed in more detail by specifying the resonant scattering process (and thus the matrix element of H C ) further. We consider a resonant scattering process in which the s core hole is filled bya4p electron during emission of the outgoing photon and the Cu related electron system remains excited via the Coulomb interaction with the 4p s exciton. This means that a momentum q=k K 2 is transferred to the valence electron system

5 SHAKE-UP VALENCE EXCITATIONS IN CuO BY PHYSICAL REVIEW B 70, 0855 (2004) M C = n H C m = s 4p,v dx dx e2 x x s 4p,v. x x 2 We assume again that the Cu 3d valence electron system is decoupled from the 4p electrons. This means that the energy of the latter remains unchanged. The Cu 3d related valence electron system is excited from its ground state i to an excited final state f, so that we can write M C = dx dx s 4p x e iqx s 4p x x f x i, 22 where we have omitted corresponding exchange terms, which means that e.g. the primed coordinate is associated solely with the exciton, the unprimed with the 3d valence system. By making use of the lattice translation symmetry x = i e 2 x R i r j, j 23 where R i are the lattice translation vectors and the sum over j is carried out over the corresponding electron (hole) sites of the elementary cell, we end up, after some Fourier transform algebra, with 4 e M C = 2 G q + G 2F s 4p G, ˆ f v q + G i. 24 The sum is over all reciprocal lattice vectors G, F s 4p is the Gth structure factor of the 4p s exciton, of course explicitly dependent on the incident polarization ˆ and v q is the qth Fourier transform of the Cu 3d related valence electron density operator v q = e iq r vj. 25 j Equation (24) is (with a slightly different argument of the structure factor F s 4p ) the expression Abbamonte et al. 22 have used to discuss the dependence on momentum transfer of their resonant scattering results on cuprates. III. EXPERIMENTAL SETUP AND THE CuO SAMPLE The RIXS experiments were performed at the HARWI (Harter Röntgen Wiggler) beamline of HASYLAB (HAmburger SYnchrotronstrahlungsLABor) and the X2 beamline at the NSLS (National Synchrotron Light Source). The HARWI beamline was equipped with a Si 5 double crystal monochromator, whose first flat crystal is water cooled and the second is sagittally focusing to a cm cm focus at the sample positioned in an evacuated chamber. The scattered radiation was energy analyzed in horizontal geometry, which means that the scattering plane was identical with the storage ring plane with the consequence that the majority of the incident radiation was polarized in the scattering plane. This setup has the disadvantage that a change of either the amount of q (by changing the scattering angle) or the direction of q (relative to the crystal) is not possible without FIG. 2. (a) Horizontal scattering geometry. In order to change the scattering angle without changing the direction of q with respect to the crystal coordinates, the sample should be rotated around an axis perpendicular to. Thus it is not possible to keep the orientation of q and ˆ fixed simultaneously. (b) Vertical scattering geometry. By rotating the sample the way shown, the orientation of both q and ˆ can be kept. changing as well the incident polarization with respect to the sample coordinates (see Fig. 2). The energy analysis of the scattered radiation was carried out by means of a Johanntype spectrometer with a spherically bent crystal R= m in horizontal Rowland geometry. Due to the rather large extension of the focus on the scattering sample, a position sensitive gas-filled detector on the Rowland circle could expose, for each spectrometer position, a certain part of the whole spectrum, which had to be assembled properly to the final spectrum. 3 The overall resolution of the whole setup, as was determined by the FWHM of the quasielastic line, was.5 ev when using a Si 444 analyzer for Cu K radiation, and.0 ev with a Si 553 analyzer crystal for the 8980 ev range. The monochromator of beamline X2 at the NSLS consists of two channel-cut Si(220) crystals in a dispersive setting (so-called four-bounce arrangement) and a double focusing mirror which collects the monochromatic beam into a 0.5 mm 0.3 mm h v spot at the sample. The incident energy resolution at 9 kev was about 0.2 ev. The attached Rowland spectrometer was equipped with a spherically bent R= m Si 553 crystal. An overall energy resolution of 0.8 ev was achieved, as determined by the FWHM of the quasielastic line. The RIXS setup at beamline X2 was mounted on a four-circle diffractometer, which allowed scattering angles between 0 and 20 to be reached in vertical

