Mixed Valence of Ce and Its Consequences on the Magnetic State of Ce9Ru4Ga5: Electronic Structure Studies

We report on X-ray photoelectron spectroscopy (XPS) and ab initio electronic structure investigations of a novel intermetallic material Ce9Ru4Ga5. The compound crystallizes with a tetragonal unit cell (space group I4mm) that contains three inequivalent Ce atoms sites. The Ce 3d core level XPS spectra indicated an intermediate valence (IV) of selected Ce ions, in line with the previously reported thermodynamic and spectroscopic data. The ab initio calculations revealed that Ce1 ions located at 2a Wyckoff positions possess stable trivalent configuration, whereas Ce2 ions that occupy 8d site are intermediate valent. Moreover, for Ce3 ions, located at different 8d position, a fractional valence was found. The results are discussed in terms of on-site and intersite hybridization effects.


Introduction
Physical properties of Ce-based intermetallic compounds are mainly determined by two competing interactions: Kondo effect, characterized by a temperature T K ∝ exp(− 1 |J f c N(E F )| ), and Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, related to T RKKY ∝ J 2 f c N(E F ). In both expressions, N(E F ) stands for density of states (DOS) at Fermi level E F , and J f c ∼ V f c is the coupling constant between 4 f and conduction (c) electron states, where V f c represents on-site hybridization energy given by f -c hybridization matrix element. According to the Schrieffer-Wolff transformation [1], where E 4 f stands for energy of 4 f level. The energy V f c determines filling of the 4 f shell, and thus governs the character of magnetic ground state. In the Ce-based intermetallics, the hybridization V f c results in a variety of intriguing properties such as heavy-fermion behavior, unconventional superconductivity, various magnetic ordering, non-Fermi liquid, and quantum critical phenomena [2].
For a number of Ce-based compounds reported in the literature, Ce ions occupy a single position in their crystallographic unit cells. If the 4 f -electron states are strongly localized, i.e., the Kondo interaction is weak, generally, a kind of magnetic ground state is expected. Most often, the compounds order antiferromagnetically, yet, a few ferromagnets are also known, e.g., Ce 2 RuGe 2 [3], CeRuPO [4],

Experimental and Computational Details
X-ray photoelectron spectroscopy (XPS) experiments were carried out on a polycrystalline sample Ce 9 Ru 4 Ga 5 used before for magnetization, magnetic susceptibility, specific heat, and resistivity measurements [17]. The XPS spectra were obtained at room temperature in vacuum of~10 −10 Torr using a Physical Electronic PHI 5700/600 ESCA spectrometer (Physical Electronics, Inc., Chanhassen, MN, USA) with monochromatized Al Kα radiation. The sample was broken in high vacuum of 6 × 10 −10 Torr, immediately before the spectra were recorded. Calibration of the spectral data was performed in a manner described in [23]. Binding energies were referenced to the Fermi level (E F = 0).
To determine theoretically the magnetic properties of the individual Ce ions in Ce 9 Ru 4 Ga 5 , the following procedure was applied. First, the PBEsol XC potential was corrected by the Hubbard-like correlation interaction using the approach developed by Anisimov et al. [28,29] with correlation energy parameter U = 1.5, 2.25, and 3 eV [30]. Then, the ab initio calculations were made within FP-LAPW approach, assuming the muffin-tin (MT) model for crystal potential. The radii of MT spheres, R MT , were taken equal 0.121 nm, 0.101 nm, and 0.111 nm for Ce, Ru, and Ga ions, respectively. The accuracy of the performed calculations was determined by the following parameters; l max = 10, G max = 14, and K max = 9/R MT 8.17 nm −1 . A number of 324 k vectors in the irreducible Brillouin zone used in the calculations was found to ensure a total energy convergence of the order of 0.01 eV. The structural data assumed in the initial calculations were taken from work in [16]; however, an atomic relaxation was performed to reach the equilibrium structure. Figure 1 shows the crystal structure of Ce 9 Ru 4 Ga 5 , which was the basis for our calculations. In the crystallographic unit cell, there are three inequivalent Wyckoff positions for cerium atoms: 2a site with Ce1 atoms, 8d site with Ce2 atoms, and another 8d site with Ce3 atoms [16]. Throughout the present paper we adopted the Ce atoms labels introduced in Table 2 in Ref. [16]. One should note, however, that in the text of the latter publication and in its figures the Ce1 atom was mistakenly switched with the Ce2 atom (we thank Dr. Elena Murashova, a coauthor of Ref. [16], for giving us comprehensive information about that error).  Table 1.

