CaCu$_3$Ru$_4$O$_{12}$: a high Kondo-temperature transition metal oxide

We present a comprehensive study of CaCu$_3$Ru$_4$O$_{12}$ using bulk sensitive hard and soft x-ray spectroscopy combined with local-density approximation (LDA) + dynamical mean-field theory (DMFT) calculations. Correlation effects on both the Cu and Ru ions can be observed. From the Cu $2p$ core level spectra we deduce the presence of magnetic Cu$^{2+}$ ions hybridized with a reservoir of itinerant electrons. The strong photon energy dependence of the valence band allows us to disentangle the Ru, Cu, and O contributions and thus to optimize the DMFT calculations. The calculated spin and charge susceptibilities show that the transition metal oxide CaCu$_3$Ru$_4$O$_{12}$ must be classified as a Kondo system and that the Kondo temperature is in the range of 500-1000 K.

We present a comprehensive study of CaCu3Ru4O12 using bulk sensitive hard and soft x-ray spectroscopy combined with local-density approximation (LDA) + dynamical mean-field theory (DMFT) calculations. Correlation effects on both the Cu and Ru ions can be observed. From the Cu 2p core level spectra we deduce the presence of magnetic Cu 2+ ions hybridized with a reservoir of itinerant electrons. The strong photon energy dependence of the valence band allows us to disentangle the Ru, Cu, and O contributions and thus to optimize the DMFT calculations. The calculated spin and charge susceptibilities show that the transition metal oxide CaCu3Ru4O12 must be classified as a Kondo system and that the Kondo temperature is in the range of 500-1000 K.
The discovery of the transition metal oxide CaCu 3 Ru 4 O 12 (CCRO) showing Kondo-like properties therefore created quite an excitement [11][12][13][14]. The crystal structure of this A-site ordered perovskite is shown in the inset of Fig. 1. However, the Kondo interpretation has also met fierce reservations. It has been argued that the specific heat coefficient γ does not deviate much from the band structure value, suggesting a minor role of the electronic correlations [15]. Other interpretations of the mass enhancement have been put forward [16,17]. Electron spectroscopy studies have also not converged on the position or even the presence of the putative Kondo peak [18][19][20].
Here we address the CCRO problem from a different perspective. Figure 1 shows the magnetic susceptibility of CCRO together with that of CaCu 3 Ti 4 O 12 (CCTO), as reproduced from Refs. [11,12,21,22] and Ref. [23],  [11,12,21,22] and of CaCu3Ti4O12 from Ref. [23]. The black dashed line shows the Curie paramagnetic behavior for a S = 1/2 Cu 2+ ion scaled by a factor of three. Inset: Crystal structure of CaCu3Ru4O12 visualized by VESTA [24]. The blue, red, gray, and indigo blue spheres represent Cu, O, Ru, and Ca atoms, respectively. respectively. One can observe that CCTO follows, far above its 25 K Néel temperature, almost a text-book Curie-Weiss law that can be understood in terms of paramagnetic S = 1/2 Cu 2+ ions. By contrast, one can also see that CCRO shows a completely different behavior with a magnetic susceptibility that is an order of mag-nitude smaller. There are also no indications at all for magnetic order. If one believes that the system is nonmagnetic and that Kondo physics does not take place in CCRO, then the Cu ions in CCRO have to be monovalent with the non-magnetic full-shell 3d 10 configuration or trivalent, which can be a nonmagnetic band insulator like NaCuO 2 .
We show here that the Cu ions are definitely divalent and thus have spin degree of freedom. We have carried out a detailed photoelectron spectroscopy study using a wide range of photon energies in order to establish the presence of correlation effects on both the Cu and Ru ions as well as to disentangle the Ru, Cu, and O contributions to the valence band. This allowed us to tune the double counting corrections in the LDA+DMFT calculations accordingly, making these calculations predictive for the low energy physics. We then were able to determine how the Cu 2+ magnetic moments can be screened. In particular, we will show that, in going from high to low temperatures, this screening takes place already at 500-1000 K, and that we thus must classify CCRO as a Kondo system with a very high Kondo temperature.