6 DÖRING et al. PHYSICAL REVIEW B 70, 0855 (2004) FIG. 3. Crystal structure of CuO. Copper atoms are the large, O atoms the smaller spheres. The two kinds of CuO 2 chains are shaded. geometry, as shown in Fig. 2(b). Both the direction of the incident polarization and the direction of q could be kept fixed when changing the scattering angle. In order to minimize absorption the sample was placed into a vacuum chamber and the beam paths of the scattered radiation were enclosed into a helium environment. It is well known (Åsbrink and Norrby 32 ) that CuO crystallizes in the monoclinic space group C2/c with four CuO units in the elementary cell a=4.68 Å,b=3.42 Å,c =5.3 Å, = =90, =99.5. In this crystal structure copper is coplanarly surrounded by four oxygens which sit on the corners of a nearly rectangular parallelogram (CuO 4 -plaquettes with sidelength 2.9 Å and 2.62 Å, and angle at the corners 90.3 and 89.7 ). As can be seen in the perspective view on the CuO crystal structure in Fig. 3, the coplanar grouping of the oxygen neighbors of each copper atom gives rise to two different kinds of chains, one propagating in [0], the other in 0 direction, where both are connected by oxygen atoms at the corners. The plane of these chains are shaded in Fig. 3. They intersect under an angle of The existence of two orientations of the CuO 4 -plaquettes in CuO prevents an orientation of the incident polarization perpendicular to both plaquettes simultaneously (pure -polarization). On the other hand, the incident polarization can be oriented parallel to the intersection of two corresponding planes (pure -polarization), where one degree of freedom for the sample orientation is left, namely the rotation around an axis parallel to the incident polarization. The single crystal of CuO used in the experiments was a plate, mm long,.5 mm wide, and 0.5 mm thick, where, according to Laue-diffraction, the b-axis was perpendicular to the mm.5 mm plane, while the a-axis coincided with the mm long sample axis. The experimental setup at the HARWI beamline (horizontal geometry) enabled a rotation of the sample around an axis perpendicular to a plane spanned by the a- and the b-axis of the crystal, so that the angle between the incident beam and the b-axis defines the direction of the incident polarization relative to the crystal coordinates. IV. CuO RIXS SPECTRA OF THE 2p s EMISSION The correct interpretation of shake-up satellites of RIXS spectra, the main goal of this paper, depends sensitively on FIG. 4. Total fluorescence yield spectrum of Cu-metal (triangles) and of CuO (dots). The solid line is the total fluorescence yield spectrum of Cu metal as deduced from an EXAFS spectrum measured in absorption (Ref. 33) taking into account energy dependent self absorption in an appropriate way. the adequate assignment of transitions which are involved in the process. The best way to detect and to classify even faint excitations, especially in the pre-edge region of the x-ray absorption spectrum is, as demonstrated by Krisch et al., 0 to study the high-resolved RIXS spectra of a re-emission channel, connected with the excitation/absorption process under investigation. Following this approach, we have measured CuO RIXS spectra (as a function of 2 ) of the s 2p K re-emission for various excitation energies ( in steps of 0.5 ev) across the Cu K-threshold (see the total fluorescence yield spectrum in Fig. 4) and for two different incident angles. This means, according to the definition of in Sec. III, different orientations of the incident polarization relative to the crystal coordinates. The measurements were performed at the HARWI beamline in horizontal geometry usingasi 444 analyzer and 90 scattering angle in order to reduce Thomson scattering. The results are shown in Figs. 5(a) and 5(b). Whereas the total fluorescence yield spectrum of Fig. 4 is rather structureless in the relevant range of the incident energy, the RIXS spectra of the 2p s emission in Figs. 5(a) and 5(b) exhibit a large variety of fine structure. In order to assign the various components of this fine structure to different excitation channels, we must have in mind the following principal properties of RIXS spectra, as thoroughly presented by Tulkki and Åberg. 34 We have to distinguish between RIXS spectra connected with a virtual transition of a core electron into a discrete level and those characterized by a transition into a level continuum. If, in the former case, the discrete level has an energy E m and the incident photon energy is by smaller than E m E i, then, due to the energy conserving -function in Eq. (), also the photon energy 2 of the re-emission is reduced by, compared with the photon energy of the regular fluorescence line. This means that the energy of the reemitted photon increases exactly by the same amount as the

