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 Ce 9 Ru 4 Ga 5 . The compound crystallizes with a tetragonal unit cell (space group I4 m m ) that contains three inequivalent Ce atoms sites. The Ce 3 d 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 2 a Wyckoff positions possess stable trivalent configuration, whereas Ce2 ions that occupy 8 d site are intermediate valent. Moreover, for Ce3 ions, located at different 8 d position, a fractional valence was found. The results are discussed in terms of on-site and intersite hybridization effects.

1. Introduction

Physical properties of Ce-based intermetallic compounds are mainly determined by two competing interactions: Kondo effect, characterized by a temperature , and Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, related to . In both expressions, stands for density of states (DOS) at Fermi level , and is the coupling constant between and conduction (c) electron states, where represents on-site hybridization energy given by f–c hybridization matrix element. According to the Schrieffer–Wolff transformation [1], is defined as , where stands for energy of level. The energy determines filling of the shell, and thus governs the character of magnetic ground state. In the Ce-based intermetallics, the hybridization 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 -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., CeRuGe [3], CeRuPO [4], CeRhSi [5], CePdAl [6], CeCrGe [7], or CePdIn [8]. However, the situation becomes less obvious when there is more than a single inequivalent Ce site in the crystal structure. Different local environments of the Ce ions can lead to dissimilar hybridization strengths, which spark the possibility of having distinctly different ground states for each individual inequivalent Ce ion. Recently, investigation of Ce-based compounds bearing multiple inequivalent Ce sites has received considerable attention, and for a few of them, diverse unusual low-temperature properties were established. Prominent examples are CeNiIn, with separate antiferromagnetic orderings in two different Ce atom sublattices [9]; CePdSi, with dipolar and quadrupolar antiferromagnetic orders associated with inequivalent Kondo sites [10]; or CePtIn and CePdIn, where two different Ce atom sublattices host antiferromagnetism and heavy-fermion superconductivity [11,12,13,14].

Another exciting case is the coexistence in a single material of long-range magnetic ordering and valence fluctuations, each phenomenon emerging in a separate Ce atom sublattice. Recently, this very rare situation was reported to occur, e.g., in CeRuGe [15], with two independent Ce atom sites in its crystallographic unit cell, and CeRuGa, which possesses as many as three inequivalent Ce atom sublattices [16,17]. Remarkably, in both compounds one of the different Ce atoms is coordinated by its Ru neighbors at a distance of Å and Å, respectively, which is much shorter than the sum of the covalent radii of the Ce and Ru atoms. As discussed in detail in a series of our previous papers on the Ce-Ru-X intermetallics (X = Ge, Ga, Al) [15,16,17,18,19,20,21,22], strong Ce–Ru bonding brings about a significant instability of the electronic shell, and thus intermediate valence behavior may arise. At the same time, the Ce ions with Ce-Ru distances of regular length remain their trivalent character that promotes localized magnetism with possible magnetic ordering at low temperatures.

The present research was aimed at verification of the electronic character of the particular Ce ions in CeRuGa by means of X-ray photoelectron spectroscopy and ab initio band structure calculations. Our results fully support the scenario of the dual nature of the electrons in this material.

2. Experimental and Computational Details

X-ray photoelectron spectroscopy (XPS) experiments were carried out on a polycrystalline sample CeRuGa used before for magnetization, magnetic susceptibility, specific heat, and resistivity measurements [17]. The XPS spectra were obtained at room temperature in vacuum of ~ Torr using a Physical Electronic PHI 5700/600 ESCA spectrometer (Physical Electronics, Inc., Chanhassen, MN, USA) with monochromatized Al K radiation. The sample was broken in high vacuum of 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 ().

The electronic band structure of CeRuGa was calculated using the full-potential linearized augmented plane waves (FP-LAPW) method [24] implemented in the WIEN2k computer code (WIEN2k_18.1, released on 30 June 2018, Institute of Materials Chemistry, TU Viena, Austria) [25] (for details on similarly made computations see, e.g., in [26]). In the performed calculations, we assumed the following electronic configurations of strongly-bound core level states (SC), weakly-bound states (WC), and valence band states in the particular atoms; Ce: [Kr]{}(); Ru: [Ar + ]{}(); and Ga: [Ne+]{}(). The fully relativistic formalism was implemented for the SC states, while local orbital (LO) states and VB electrons were treated within the scalar-relativistic Kohn–Sham approach. The spin-orbit (SO) interaction was applied within the second variational approach [24] for calculation of the VB and LO states. The revised Perdew–Burke–Ernzerhof (PBEsol) generalized gradient approximation (GGA) [27] was applied for the exchange correlation (XC) potential.