Hard x-ray photoemission (HAXPES) measurements were carried out at the Max-Planck-NSRRC HAX-PES end-station [25] at the Taiwan undulator beamline BL12XU of SPring-8 in Japan. The photon energy was set to hν = 6.5 keV and the overall energy resolution was ≈ 270 meV as determined from the Fermi cutoff of a gold reference sample. Soft x-ray (resonant) photoelectron (PES) and absorption (XAS) spectroscopy experiments were performed at the NSRRC-MPI TPS 45A Submicron Soft x-ray Spectroscopy beamline at the Taiwan Photon Source (TPS) in Taiwan. The overall energy resolution when using 1.2 keV, 931 eV, and 440 eV photons was ≈ 150 meV, 125 meV, and 60 meV, respectively. Photoemission measurements in the vicinity of the Ru 4d Cooper minimum, i.e. at photon energies of 200 eV, 150 eV, and 100 eV, were performed at the PLS-II 4A1 micro-ARPES beamline of the Pohang Light Source (PLS) in Korea. The overall energy resolution was ≈ 55 meV. Polycrystalline samples of CCRO were synthesized by solid-state reactions [26]. Clean sample surfaces were obtained by cleaving sintered samples in situ in ultra-high vacuum preparation chambers with pressures in the low 10 −10 mbar range. The measurements at SPring-8 and TPS were carried out at 80 K, and the measurements at PLS at 100 K. We have used three different batches of CCRO samples for our spectroscopic measurements, all giving the same results, providing confidence in the reliability of the data, see Appendix A.
Our calculations employ the LDA+DMFT scheme [27][28][29]. We start with density functional calculations for the experimental crystal structure [12], see Fig. 1, using the Wien2k code [30] and construct the multi-band Hubbard model on the basis spanned by the Cu 3d, Ru 4d, and O 2p Wannier functions obtained with the wan-nier90 package [31,32]. The on-site Coulomb interactions on the Cu and Ru sites are approximated with the 2 4 2 2 2 0 1 8 1 6 1 4 1 2 1 0 8 6 4 2 0 I n t e n s i t y ( a r b . u n i t s ) B i n d i n g e n e r g y ( e V ) C a C u 3 R u 4 O 1 2 C u 2 p r e s o n a n t P E S R e s o n a n t O f f -R e s o n a n t 9 3 0 9 4 0 9 5 0 9 6 0 I n t e n s i t y ( a . u . ) P h o t o n e n e r g y ( e V ) C a C u 3 R u 4 O 1 2 X A S , 3 0 0 K C u L 2 , 3
density-density form with parameters (U , J)=(8.5 eV, 0.98 eV) for Cu 3d electrons and (3.1 eV, 0.7 eV) for Ru 4d electrons, which are typical values for Cu and Ru systems [19,[33][34][35]. The continuous-time quantum Monte Carlo (CT-QMC) method with the hybridization expansion [36][37][38] was used to solve the auxiliary Anderson impurity model (AIM). The double-counting corrections arising in LDA+X methods [29,39], which fix the charge transfer energies on the Cu and Ru sites, were treated as adjustable parameters and their values fixed by comparison to the present valence band and core-level photoemission data as well as previous angle-resolved PES (ARPES) data. The valence spectra were obtained by analytic continuation of self-energy using the maximum entropy method [40,41]. The Cu 2p and Ru 3d core-level XPS were calculated using the method of Refs. 42-44. We have carried out XAS and valence band resonant PES measurements in the vicinity of the Cu L 2,3 edge, see Fig. 2. The peak positions and line shape of the spectra are characteristic for divalent Cu [19,[45][46][47]. We can exclude that the Cu in CCRO is monovalent or trivalent since the spectral features of Cu 1+ /Cu 3+ oxides are positioned at quite different energies [19,46,48]. We thus can conclude that the Cu ions in CCRO possess a spin degree of freedom and that some form of screening must take place as to make their magnetic susceptibility to deviate dramatically from the Curie-Weiss law. Figure 3 shows the Cu 2p core level HAXPES spectrum of CCRO together with that of Li 2 CuO 2 as reproduced from Ref. [49]. Here we took Li 2 CuO 2 as a reference system which contains CuO 4 plaquettes that are weakly coupled [49] and thus can serve for a comparison with CCRO which also contains rather isolated CuO 4 plaquettes, see Fig. 1. The spectra share the gross features: the main peak (B) at around 932-933 eV binding energy and the satellite (C) at 942 eV for the Cu 2p 3/2 component. However, the fine structure differs considerably: the main peak of CCRO consists of two peaks (A and B) [18,50] unlike the single peak (B) of Li 2 CuO 2 .