7 SHAKE-UP VALENCE EXCITATIONS IN CuO BY PHYSICAL REVIEW B 70, 0855 (2004) FIG. 5. (a) Cu K fluorescence spectra of CuO for different excitation energies as indicated with incident angle =30. The spectra are vertically off set. The structures A, B, and C (see text) are marked at one spectrum. (b) The same as (a) but for =54. incident photon energy (the so-called Raman shift, first observed in the x-ray case by Eisenberger et al. 35 ). Passing with the incident photon energy through =0, the intensity of the corresponding re-emitted radiation runs through a maximum, as a consequence of the resonance denominator of Eq. () and fades away, whereas the Raman shift persists. The spectral distribution of the re-emission is determined by the convolution of the spectral shape of the involved unoccupied discrete level with a Lorentzian, which describes the lifetime broadening of the core level left behind in the re-emission process, convoluted with the experimental resolution. Let, on the other hand, the core electron to be excited into a continuum of unoccupied levels with a step-like threshold at the Fermi edge. If we are working with increasing incident photon energy by starting with smaller than the binding energy of the involved core level, then we observe again a decreasing Raman shift of the re-emitted photon energy, together with an increasing intensity of the re-emission. In that case the spectral distribution of the re-emission is the convolution of the tail of a Lorentzian representing the lifetime broadening of the intermediate core hole as cut by the threshold and another Lorentzian, which stands for the lifetime broadening of the core hole left behind in the re-emission process, convoluted with the experimental resolution. However, if the increasing incident photon energy reaches the binding energy of the core level, the re-emitted photon energy stays fixed at the value of the regular fluorescence. The linewidth of the re-emission becomes narrower than that of the regular fluorescence, since the Lorentzian which describes the intermediate core-hole lifetime is exactly bisected by the threshold. With further increase of the incident photon energy, the spectral distribution of the re-emission turns into the line shape of a regular fluorescence. Having these general properties of RIXS spectra in mind, we can distinguish three structures in Figs. 5(a) and 5(b) which have been denoted A, B, and C at one of the reemission spectra where they are visible simultaneously. In order to study position and intensity of these structures with increasing incident photon energy, we have fitted a Voigt function to structure A and a Gaussian to structures B and C each with variable width, position, and amplitude, neglecting partly the rather large diversity of possible shapes discussed previously, and assuming that the experimental resolution is dominant. In Fig. 6 the amplitude of the structures A, B, and C is plotted as a function of the incident photon energy, exemplarily for =30. The energy positions of the structures A, B, and C, as obtained by this fitting procedure are shown in Fig. 7. The peak widths obtained by the fit are found to be constant within the error limits for all incident photon energies. A resonating amplitude of the structure A in Fig. 6 and a Raman shift beyond the resonance observed for the same structure in Fig. 7 are safe indicators that the structure A must be traced back to a transition of the s-electrons into a discrete unoccupied 3d level, since the energy position of the resonance coincides at ev with a little hump in the total fluorescence yield spectrum of Fig. 4, which was attributed by Bocharov et al. 8 to a quadrupolar s 3d transition. A saturation behavior of the energy position as observed for structure B in Fig. 7 along with an indication that the amplitude of structure B exhibits a resonance around 8982 ev (see Fig. 6) admits the following interpretation: This structure reflects a 4p s transition into a narrow continuum, not much wider than 3 ev, and should be situated in

8 DÖRING et al. PHYSICAL REVIEW B 70, 0855 (2004) FIG. 6. Amplitude of the structures A (dots), B(triangles), and C (squares) of Fig. 5(a) as a function of incident photon energy for =30. The lines indicate fits of Gaussians to the data points for structures A (dashed line), B(solid line), and C (dashed dotted line). the total fluorescence spectrum 3 ev below its inflection point. It is characteristic for the superiority of RIXS over XAS that the total fluorescence spectrum of Fig. 4 exhibits no structure at this energy range. This interpretation of structure B is largely in agreement with that given by Döring and Hayashi et al. 2 Structure C exhibits a linear Raman shift as shown in Fig. 7 within the range of excitation energy which was used for fitting all three structures. According to further measurements not shown here, the energy position of the then solely existing structure saturates beginning at 8985 ev incident energy. From there, the amplitude follows the total fluorescence yield shown in Fig. 4. Therefore, it seems to be justified to assign structure C to transitions into a 4p-like continuum. FIG. 8. Cu K fluorescence spectra of CuO for an excitation energy of ev for seven different incident angles as indicated. The vertical off set was chosen proportional to. Let us now discuss the -dependence, this means the polarization dependence of the 2p s RIXS spectra, as shown for seven different angles in Fig. 8 for one