Yb₅Rh₆Sn₁₈: a valence fluctuating system with ultra-low thermal conductivity

Abstract

Yb₅Rh₆Sn₁₈ crystallizes with a unique structural arrangement [space group P4₂/nmc, a = 9.6997(4) Å, c = 13.7710(7) Å], which is related with primitive cubic Yb₃Rh₄Sn₁₃ and body-centered tetragonal (Sn_1−xTb_x)Tb₄Rh₆Sn₁₈ types. X-ray absorption spectroscopy showed that Yb atoms exhibit temperature-dependent valence fluctuations (VF) (i.e., intermediate valence state). Its complex mechanism is corroborated by the fact that the well-pronounced maximum in magnetic susceptibility can only be fairly described by the Bickers–Cox–Wilkins model developed for a J = 3/2 multiplet, atypical for Yb ions. Both Hall and Seebeck coefficients revealed a switch of the sign, indicating the change of charge carrier type from electrons to holes between 120 and 220 K. Both these effects together with electrical resistivity and theoretical DFT calculations confirm Yb₅Rh₆Sn₁₈ to be a metal, which disobeys the free electron gas theory. ‘Rattling’ motion of Sn1 atoms within the enlarged 16-vertices distorted Frank–Kasper polyhedra, concluded from the specific heat measurements, is argued to be the main reason for the appearance of a phonon resonance behavior, resulting in an ultra-low thermal conductivity in the studied stannide.

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1 Introduction

Intermetallics containing 4f rare-earth elements are known to tend towards magnetically ordered states defined by the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. Its strength depends on the number of 4f electrons (n_f) as well as their position relative to the Fermi level (E_F), which determines the degree of localization. The competing scenario is the formation of a non-magnetic Fermi liquid ground state commonly referred to as the Kondo effect. Obviously, the decisive role in such a competition is played by the level of hybridization of 4f electrons with the conduction band states. Frequently it results in some exciting features like heavy fermions, quantum criticality, non-Fermi liquid state, spin and valence fluctuations [i.e., intermediate valence state (IVS)], etc.^1–4

The latter effect is observed in intermetallics containing rare-earth ions, revealing both non-magnetic (e.g., Ce⁴⁺, Eu³⁺, and Yb²⁺) and magnetic (e.g., Ce³⁺, Eu²⁺, and Yb³⁺) states. In some cases, such an ion can switch between configurations, resulting in a non-integer valence that is usually temperature dependent.^5,6 All these indicate the change of the ground state of the ion and its degeneracy (N), which is reflected in well-pronounced maxima in magnetic susceptibility and specific heat capacity. To simulate them, some theories, based on the Anderson and Kondo impurity models accounting for the hybridization strength,^7,8 are normally applied. For instance, using a unified approach for the former one, which assumes a simplified concept of large angular momentum degeneracy, Bickers, Cox and Wilkins⁹ successfully simulated static properties. The Bethe ansatz, allowing determination of exact wave functions for quantum many-body models,¹⁰ is a powerful tool in the solution of both the abovementioned models. So, generalization of the Kondo impurity approach for an N-fold degenerate state and application of the Bethe ansatz allowed the solution of the SU (N) Coqblin–Schrieffer model, which described well the properties of such impurities in a metal as Ce³⁺ (N = 6) and Yb³⁺ (N = 8).^11,12 Furthermore, it was extended to a complete range of temperatures and fields.¹³ However, despite all these successes, none of the discussed models accounts for crystal electric field (CEF) multiplets splitting at non-zero temperatures. Being in the order of a few eV,¹⁴ the splitting energy influences strongly the magnetic susceptibility and specific heat capacity. There is an attempt of theoretical simulation of such an effect; however, it is done only for the ions with a degeneracy of N = 4.¹⁵ Among the known examples, where the theories fail in the description of temperature dependence of magnetic susceptibilities, we would like to mention YbCuAl,^16–18 YbB₁₂,¹⁹ YbNi₂Ge₂ and YbPd₂Si₂.²⁰

In the past decades, numerous compounds have been found to be isostructural to the primitive cubic Yb₃Rh₄Sn₁₃ structure type discovered by Remeika and co-workers.²¹ Furthermore, several intermetallics are shown to be its distorted variants with the same composition (such a structural relationship is normally explained within the group–subgroup scheme^22–27) or with a close composition.^28–30 The majority of the latter compounds crystallize with the body-centered tetragonal (Sn_1−xTb_x)Tb₄Rh₆Sn₁₈ type.²⁸ Interestingly, among stannides of this series, mainly those containing non-magnetic Sc, Y or Lu attract much attention, which is due to their exciting superconducting properties.^31–37 A further important observation is that such compounds are unknown with Ce, Eu or Yb, which could reveal VF. The only exceptions in this respect would be primitive tetragonal Yb_6.6Ir₆Sn₁₆,²⁹ face-centered cubic Yb₅Pt₆In₁₆Bi₂ ³⁸ and body-centered tetragonal Yb₅Ru₆Sn₁₈ ³⁹ revealing a mixed valence state (i.e., non-integer, but temperature independent) for Yb ions.