The Cu 2p core level spectrum of Li 2 CuO 2 is typi- cal for a Cu 2+ oxide [51]. Thanks to the weak coupling between the CuO 4 plaquettes, it can be explained accurately with a full multiplet single CuO 4 -cluster calculation [52,53], as shown in Fig. 3b. In contrast, the two-peak structure (A and B) of the CCRO main peak cannot be captured by the cluster model. This is indicative of a screening process [18,50,54,55] which is present in CCRO but absent in Li 2 CuO 2 . Since the CuO 4 plaquettes in CCRO are quite isolated from each other, a non-local screening mechanism due to inter-Cu-cluster hopping is not expected to play an important role. We rather relate the screening process to the metallic state of CCRO. The LDA+DMFT calculations shown in Fig. 3d, reproduce the fine structure of the main peak very well. We thus indeed can infer that CCRO contains correlated magnetic Cu 2+ ions, which experience screening by conduction electrons. How strong or complete the screening is will be discussed below. Figure 4a shows the experimental valence band spectra of CCRO in a broad energy range measured at various photon energies. Our motivation here is to make use of the different photon-energy dependence of the photoionization cross sections [56][57][58] to distinguish the Ru 4d, Cu 3d, and O 2p contributions to the spectra. The photon energy of 6.5 keV in HAXPES is much higher than the previously used photon energies of 1486.6 eV and 920 eV [18,19,50] and makes Ru 4d have the largest cross section relative to Cu 3d and O 2p. The low photon energies of 200, 150, and 100 eV are close to the Cooper V a l e n c e b a n d I n t e n s i t y ( a r b . u n i t s ) x y y z 3 z 2 -r 2 z x minimum of the Ru 4d cross section [58], so that with these photon energies the Ru 4d signal gets maximally suppressed, enabling us to observe better the Cu 3d contribution. The ratio between the O 2p and Cu 3d cross sections also becomes continuously larger with lowering the photon energy [58]. See also Appendix B displaying the photon energy dependence of the cross sections in more detail.
In the set of valence band spectra, Fig. 4a, we can identify features labeled A, B, C, D, and E. Features A and D are very strong at 6.5 keV and diminish almost completely between 100-200 eV. This strongly suggests that A and D are derived from the Ru 4d orbitals. Features B and E are visible throughout the set, while the intensity of C is enhanced at 100 eV, the lowest photon energy of the set. This observation indicates that the features B and E are related to the Cu 3d while C is likely of O 2p origin. We point out that the positions of these five features do not change with the photon energy and coincide with the previously reported soft x-ray studies [18,19,50].
The LDA+DMFT results are presented in Fig. 4b. They corroborate the above assignment. The calculated Ru 4d spectrum matches the experimental features A and D, while the theoretical Cu 3d spectrum explains well the features B and E, the former being the Cu |d 9 L and the latter the Cu |d 8 final state [19,[45][46][47]. The feature C is captured by the theoretical O 2p spectra. The calculations reveal that the spectrum around the Fermi level is dominated by hybridized Ru 4d and O 2p bands. In Fig. 4c, the Cu 3d spectrum is decomposed into xy, 3z 2r 2 , x 2 -y 2 , yz, and zx components. As shown in the inset, the Cu 3d xy orbital points to the surrounding four oxygen sites. Therefore, the Cu 3d xy orbital hybridize with the Ru 4d orbitals (via the O 2p orbitals) most strongly among the five Cu 3d orbitals. The Cu 3d xy spectral density near the Fermi level is quite low compared to the Ru 4d and O 2p one, see Fig. 4b, and most of its weight is above the Fermi level.