XPS Results
The X-ray absorption near-edge structure (XANES) spectroscopy, performed for Ce 9 Ru 4 Ga 5 near its Ce L 3 edge, revealed a mixed valence state of the Ce ions, giving an average valence of Ce ions to be about 3.1 at room temperature [16]. In order to corroborate that finding, we measured Ce 3d and Ce 4d core-level XPS spectra and analyzed the results in terms of the Anderson theory [31]. For a system with partial filling of the Ce 4 f shell, the theory predicts the appearance of the f 0 and f 2 final states as a result of intra-atomic hybridization between 4 f and conduction band states. The 3d XPS spectrum recorded at room temperature is presented in Figure 2a. The main lines correspond to the 3d 9 5/2 4 f 1 and 3d 9 3/2 4 f 1 final states, separated by spin-orbit (SO) interaction ∆ SO = 18.6 eV. Most remarkably, the spectrum also shows satellites 3d 9 5/2 4 f n and 3d 9 3/2 4 f n with n = 0 and 2, separated by the same energy ∆ SO .
According to the Gunnarsson-Schönhammer (GS) model [32,33], the 3d4 f 0 line arises due to the intermediate valence effect, whereas 3d4 f 2 reflects the on-site hybridization strength, which is expressed by the energy ∆ f c = πV 2 f c N(E F ) [31]. It is possible to separate of the overlapping peaks on the basis of the Doniach-Šunjić theory [34], and ∆ f c can be estimated from the intensity ratio I( f 2 )/[I( f 1 ) + I( f 2 )] of the respective Ce 3d XPS lines [33]. In turn, the intensity ratio r = gives an estimate for the 4 f shell mean occupation number n f [33]. The accuracy of determining ∆ f c and n f is usually less than 20% [35,36] (the limitations were discussed in details, e.g., in [33]). Moreover, one should note that these two quantities are interrelated.
In the case of Ce 9 Ru 4 Ga 5 , we found from the GS approach ∆ f c ≈ 210 meV. In order to determine the ground-state 4 f occupation, we used the theoretical method proposed by Fuggle et al. in Ref. [33], where the r ratio is calculated as a function of the initial f occupation and c ( f 0 ) equal to wave function amplitude of the initial f 0 configuration state [33], we derived the fractional 4 f electron count n f ≈ 0.88.
The fractional valence of Ce ions in Ce 9 Ru 4 Ga 5 was further corroborated by inspection of the Ce 4d XPS spectrum (see Figure 2b), which exhibits two lines near 120 and 124 eV, characteristic of the Ce 4+ states [33]. The fractional valence of Ce ions in Ce 9 Ru 4 Ga 5 was further corro 108 4d XPS spectrum (see Fig. 2b), which exhibits two lines near 120 and 1 109 states [34].  Table 1, and the so-de 113 Ce1, Ce2, and Ce3 atoms are given in Table 2. All the respective intera 114 to those reported in the literature [16] (see also Ref. citeremark).