Stannides with primitive tetragonal (a ≈ 9.7 Å, c ≈ 13.7 Å), face-centered cubic (a ≈ 13.7 Å) and distorted body-centered cubic (a ≈ 19.4 Å) structures and compositions close to 3 : 4 : 13 in the Yb–Rh–Sn system were reported to exist by Espinosa.⁴⁰ However, in further studies,^41,42 this information has not been confirmed. Furthermore, detailed single-crystal diffraction studies performed on SnM₃Rh₄Sn₁₃ (M = La–Gd, Yb, Ca, Sr and Th) explained some earlier observed distortions by anomalous thermal displacements of Sn2-atoms and structural disorder.⁴³ Finally, a face-centered cubic phase with a unit cell parameter a ≈ 13.7 Å and the (Yb_xSn_1−x)Yb₄Rh₆Sn₁₈ composition (no refinements were provided) was reported to exist in ref. 44. For unclear reasons, the same work claims Yb atoms to be in an intermediate valence state based on measurements of magnetic susceptibility (not presented) and room-temperature X-ray absorption spectra at the YbL_III edge. Considering this information and taking into account unusual valence fluctuations reported for Yb₃Co₄Ge₁₃,^45,46 we decided to shed more light on the properties of a stannide with a composition close to the Yb₃Rh₄Sn₁₃,⁴⁷ recently studied.

In this work, we show that Yb₅Rh₆Sn₁₈ crystallizes with a unique structural arrangement, and investigate its magnetic, electrical and thermal transport properties. They reveal a complex temperature-dependent behavior of the Yb ion valence. The observed ‘rattling’ effect is found to result in a phonon resonance behaviour, resulting in ultra-low thermal conductivity.

2 Experimental

The Yb₅Rh₆Sn₁₈ sample of a total mass of 1 g was prepared in a glassy carbon crucible by high-frequency melting from stoichiometric amounts of ytterbium (Ames, 99.95 wt%), rhodium granules (ChemPur, 99.9 wt%), and tin foil (ChemPur, 99.995 wt%). The total weight losses were <0.5 wt%. To avoid oxygen and moisture contaminations, all sample handling and manipulations were performed inside a glove box system under a protective argon atmosphere [p(H₂O), p(O₂) < 1 ppm]. To perform heat treatment, the obtained button was placed in a glassy carbon crucible, sealed in a Ta-tube and enclosed in an evacuated silica ampoule. Further, it was annealed at 1070 K for 720 hours and finally quenched in cold water.

Powder X-ray diffraction (PXRD) was carried out on a Huber G670 Guinier camera (CuK_α1 radiation, λ = 1.54056 Å). Phase analysis was performed using the WinXpow software package,⁴⁸ and Rietveld refinement using WinCSD.⁴⁹ Single-crystal diffraction (SCXRD, two-circle diffractometer Stoe IPDS-2T, MoK_α1 radiation, λ = 0.71073 Å) was performed on a crystal mechanically extracted from the crushed sample. The obtained data were refined using the SHELXL-2019/3 program⁵⁰ within the WinGX package.⁵¹

The Yb₅Rh₆Sn₁₈ sample was embedded in a conductive resin and polished, and the surface was investigated with a scanning electron microscope (SEM) (Jeol JSM – 7800F) and its chemical composition Yb_5.0(1)Rh_5.9(1)Sn_18.1(1) (in good agreement with the nominal and refined ones) was determined by the mapping procedure with energy dispersive X-ray spectroscopy (EDXS) (Quantax 400 EDXS system, Bruker). Interestingly, in the mapped surface (an area of ≈600 × 600 μm), no traces of Yb₂O₃ were detected (Fig. S1†). This finding confirms, if any, only minor quantities of such an impurity in the studied sample.

The X-ray absorption spectra (XAS) near the YbL_III edge of Yb₅Rh₆Sn₁₈ were measured in fluorescence mode at 100 K and 293 K at the Rossendorf Beamline (BM20) of ESRF with the help of a Canberra photodiode.⁵² The spectra were recorded at a scattering angle of 90° in the horizontal plane.⁵³ The XAS spectra were recorded in the energy region from 8920 to 9000 eV with a step size of 0.5 eV. The intensity was normalized to the incident flux. The standard normalization routines were applied to the spectra with the help of the PyMCA program.⁵⁴

To investigate the temperature and field dependencies of magnetic susceptibility, specific heat capacity, electrical resistivity, Seebeck coefficient, thermal conductivity and Hall coefficients for Yb₅Rh₆Sn₁₈ VSM, ⁴He heat capacity, TTO and Van der Pauw options of a DynaCool-12 instrument (Quantum Design) were used.

The electronic band structure and density of states were calculated with density functional theory (DFT) [local density approximation (LDA)] using the full-potential FPLO code (version 18.00–52.38).⁵⁵ The exchange–correlation potential by Perdew and Wang⁵⁶ was used to perform the scalar relativistic calculation. The k-mesh included 8000 points in the first Brillouin zone.