In order to experimentally detect the Cu contribution around the Fermi level, we focus on the larger peak at positive energies and the spectra taken at 150 eV which is almost the photon energy to minimize the Ru 4d signal. The top panel of Fig. 5 displays a close-up of the spectra along with the corresponding gold reference spectrum taken under the same conditions. In order to look for the possible presence of states above the Fermi level, we divide the CCRO spectrum by the corresponding gold spectrum. The results are shown in the bottom panel of Fig. 5. We can identify clearly the presence of a sharp peak at about 0.07-0.08 eV above the Fermi level, very consistent with the results of the LDA+DMFT calculations.
Zooming to the vicinity of the Fermi level [59], the calculated Cu 3d xy spectrum in Fig. 6a reveals sharp temperature-dependent peaks. To put the present results in the context of Anderson impurity and periodic Anderson models we show the Cu 3d xy hybridization densities in Fig. 6b. The global view in the inset shows a strongly asymmetric situation with the Fermi energy located in the tail of the hybridization density. The hybridization density exhibits only a minor temperature dependence. A sharp hybridization peak below the Fermi level, responsible for the peak around −0.08 eV in the Cu spectra also observed with angle-resolved PES (ARPES) around the H point in the Brillouin zone [20], is an exception. This explains the different temperature behavior of the peaks in Fig. 6a. Damping of the peaks above the Fermi level with increasing temperature is not accompanied by changes of the hybridization function and thus reflects the Anderson/Kondo impurity physics controlled by the Kondo temperature. Damping of the −0.08 eV peak arises from temperature-induced changes of the hybridization function, i.e., involves the feedback from the Cu ions (the temperature dependence of Ru contribution is negligible in the studied range). It is therefore an Anderson/Kondo lattice effect related to somewhat lower coherence temperature [60,61].
The inset of Fig. 6a shows the inverse quasi-particle lifetime. The region of long lifetime (dip in the inverse lifetime) between -0.1 eV and +0.05 eV around the Fermi level found at lower temperatures, marks limits of the Fermi liquid theory. Between 600-700 K the scattering rate around Fermi level increases forming a peak in Im Σ. Eventually (1500-2000 K) the scattering rate grows quasi-uniformly in entire low-energy regime (-0.5-0.5 eV) with increasing temperature.
Next, we discuss the local spin susceptibility χ loc at the Cu site. The top panel of Fig. 7 displays the χ loc obtained with LDA+DMFT. It exhibits the Curie behavior at high temperatures, and turns into a broad maximum at lower temperatures which is characteristic for Kondo screening. The deviation from the Curie behavior starts around 1000 K, suggesting a relatively high Kondo scale T K . The calculated χ loc reproduces quite well the experimental susceptibility [11] although an exact match can not be expected between the local and the uniform spin susceptibility, due to the contribution of the itinerant Ru 4d -O 2p states. The dynamical local susceptibility χ loc (ω) is, nevertheless, directly related to the spinlattice relaxation rate 1/T 1 measured in nuclear magnetic resonance (NMR) experiments [12,21] 1 The calculated temperature dependence of 1/T 1 is shown in Fig. 7. It shows an approximately linear increase at low-temperatures, which flattens into a constant behavior at around 1000 K. It is well-known that the former is characteristic of a Fermi liquid, while the latter of a fluctuating local moment. For CCRO, the NMR experiment was performed up to 700 K so far, where the absence of the constant behavior was brought up as an evidence against the Kondo physics in this compound. Our result however suggests that CCRO manifests its local moment signature in NMR above the reported temperature. The corresponding spin-spin correlation functions χ spin (τ ), see Appendix C, reflect the presence of an instantaneous Cu 3d moment at all temperatures, which rapidly disappears on a short-time scale at temperatures below T K . In contrast, the charge correlations are temperature independent. Rapid charge fluctuations present at all temperatures rather reflect the Cu-O bonding, not the hybridization with the states in the vicinity of the Fermi level. The presence of magnetic Cu ions immersed in an itinerant band