incident photon energy = ev : The amplitude of structure C increases with increasing angle. To the contrary, the amplitude of structure A decreases with increasing up to a minimum at 50 to increase again with increasing. The amplitude of the structure B exhibits only a weak dependence on in the sense that near the resonance the =54 B-type spectra are somewhat more intense than the =30 ones (not explicitly shown in a special plot). In order to calculate the partial spectral weights [defined in Eqs. (7) and (8)] as a function of the incident angle, we will choose a coordinate system which fits the symmetry of the CuO crystal, namely the CuO 4 plaquettes. Figure 9 shows a suitable orthogonal coordinate system, whose z-axis FIG. 7. Peak emission energy of the structures A (dots), B(triangles), and C (squares) of Fig. 5(a) for =30. The lines indicate linear fits to the energy position data points for structures A (dashed line), B(solid line), and C (dashed dotted line). FIG. 9. Orthogonal coordinate system chosen to represent the p- and d-symmetries in its orientation relative to a CuO 4 -plaquette. Smaller spheres are the oxygens, the central larger sphere is the copper

9 SHAKE-UP VALENCE EXCITATIONS IN CuO BY PHYSICAL REVIEW B 70, 0855 (2004) FIG. 0. Partial spectral weights with p x - (dashed line), p y - (dotted line), and p z - (solid line) symmetry as a function of incident angle. is perpendicular to the plane of the plaquettes, and whose x-axis bisects the O-Cu-O bond angle of This coordinate system has also been used by Bocharov et al. 8 in their symmetry analysis of x-ray absorption spectra of CuO. On the basis of this coordinate system the partial spectral weights as a function of were calculated for each of the two types of CuO 4 -plaquettes independently and then averaged. The partial spectral weights with p-symmetry, as a function of, are presented in Fig. 0 and those with d-symmetry in Fig.. Comparing the calculated p-symmetry partial spectral weights with the experimental results one can easily verify that the excitations connected with the structure C must be FIG.. Partial spectral weights with d xy - (solid line), d xz - (solid line with triangles), d yz - (dotted-dashed line), d x 2 y2- (dashed line), and d 3z 2 - (dotted line) symmetry as a function of incident angle. attributed to transitions into an unoccupied p z -like continuum, more precisely into 3d 0 L 4p z, since it is commonly accepted that, due to the reduced screening of the s core hole, the s 3d 0 L 4p transition energy is by a few ev smaller than the s 3d 9 4p transition energy, where L means a hole at the ligand as the result of a charge transfer from the ligand O 2p to the central metal Cu 3d. The amplitude of structure B exhibits only a weak dependence on with a slight preponderance of the =54 spectra. Therefore, we must conclude, if we are admitting of only dipole allowed transitions, that structure B reflects transitions into a narrow 4p-like continuum with both p z - and p x -symmetry, where the p z -symmetry is slightly overrepresented. Possibly these are the 4p-orbitals admixed to a minority of 3d-orbital (according to calculations of Bocharov et al. 8 orbitals with d xy -symmetry), which are shifted by 2 3 ev to higher energies (.5 ev in the calculation of Ref. 8) with respect to the majority of 3d-holes, where transitions into the latter give rise to structure A. Comparing the calculated d-symmetry partial spectral weights with the experimental results (minimum intensity of the structure A at =54 ) excitation into 3d orbitals with at least strong admixture of d yz -, d x 2 y 2-, and d 3z 2 -symmetry can be held responsible for structure A. V. SHAKE-UP SATELLITES IN THE RIXS SPECTRA OF CuO In what follows we discuss shake-up satellites in RIXS spectra of CuO, which originate from Coulomb interaction of the s core hole with the system of valence electrons within the intermediate state under experimental conditions, where we look at the re-emission photon energies near the incident energy. In other words we concentrate on the re-emission due to the decay of the s 4p exciton. The measurements of the energy position, shape and amplitude of these shake-up satellites in the RIXS spectra of CuO as a function of the incident photon energy were performed at the NSLS beamline in vertical scattering geometry with a scattering angle of 90. The incident polarization ˆ was oriented parallel to the intersecting line of both CuO 4 -plaquettes, so that, according to Eq. (7), dipolar transitions of the absorption process could occur only into unoccupied orbitals, with p x - or p y -symmetry. Figure 2 presents the results of these measurements for a set of incident photon energies as indicated, where we have plotted the RIXS spectra as a function of energy loss. The (quasi)elastically scattered line is at the zero position, and for incident energies larger than 898 ev one can see a broad structure shifting its peak position from 0 ev energy loss to 25 ev with increasing the incident energy by 5 ev. We interpret this structure to be the valence fluorescence in its saturating position. Between the elastic line and the valence fluorescence one sees (indicated with lines in Fig. 2) for incident energies between 898 and 8988 ev a peaklike structure with an energy position shifting from 3 ev to 6 ev, and for incident energies between 899 and 900 ev a peaklike structure, which stays more or less fixed at 5.5 ev energy loss. We will show that these structures can be attributed to excitations within the Cu 3d related