Calculated Electronic Structure
The atomic positions in the crystallographic unit cell of Ce 9 Ru 4 Ga 5 , obtained as a result of minimizing interatomic forces, are presented in Table 1, and the so-derived local environments of the Ce1, Ce2, and Ce3 atoms are given in Table 2. All the respective interatomic distances are very similar to those reported in the literature [16] (see also Ref. citeremark ).
The electronic bands in Ce 9 Ru 4 Ga 5 , calculated assuming the correlation energy U = 1.5 eV and 2.25 eV, are shown in Figure 3 in a form of the total density of states (TDOS). In addition, the calculations were performed for a model in which different U values were attributed to distinct Ce atoms, and Figure 3 displays the result obtained setting U = 3 eV for the Ce1 atom and U = 2.25 eV for the Ce2 and Ce3 atoms. As can be inferred from the figure, the DFT data hardly depend on U, except a narrow range of binding energies −1 eV < E < E F . Figure 4a shows the spin-resolved TDOS calculated for the latter values of U compared with the valence band of of Ce 9 Ru 4 Ga 5 determined experimentally. Figure 4b, with an expanded energy scale, presents the same theoretical and XPS data together with the partial TDOS due to the particular atoms in the unit cell. Clearly, all the features present in the XPS spectrum are properly reproduced in the computed data. The main contribution due to the Ru 4d states is distributed between E F and the binding energy of 4 eV. In turn, the Ga 4p states form bands located near the binding energy of about 6 eV. The Ce 4 f states are responsible for a broad and fairly weak feature near E F . At the binding energy of about 17 eV and 19 eV, the calculated Ce 5p electronic states show SO-separated features, which are displaced in respect to the measured data by~1 eV. The discrepancy can be attributed to Ce 5d-electron correlations, which usually shift the calculated Ce 5p states to lower binding energies [26,37,38].  The main numerical results of the performed PBEsol+U calculations are listed in Table 3. For various U, the mean occupancy of the 4f shell of the Ce1 atoms is close to 1, while n f computed for both the Ce2 and Ce3 atoms is notably smaller than 1. The effect of U on the obtained 4f electron count is almost negligible. Taking into account the multiplicity of the particular Ce sites, one obtains an average filling of the 4f shell in Ce 9 Ru 4 Ga 5 equal to 0.86-0.89, in perfect agreement with the experimental result n f ≈ 0.88 (see above). The calculated total magnetic moment m Ce1 amounts to about 1 µ B , regardless of the value of U, whereas the magnetic moment found at the Ce2 and Ce3 sites is significantly smaller, namely, m Ce2 ≈ 0.3 µ B and m Ce3 ≈ 0.5 µ B .   The fractional valence of the Ce2 and Ce3 ions likely results from the on-site hybridization effect as well as some intersite hybridization between the Ce 4 f and Ru d-electron states. As can be inferred from Figure 5a-c, the on-site f-c hybridization causes a significant increase in the number of Ce2 and Ce3 5d-electron states, whereas the Ce1 4f electrons remain well localized at the binding energy of about 1 eV. At the same time, the DFT calculations clearly revealed strong inter-band hybridization of the Ce 5d and Ru 4d electron states for the Ce2 and Ce3 atoms, whereas the latter effect is negligibly small for the Ce1 atom (see Figure 5d-f).
Version May 16, 2020 submitted to Materials 7 of 10 5d and Ru 4d electron states for the Ce2 and Ce3 atoms, whereas the latter effect is negligibly small for 145 the Ce1 atom (see Fig. 5d − f ). In order to visualize the intersite hybridization and the atomic bonds in the unit cell of Ce 9 Ru 4 Ga 5 , we calculated the charge densities (setting U Ce1 = 3 eV and U Ce2,Ce3 = 2.25 eV). Figure 6 displays the electron density map within the crystallographic (010) plane. The map clearly shows almost isotropic distribution of valence electrons around the Ce1 and Ga atoms. In contrast, the charge distribution near the Ce2, Ce3, and Ru atoms is strongly anisotropic with strong accumulation of the electronic density along the bonds Ce2-Ru and Ce3-Ru. The strongest covalent bonding occurs between the Ce2 and Ru atoms, in concert with the crystal structure refinement, which revealed abnormally short Ce2-Ru interatomic distance [16]. Thus, the DFT calculations fully corroborated the scenario developed before [16,17], in which the IV behavior in Ce 9 Ru 4 Ga 5 , evidenced in the spectroscopic and thermodynamic properties of the compound, can be associated primarily with the Ce2 atoms.

Conclusions
The XPS experiment performed for Ce 9 Ru 4 Ga 5 confirmed the fractional valence of the Ce ions, noticed before in the L 3 XANES spectroscopy [16] and bulk thermodynamic measurements [17]. The compound forms with a crystallographic unit cell that hosts three inequivalent Wyckoff positions for Ce atoms, thus the experimentally derived filling of the 4 f shell (n f ≈ 0.88) was an average over those three sites. The DFT calculations allowed for inspecting the 4 f electron counts at each Ce atom. The results indicated that the Ce1 ion located at the 2a site is trivalent (n f is close to 1). In contrast, the Ce2 and Ce3 ions, placed at the 8d sites, were found intermediate valent with n f notably smaller than 1. The ab initio calculated mean occupation of the 4 f shell in Ce 9 Ru 4 Ga 5 is 0.86-0.89, which is in very good agreement with the experimental finding.
The electronic instability of the 4 f shell in the Ce2 ion gives rise to the IV character of the compound, established before in the study on its low-temperature bulk physical properties [17]. Most remarkably, the IV features were found to coexist with a long-range antiferromagnetic (AFM) ordering that sets in below T N = 3.7 K [17]. As suggested by our group in an earlier study [17] these two phenomena are spatially separated, i.e., they develop in different Ce ions sublattices. The present DFT results have corroborated such a scenario. Due to dissimilar strength of the intra-site band hybridization, the calculated magnetic moments are distinctly different for Ce1 (∼1 µ B /atom), Ce2 (∼0.3 µ B /atom), and Ce3 (∼0.5 µ B /atom). Therefore, it is reasonable to attribute the AFM state principally to the Ce1 ions, with a possible contribution due to the Ce3 ions, while the Ce2 ions remain nonmagnetic. The energy ∆ f c ∼ 200 meV is for Ce 9 Ru 4 Ga 5 quite large, however, V f c = ( ∆ f c πN(E F ) ) 1/2 ∼ 44 meV. One can also estimate the coupling constant J f c = 2V 2 f c |E f −E F | ≈ 5 meV [1] between the nearest Ce magnetic moments, which well correlates with low T N temperature-scale. In order to verify this tempting conjecture, neutron diffraction experiment is compulsory. Undoubtedly, the coexistence of intermediate valent and trivalent cerium ions makes Ce 9 Ru 4 Ga 5 an interesting material for further comprehensive experimental and theoretical investigations.