leads indeed to the emergence of Kondo physics, as demonstrated by our LDA+DMFT results. An important aspect for the long standing discussions about CCRO is our finding that the Kondo temperature is quite high, namely at around 700 K (between 500 and 1000 K). We would like to note that the parameters and double-counting corrections in our LDA+DMFT calculations have been tuned as to reproduce the available experimental (HAX)PES and the recent ARPES [20] spectra, details can be found in Appendix D. In this respect it is worth mentioning that we have also included on-site Coulomb interactions at the Ru site. This was necessary to explain the presence of a satellite structure in the Ru 3d core level PES spectrum, see Appendix D. It turned out that the inclusion of U Ru in the Ru 4d shell has also a substantial influence on the low energy properties of CCRO. We have calculated the local spin susceptibility χ loc at the Cu site with U Ru = 3.1 eV and U Ru = 0.0 eV, see Appendix D, indicating that correlations on the Ru site influence the screening process on the Cu site. This can be traced back to the influence of the Ru U on the shape of the Ru 4d -O 2p band. In Appendix D, we show the Cu xy spectral intensities calculated with U Ru = 0 eV on the Ru 4d shell, where the agreement to the available PES data is surrendered.
The high Kondo temperature we have found from our LDA+DMFT calculations implies that the contribution of the Kondo screening process to the low temperature specific heat is modest, thus explaining why band structure calculations can seemingly reproduce the experimentally observed γ value of the specific heat rather well since the main contribution comes from the uncorrelated Ru-O derived bands. The high Kondo temperature also implies that one needs to go to very high temperatures to see the appearance of local Cu moments, as demonstrated in Fig. 7. We can thus infer that upon going from high to low temperatures the Kondo screening process is practically completed already at 300 K, and that lowering the temperature further would not produce significant changes in the electronic and magnetic properties. It is understandable that there has been a controversy concerning Kondo physics in CaCu 3 Ru 4 O 12 since its signatures in low temperature measurements are weak.
Nevertheless, Kondo physics is present, as evidenced by the low-temperature disappearance of the Cu 2+ magnetic susceptibility for which we were able to provide a quantitatively explanation using our LDA+DMFT calculations that include fine tuning of the parameters from a detailed comparison to bulk sensitive photoemission data. Our findings indicate that the material class CaCu 3 M 4 O 12 indeed provides a unique opportunity to explore Kondo  In order to ensure that the experimental spectra and their features are intrinsic to the material, we have utilized three different batches of samples synthesized by three different groups: A.C. Komarek's group from Max Planck Institute for Chemical Physics of Solids, Dresden, A. Günther from the University of Augsburg, and Y. Shimakawa's group from the Kyoto University. Figure 8 shows the comparison of the valence band PES and XAS spectra taken from these three batches of samples. The results match perfectly, and thus confirm that the data presented in this paper are not sample specific or due to extrinsic contributions  9 shows the photon energy dependence of the photoionization cross sections as interpolated from the data tabulated in Refs. [56][57][58]. The vertical lines highlight the photon energies used in this study. At high photon energies (HAXPES, 6.5 keV), the Ru 4d provides a larger signal than the Cu 3d and the O 2p. Going to lower photon energies, the Cu 3d becomes gradually stronger, and with a crossover at 1.2 keV, it becomes the dominant contributor. At around 100 eV, there is a minimum in the Ru 4d cross section, also known as the Cooper minimum, which provides the ideal condition for studying the Cu 3d contributions. At this photon energy, the Cu 3d signal is enhanced by a factor ≈ 70 with respect to the Ru 4d. For lower photon energies, the Ru 4d quickly gains strength and becomes again the dominant contributor, making these low photon energies less useful when searching for the Cu 3d signal close to the Fermi level. As for the O 2p contribution, we observe that it is highly suppressed for high energies, but becomes only competitive with the Cu 3d for energies below ≈ 50 eV.