10 DÖRING et al. PHYSICAL REVIEW B 70, 0855 (2004) FIG. 2. Energy loss spectra of CuO measured at different incident energies as indicated in vertical scattering geometry with a scattering angle of 90. The spectra are vertically off-set. The peak energy loss dispersion of the shake-up satellites are marked by lines as a guide to the eye. valence electron system, more precisely to an excitation from the b g bonding ground state, as calculated by Eskes et al. 36 for a CuO 4 6 cluster, into one of the antibonding states with a g,b 2g,b g or e g symmetry, between 5 and 6 ev apart from the ground state. The model used by Eskes et al. 36 takes into account both the hybridization between the 2p orbitals at the oxygen sites and the hybridization between the 2p oxygen and the 3d copper orbitals. This way one obtains bonding m and antibonding combinations of the hole basis functions d and m p with the same symmetry m, whose creation operators b + m (bonding) and a + m (antibonding) read b + m = * m d + m + * m p + m, 26 a + m = m d + m m p + m. 27 The exchange of weighting factors m and m between bonding and antibonding states which insures orthogonality, justifies to speak of charge transfer transition between these states. Since the q vector does not point into a highsymmetry direction of the two CuO 4 -plaquettes, the matrix element of Eq. (24) does not exclude one of the above antibonding symmetries. In order to understand the unusual dependence of the satellites on the incident energy, as presented in Fig. 2, especially when compared with the behavior of the valence fluorescence spectra, we have to consider the following two requirements for the occurrence of shake-up satellites according to Eq. (7): The first condition is to have unoccupied density of states with a symmetry, that allows resonant absorption of the incident photon by a dipole transition from the s level. According to our special orientation of the incident polarization, contributions with p x -orp y -symmetry to the unoccupied states are necessary. Secondly, largely 3d 9 configuration must contribute to the bonding intermediate state m [see Eq. (7)] reached in the resonant absorption process to bring about transitions into the antibonding states with mainly 3d 0 L configuration. Intermediate states of s 3d 0 L 4p configuration reached in the resonant absorption process cannot give rise to charge transfer shake-up satellites. This is certainly the reason for the absence of shake-up satellites between 8988 and 899 ev incident energy in Fig. 4, while the intensity of the valence fluorescence is not decreasing in this energy range. This very straightforward explanation for the absence of shake-up satellites for a certain incident energy range presumes a third-order treatment of the scattering process. The second-order treatment on the basis of the Anderson impurity model relies on the stronger delocalization of the so-called Zhang-Rice singlet emerging from a 3d 0 L configuration. The absence of shake-up satellites within a certain range of incident energy is then traced back to the reduced overlap of the Zhang-Rice singlet 22 with the much stronger localized 3d 9 configuration. What still remains to be explained is the dependence of amplitude, shape and energy position of the shake-up satellites on the incident energy, while we assume that the underlying process in the Cu 3d related valence electron system is the same for all incident energies. It is again the third-order treatment with the appearance of the double resonance denominator in Eq. (7), which provides a rather straightforward explanation. We are starting with Eq. (9). If we assume the absorption and re-emission matrix elements f b 2 n and m b i to be independent on energy, then the intensity of the shake-up satellites is given by I mod L E ex L 2 E ex 2 n H C m 2 E f E i, 28 where L,2 E stands for the Lorentz function L,2 E = E ,2 Since we are concerned with satellites of the elastically scattered line, we have set E ex, = E ex,2 = E ex 30 in Eq. (9). According to Eqs. (24) and (25) the energy dependent part of M C 2 E f E i = n H C m 2 E f E i 3 is given by f v q + G i 2 E f E i 32 G being nothing else than the sum over G of all valence electron dynamic structure factors, which reflect the spectrum of all symmetry allowed excitations in the Cu 3d related va