To determine theoretically the magnetic properties of the individual Ce ions in CeRuGa, 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, , 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; , , and . A number of 324 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 CeRuGa, which was the basis for our calculations. In the crystallographic unit cell, there are three inequivalent Wyckoff positions for cerium atoms: site with Ce1 atoms, site with Ce2 atoms, and another 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).

Figure 1

Tetragonal unit cell of CeRuGa (space group I4mm, No 107). The structure details are given in Table 1.

3. XPS Results

The X-ray absorption near-edge structure (XANES) spectroscopy, performed for CeRuGa near its Ce 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 and Ce core-level XPS spectra and analyzed the results in terms of the Anderson theory [31]. For a system with partial filling of the Ce shell, the theory predicts the appearance of the and final states as a result of intra-atomic hybridization between and conduction band states. The XPS spectrum recorded at room temperature is presented in Figure 2a. The main lines correspond to the and final states, separated by spin-orbit (SO) interaction eV. Most remarkably, the spectrum also shows satellites and with n = 0 and 2, separated by the same energy .

Figure 2

(a) Experimental Ce core-level X-ray photoelectron spectroscopy (XPS) spectrum of CeRuGa (blue points) and its Gunnarsson–Schönhammer (GS) modeling (orange line). The contributions and (with n = 0, 1, and 2) are presented in green and red, respectively. The estimated SO splitting is 18.6 eV. The components , , and are marked by solid, dashed and thick curves, respectively. The brown line represents the calculated background. Panel (b) shows Ce XPS spectrum of CeRuGa compared with the respective spectra of the similar intermediate valent compounds CeRuGe and CeRuAl. For each compound, two features located at 120 and 124 eV (marked by vertical dotted lines) signal mixed valence of Ce ions.

According to the Gunnarsson–Schönhammer (GS) model [32,33], the line arises due to the intermediate valence effect, whereas reflects the on-site hybridization strength, which is expressed by the energy [31]. It is possible to separate of the overlapping peaks on the basis of the Doniach–Šunjić theory [34], and can be estimated from the intensity ratio of the respective Ce XPS lines [33]. In turn, the intensity ratio gives an estimate for the shell mean occupation number [33]. The accuracy of determining and 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 CeRuGa, we found from the GS approach meV. In order to determine the ground-state 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 for different values of . Assuming and equal to wave function amplitude of the initial configuration state [33], we derived the fractional electron count .

The fractional valence of Ce ions in CeRuGa was further corroborated by inspection of the Ce XPS spectrum (see Figure 2b), which exhibits two lines near 120 and 124 eV, characteristic of the Ce states [33].

4. Calculated Electronic Structure

The atomic positions in the crystallographic unit cell of CeRuGa, 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 ).

Table 1

Relaxed atomic positions in CeRuGa. Calculations were performed assuming the experimental lattice parameters Å and Å.

Wyckoff	Atom		Coordinates
Position		x	y	z
2a	Ce1	0.000000	0.000000	0.141812
8d	Ce2	0.290732	0.000000	0.498706
8d	Ce3	0.287464	0.000000	0.871542
8d	Ru	0.347568	0.000000	0.203546
2a	Ga1	0.000000	0.000000	0.699153
8c	Ga2	0.217667	0.217667	0.178952

Table 2

Interatomic distances (Å) in the crystal structure of CeRuGa.

Ce1	Atom	Distance	Ce2	Atom	Distance	Ce3	Atom	Distance
	4Ga2	3.1355		Ru	2.4237		3Ru	2.8233
	Ga1	3.3769		Ce3	3.0114		Ru	2.9531
	4Ce3	3.4661		2Ru	3.0792		Ce2	3.0114
	4Ru	3.6321		2Ce3	3.1721		2Ce2	3.1721
	4Ce2	4.3802		2Ga2	3.2187		Ga1	3.2437
	Ga1	4.7443		Ga1	3.3633		2Ga2	3.2791
				2Ga2	3.5047		2Ga2	3.3838
				Ce2	4.1355		Ce1	3.4661
				2Ce2	4.2269		2Ce3	4.0909
				Ru	4.2978		Ce3	4.3277
				Ce1	4.3802		Ru	4.5993

The electronic bands in CeRuGa, calculated assuming the correlation energy 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 eV for the Ce1 atom and 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 eV . Figure 4a shows the spin-resolved TDOS calculated for the latter values of U compared with the valence band of of CeRuGa 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 states is distributed between 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 states are responsible for a broad and fairly weak feature near . At the binding energy of about 17 eV and 19 eV, the calculated Ce 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 -electron correlations, which usually shift the calculated Ce states to lower binding energies [26,37,38].