Appendix C: LDA+DMFT computational method
Below we describe the LDA+DMFT scheme [27][28][29] employed to analyze the experimental data. We start with density functional calculations for the experimental crystal structure of CCRO [12] using the Wien2k code [30]. Then we construct the multi-band Hubbard model on the basis spanned by the Cu 3d, Ru 4d, and O 2p Wannier functions from the LDA bands using wien2wannier and wannier90 packages [31,32]. The multi-band Hubbard model is augmented with the local electron-electron interaction within the Cu 3d and Ru 4d shells giving the Hamiltonian, Here, c † k is an operator-valued vector whose elements are Fourier transforms of c γi , that annihilate the Cu 3d electron in the orbital γ in the i-th unit cell. Similarly r † k and p † k are those for Ru 4d and O 2p electrons. The onsite Coulomb interaction W dd i,Cu and W dd i,Ru on Cu and Ru sites is approximated with the density-density form with parameters (U , J)=(8.5 eV, 0.98 eV) for Cu 3d electrons and (3.1 eV, 0.7 eV) for Ru 4d electrons, which are typical values for Cu and Ru systems [19,[33][34][35]. The doublecounting terms µ Ru dc , µ Cu dc , which correct for the d-d interaction present in the LDA step [29,39], renormalize the p-d splitting and thus the charge-transfer energy. We have fixed the double-counting values to µ Cu dc = 70.1 eV and µ Ru dc = 16.2 eV (µ Ru dc = 13.4 eV for the model without Ru e g states) by comparison to the photoemission spectroscopy data discussed in Appendix D. The CT-QMC method with the hybridization expansion [36][37][38]62] was used to solve the auxiliary Anderson impurity model (AIM) in the DMFT self-consistent calculation. The valence spectra were obtained by analytic continuation of self-energy Σ(ε) using the maximum entropy method [40,41]. The hybridization function ∆(ε) for the Cu 3d orbital γ (|d γ ) is given by [28,29] where G(ε) and h 0 are the local Green's function and the one-body part of the on-site Hamiltonian at the Cu site, respectively. ∆(ε) does not depend on the spin, and small off-diagonal elements between different orbitals are neglected. The Cu 2p core-level spectrum was calculated using the method of Refs. 42 and 43, where the AIM with the DMFT hybridization density is extended to include the Cu 2p core orbitals [42,44]. The configuration interaction scheme with 25 bath states representing the DMFT hybridization density is employed to evaluate the 2p spectra. Figure 10 shows the local spin χ spin (τ ) and charge χ charge (τ ) correlation functions at the Cu site. χ spin (τ ) is given as χ spin (τ ) = m z (τ )m z (0) [63,64] and χ charge (τ ) is given by χ charge (τ ) = δn(τ )δn(0) [65], where δn = n − n with the Cu d-occupation number operatorn.  A rigid instantaneous Cu 3d spin moment is present for all temperatures, as imprinted in the temperatureindependent value of χ spin (τ = 0), see Fig. 10. The spin moment survives on a long-time scale at high temperatures (see e.g. at 2000 K), giving the Curie behavior in χ loc shown in the main text, while it disappears on short timescales at low temperatures due to the Kondo screening by the Ru 4d -O 2p bands, giving the Pauli-like behavior in χ loc . In contrast to the temperature dependence in χ spin (τ /β), the local charge correlation function χ charge is substantially suppressed at all temperatures, see the inset of Fig. 10, indicating the frozen charge fluctuation of the Cu 3d electrons in CCRO, as expected in the Kondo regime. Thus the calculated spin and charge correlation functions support the Kondo behavior of Cu 3d electrons in the studied material. We point out that the χ spin (τ = 0) value is reduced to 0.7 from 1.0 which corresponds to the S = 1/2 Kondo limit. The reduction comes from the mixture of the |d 10 L configuration by the Cu-O hybridization, where |L represents an O 2p hole, but it does not show a temperature dependence. This behavior characterizes the effect of charge-transfer in the Cu-O subsystem. The energy scale of the Kondo physics therefore builds on the ligand-metal hybridization in addition to the Coulomb interaction U at the Cu site.