11 SHAKE-UP VALENCE EXCITATIONS IN CuO BY PHYSICAL REVIEW B 70, 0855 (2004) lence system. We will assume that this spectrum can be represented by a single Gaussian, centered at the energy loss of and with a lifetime broadening of S 33 2 G S = exp 2 2 S so that the total model spectrum of the shake-up satellite reads I mod = L E ex L 2 E ex 2 G S. 34 One can easily verify that the model spectrum according to Eq. (34), plotted as a function of the energy loss, in case for a certain value of incident energy, can exhibit peaklike structures, which reach a maximum value (we will speak of a resonance) at two different incident energies. With increasing incident energy, the energy position of the first peaklike structure remains unaltered with a peak at = and attains a resonance whereas the peak position of the second structure undergoes a shift, so that peak = E ex. The second structure achieves resonance at =E ex +. Certainly this double resonance will occur only if the width parameters, 2, and S are sufficiently small compared to. Otherwise the two resonances will join. But it can also happen that one resonance nearly disappears, namely if either or 2 is small compared to and S. Then the remaining resonance determines the behavior of the peaklike structure as a function of the incident energy. As already mentioned, Fig. 2 shows a peaklike shake-up satellite structure, which we have attributed to excitations of the Cu 3d related valence electron system, and which, as a function of incident photon energy, reaches two resonances with different dispersion behavior. The resonance at smaller incident energies is characterized by a positive peak shift with increasing energy, whereas the resonance at larger incident energies exhibits a stable peak position with varying incident energy. This behavior is exactly opposite to that suggested by the above model. Therefore, we have to conclude that the experimental results of Fig. 2 cannot adequately be described by this simple model with only one excitation energy E ex. On the contrary, it must be assumed that the two observed resonances must be ascribed to two different intermediate states with distinct excitation energies E A B ex and E ex thus writing I mod = I mod A + I mod B = L A, E A ex L A,2 E A ex 2 G S + L B, E B ex L B,2 E B ex 2 G S. 35 In order to adapt this model with equal weights of both excitation channels to the experiment, first the elastic line and the valence fluorescence of the experimental spectra were subtracted by fitting appropriate functions to both (a Gaussian to the elastic line and a Voigt function to the fluorescence line). Then, as well in the experiment as in the model, the peak intensity of both resonances was normalized to one. It turned out that the model fits well the experiment using the following model parameters: E A B ex =898.4 ev, E ex = ev, =5.4 ev, and S =.8 ev. Figure 3 presents the comparison between experiment [Fig. 3(a)] and model [Fig. 3(b)] in the form of a level diagram. One realizes a FIG. 3. Dependence on the incident energy of the intensity of the shake-up sattellites (peak intensity normalized to one), (a) experiment and (b) simulation. The distance of the level lines is 0.2. very good agreement both with respect to the peak shift of the first and the stable peak position of the second resonance. Comparatively accurate values for the Lorentzian halfwidths could not be obtained, due to the rather poor statistics. Nevertheless, the following qualitative estimations seem appropriate: A,2 is evidently smaller than A, 2.5 ev. This relation between the A s suppresses the first resonance of I A mod as discussed above and leaves behind only that with the shifting peak positions, where the resonance occurs roughly at E A ex + = ev. On the contrary B, and B,2 are nearly equal in size, both 3.5 ev. Since this value is not too far from and is much larger than S, the two resonances of I mod B join, giving rise to the broad structure in Fig. 3 with only one resonance around E B ex + /2 = ev. According to the energy position and the symmetry assignment, the excitation corresponding to E ex A = ev must be attributed to transitions into the narrow unoccupied 4p-like continuum with partly p x -symmetry, which gave rise to the structure B in the analysis of the 0855-

12 DÖRING et al. PHYSICAL REVIEW B 70, 0855 (2004) resonantly excited s 2p emission with its peak at 8982 ev incident energy. Moreover, if our assumption about the origin of structure B in Sec. IV is correct, the corresponding transition is connected with an open 3d-shell at the copper site, so that the condition for shake-up satellites laid down above is satisfied. The rather small width of A,, when compared with B,/2, is an additional indication of a narrower unoccupied 4p-like continuum involved, since A, is not far from the inverse lifetime of the Cu s core hole. Looking for a physically plausible reason for the still smaller width of A,2 characterizing the re-emission process, one must be aware that, by assuming hybridization between 4p- and 