Figure 3

Total spin-resolved density of states in CeRuGa calculated for different correlation energy parameter U.

Figure 4

(a) Valence band XPS spectrum of CeRuGa (brown points) compared to the spin-resolved total density of states (blue line) calculated for eV and eV. (b) Total and partial DOS in CeRuGa, calculated as in panel (a), compared to the experimental XPS data.

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 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 CeRuGa equal to 0.86–0.89, in perfect agreement with the experimental result (see above). The calculated total magnetic moment amounts to about 1 , regardless of the value of U, whereas the magnetic moment found at the Ce2 and Ce3 sites is significantly smaller, namely, and .

Table 3

Results of the LSDA+U calculations performed for CeRuGa with different values of the correlation energy U, where stands for the number of electrons and m is the total magnetic moment per atom.

U (eV)		U_Ce1,Ce2,Ce3 = 1.5		U_Ce1,Ce2,Ce3 = 2.25		U_Ce1 = 3 U_Ce2,Ce3 = 2.25
N(E_F) (eV f.u.)^−1		31.32		31.32		34.26
Atom	n_f	m (μ_B)	n_f	m (μ_B)	n_f	m (μ_B)
Ce1	0.9833	0.9644	0.9880	0.9951	0.9873	1.0149
Ce2	0.8860	0.2589	0.8446	0.2808	0.8451	0.3099
Ce3	0.8732	0.4031	0.8309	0.4602	0.8423	0.5295
Ru		−0.0396		−0.0446		−0.0477
Ga1		0.0014		0.0023		0.0027
Ga2		0.0013		0.0024		0.0035

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 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 and Ru electron states for the Ce2 and Ce3 atoms, whereas the latter effect is negligibly small for the Ce1 atom (see Figure 5d–f).

Figure 5

Spin-resolved density of states in CeRuGa due to (a) Ce1,(b) Ce2, and (c) Ce3 atoms. The density of states (DOS) calculations were performed for the correlation parameters U = 3 eV for Ce1, and U = 2.25 eV for Ce1 and Ce2. The insets present in details the contributions. The right panels (d–f) show partial density of states in CeRuGa due to Ce and Ru electrons at Ce1 (d), Ce2 (e), and Ce3 (f) sites. The DOS calculations were performed for the correlation parameters U = 3 eV for Ce1 atom, and U = 2.25 eV for Ce2 and Ce3 atoms.

In order to visualize the intersite hybridization and the atomic bonds in the unit cell of CeRuGa, we calculated the charge densities (setting eV and 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 CeRuGa, evidenced in the spectroscopic and thermodynamic properties of the compound, can be associated primarily with the Ce2 atoms.

Figure 6

Electron densities (in (au)) visualized for the crystallographic plane (010) of the crystal structure of CeRuGa. The projection of the unit cell is outlined by gray lines.

5. Conclusions

The XPS experiment performed for CeRuGa confirmed the fractional valence of the Ce ions, noticed before in the 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 shell () was an average over those three sites. The DFT calculations allowed for inspecting the electron counts at each Ce atom. The results indicated that the Ce1 ion located at the site is trivalent ( is close to 1). In contrast, the Ce2 and Ce3 ions, placed at the sites, were found intermediate valent with notably smaller than 1. The ab initio calculated mean occupation of the shell in CeRuGa is 0.86–0.89, which is in very good agreement with the experimental finding.

The electronic instability of the 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 = 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 /atom), Ce2 (∼0.3 /atom), and Ce3 (∼0.5 /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 meV is for CeRuGa quite large, however, meV. One can also estimate the coupling constant meV [1] between the nearest Ce magnetic moments, which well correlates with low 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 CeRuGa an interesting material for further comprehensive experimental and theoretical investigations.