Appendix D: Parameters in the LDA+DMFT simulation
In the calculations we used two versions of the model described in Appendix C. First model contains all Ru 4d states, while the second contains only the Ru t 2g states present at and below the Fermi level. The first model, which we consider as the primary one, is needed to capture the full spectrum, in particular the feature D of Fig. 4a. The second model is used for numerical conve- nience as it allows us to perform LDA+DMFT calculations at lower temperature (below 300 K). It is employed to compute the low-energy Cu spectral intensities (Fig. 6) and response functions (Fig. 7) including temperatures below 300 K. Its results are validated by comparison to the first model at the higher temperatures.
The parameters µ Cu dc and µ Ru dc of our theory (Appendix C) are fixed to reproduce (i) positions of the peaks A-E in the valence valence photoemission spectra of Fig. 4, (ii) positions of the low-energy peak at 0.07-0.08 eV above, Fig. 5, and around 0.08 eV below [20] the Fermi energy, (iii) the splitting of the Cu 2p 3/2 XPS main line, see Fig. 3, and (iv) observation of a shoulder in Ru 3d core-level XPS spectrum, see Fig. 12.
Before analysing µ Cu dc and µ Ru dc we briefly discuss why we need to include the electron-electron interaction on the Ru site [66][67][68]. To this end we have compared calculations with U Ru = 3.1 eV from previous DFT studies for Ru oxides (with the same formal Ru valence as in CCRO) [34,35] and with U Ru = 0.0 eV. Fig. 12a shows that finite U Ru is needed to yield the shoulder/satellite feature (iv) in the Ru 3d core-level XPS spectra. However, finite U Ru affects also the behavior of the Cu xy state, see dashed lines in Figs. 11ab, through dynamical renormalization (band narrowing) of Ru bands, which modifies the environment (hybridization density) of Cu, see Fig. 11d. With U Ru = 0.0 eV we could not find µ Cu dc to fulfill (i-iii). We would like to note that the Ru spin response is far from the Curie form despite the presence of correlations in the Ru 4d shell, see Fig. 12b. Correspondingly, also the screened (effective) Ru moment is substantially smaller than the atomic value (∼ 2µ B ) for a Ru 4+ ion. The value of µ Cu dc affects the positions of the Cu d 8 satellite, feature E, and feature B in Fig.4 as well as the low-energy peaks (ii). All of these are well reproduced by its chosen value of µ Cu dc = 70.1 eV, see Fig. 11a and Fig. 4. The small screened spin susceptibility due to the Kondo screening is found around the optimal µ Cu dc value, see Figs. 11bc. The µ Cu dc also affects the Cu-O charge-transfer energy and thus the splitting between the main line and charge-transfer satellite in Cu 2p XPS, see Fig. 13b. On the other hand, the behavior of Cu, including variation of µ Cu dc , has minor effect on its hybridization function, except for the peak just below Fermi energy. As a result the splitting of Cu 2p XPS main line (A,B) is independent of µ Cu dc , see Fig. 13b.
The same is not true for µ Ru dc . Affecting primarily the Ru-O charge-transfer energy, its variation modifies the Cu 3d xy hybridization function around Fermi level, which has a sizable effect on the splitting in the Cu 2p XPS main line (A,B), see Fig. 13a. We can thus use (iii) together with the Ru features in the global spectrum (i) to establish µ Ru dc = 16.2 eV.
Finally, we discuss the relationship of models with and without Ru 4d e g states. While µ Cu dc is the same for the two models, µ Ru dc must be different since the models include interaction between all Ru 4d orbitals in one case and between the t 2g only in the other one. The matching µ Ru dc for the model without Ru 4d e g states is 13.2 eV. Both models yield the very similar temperature dependencies of the Cu spectral functions as well as hybridization densities and local susceptibilities, as shown in Fig. 14.