3d-orbitals, one prerequisite of our simple model with a single excitation energy E ex and a single width cannot be satisfied, namely the decoupling between the 3d-related valence system and the 4p electrons. Therefore, the physical meaning of the s should not be overestimated. According to the general principles stated above and due to the incident polarization of our experiment, the excitation corresponding to E B ex = ev must be attributed to transitions into unoccupied states with 3d s 9 4p x,y configuration. In their symmetry selective evaluation of polarized x-ray absorption spectra of CuO, Bocharov et al. 8 have found the peak DOS with p x - and p y -symmetry around ev in rather good agreement with our assignment. Of course, the above study finds also p x - and p y -symmetry contributions (shoulderlike structures of the DOS) at lower energies. But it is commonly believed (see, e.g., Refs. 9 and 20) that these are due to states with s 3d 0 L 4p x,y configuration, which cannot give rise to shake-up satellites of the kind under investigation. Concluding, we can state that the third-order perturbation treatment of the shake-up satellites can explain all details of the incident energy dependence of their energy loss position and their resonance behavior. Moreover, this treatment can account for the absence of the satellites in certain ranges of the excitation spectrum as well as for the selectivity brought about by a definite incident polarization. Finally, we will study the q-dependence of the shake-up satellites. VI. MOMENTUM TRANSFER DEPENDENCE OF THE CuO SHAKE-UP SATELLITES All our previous considerations of the CuO shake-up satellites started with the assumption that after the re-emission of the photon the Cu 3d related valence system remains excited via Coulomb interaction with the s core hole, so that the momentum q=k K 2 is transferred to the valence electron system. This means that the energy loss position of a shake-up satellite should change with the transferred momentum q, whenever the corresponding excitation of the Cu 3d related valence electron system exhibits dispersion. Such a q-dependence has indeed been observed for a certain class of excitations. 22,25,26 On the other hand, the charge transfer excitation showing =5.4 ev we have figured out as being responsible for the shake-up satellites in our RIXS spectra of CuO, is strictly local, so that a dispersion of these features is not expected. This assumption could be verified by means of a measurement, whose results are presented in Fig. 4. The FIG. 4. Energy loss spectra of CuO measured at different scattering angles as indicated in vertical scattering geometry. CuO shake-up satellite was measured in vertical scattering geometry at beamline X2 with an incident energy of 8986 ev and with ˆ parallel to the intersection line of both kinds of CuO 4 -plaquettes as in Sec. V. By changing the scattering angle between 90 and 30 it was possible to vary the amount of q without changing either the direction of q or the direction of ˆ relative to the crystal coordinates. No dispersion of the energy loss position of the shake-up satellites was discovered. But also the intensity of the satellite structure does not exhibit a distinct dependence on q, although Eq. (24) seems not to exclude such a dependency. Apparently, the summation over all reciprocal lattice vectors G in Eq. (24) wipes away such effects. Additionally we have tested whether varying the direction of q but leaving the amount of q unchanged has any influence on the intensity of the satellites. We could not observe any effect outside the error limits. However, it might be of interest for further studies of excitations connected with intermediate state interactions to present the results of another experiment in horizontal scattering geometry, represented in Fig. 5 and to find an explanation of the rather unexpected results: The CuO shake-up satellite was measured in horizontal scattering geometry at the HARWI beamline with an incident energy of 8986 ev and with an incidence angle fixed to 54, so that also the orientation of the incident polarization was fixed. Under these constraints the horizontal scattering geometry does not allow for changing the scattering angle (or the amount of q) without changing the direction of q relative to the crystal. Thus in this experiment different scattering angles as marked in Fig. 5 means not only different amounts of q but also different orientations of q. Apparently, this constellation leads to a remarkable change of the satellite intensity, whereas the intensity of the fluorescence line remains unchanged. As in the above experiments, also here the energy loss position of the satellite is not influenced by the variation of q, again an indication of the zero dispersion of the contributing excitation. Of course, also the intensity of the elastic line undergoes a drastic variation, a consequence of the horizontal scattering geometry which predicts for the Thom