3 Results and discussion

3.1 Crystal structure

The single-crystal XRD (SCXRD) data set of Yb₅Rh₆Sn₁₈ was collected from a crystalline shard of an approximate size of 0.01 × 0.02 × 0.03 mm³. The positions of the strongest reflections were in accordance with a cubic crystal system (cF, a ≈ 13.8 Å), but additional weak reflections indicated a superstructure (cP, a ≈ 13.8 Å or tP, a ≈ 9.7 Å, c ≈ 13.8 Å). As the intensities of symmetry-equivalent reflections did not adhere to cubic symmetry (shown in Fig. S2†), it was concluded that the diffraction pattern was produced by a three-domain twin of the tetragonal system with domains of noticeably different diffraction powers. [The twin domains are related to one another by 120° and 240° rotation about [0 1 2]; data of domains 2 and 3 are related to domain 1 according to (−0.5 0.5–0.5) (−0.5 0.5 0.5) (1 1 0) and (−0.5–0.5 0.5) (0.5 0.5 0.5) (−1 1 0), respectively.] For structure solution, the data set of the predominant domain was integrated, and the solution (intrinsic phasing method, ShelXT) was performed with the HKLF4 format data set obtained therefrom. For the final refinement, data integration was performed for all three domains to account for spot shape distortions of the overlapping reflections, and an HKLF5 format data set of all reflections of domain 1 (with overlaps of domains 2 and 3) was created. A numerical absorption correction (cylinder) was applied afterwards in WinGX. In the structure refinement, the population of the twin domains was refined to 0.635(6), 0.293(6), and 0.072(6) for domains 1, 2 and 3, respectively.

In agreement with the model obtained from the refinement of SCXRD data, all peaks observed in the PXRD pattern of Yb₅Rh₆Sn₁₈ could be indexed with the unit cell parameters presented in Table 1 (the difference in the given values obtained from SCXRD is due to diverse methods of their estimation and not due to a negative thermal expansion). The analysis of systematic absences (h + k = 2n, l = 2n and k = 2n for hk0, hhl, 00l and 0k0, respectively) confirmed the space group (SG) P4₂/nmc. Further refinement of atomic coordinates and thermal displacement parameters (Table 2 and 3) converged with low reliability factors (Table 1) confirming its correctness. The obtained profiles are depicted in Fig. 1. Noteworthily, the main peaks in the PXRD pattern of Yb₅Rh₆Sn₁₈ could be indexed within a primitive cubic SG Pm3̄m with a ≈ 9.8 Å, whereas some of their splittings could be indexed within a primitive tetragonal SG P4/mmm with unit cell parameters a ≈ c ≈ 9.8 Å (inset in Fig. 1). These observations suggest a structural relationship of the herein studied phase with the Yb₃Rh₄Sn₁₃ Remeika prototype.²¹ In this regard, we would like to refer to our previous discussions in ref. 23, 24 and 27.

Fig. 1 Powder X-ray diffraction patterns of Yb₅Rh₆Sn₁₈. Inset: selected range of PXRD patterns with possible indexing within cubic (SG Pm3̄m, a ≈ 9.8 Å), tetragonal (SG P4/mmm, a ≈ c ≈ 9.8 Å) and the model (SG P4₂/nmc) refined in this work.

Table 1 Crystallographic data collected for Yb₅Rh₆Sn₁₈ at different temperatures

a w = 1/σ^2(Fo)^2 + (0.0757p)^2 with p = (Fo^2 + 2Fc^2)/3.
Temperature, T (K)	200	293
Method	Single crystal	Powder
Space group (No.)	P42/nmc (137)
Formula per unit cell, Z	2
Unit cell parameter:
a (Å)	9.7018(5)	9.6997(4)
c (Å)	13.793(1)	13.7710(7)
V (Å^3)	1298.3(2)	1295.6(2)
Density, ρ (g cm^−3)	9.26(1)	9.3(1)
Crystal size (mm^3)	0.03 × 0.02 × 0.01	—
Abs. coef., μ (mm^−1)	38.4	203.3
F(000)	3040
Radiation	MoKα	CuKα1
Diffractometer	STOE IPDS 2T	Huber G670
2θmax (°)	58.0	105.5
(sin θ/λ)max	0.682	0.517
Minimum h, k, l	−13, −13, −18	0, 0, 0
Maximum h, k, l	13, 13, 18	7, 9, 14
Abs. correction	Numerical	—
Collected reflections	14 544	393
Independent reflections	961 (Rint = 0.070)	—
Refined parameters	45	31
Refl. with Ihkl ≥ 2σ(I)	920 (Rσ = 0.027)	—
Refinement on	F ^2	—
R/wR^a; RI/RP		0.056/0.121
(I ≥ 2σ(I))	0.042/0.108
(All data)	0.057/0.116	—
Goodness of fit	1.022	1.011
Min. and max. residual
Resid. electr. density (e^− Å^−3)	−2.99; +2.04	−1.12; +1.35

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Table 2 Atomic coordinates and thermal displacement parameters (U_eq/U_iso × 10⁻³ Ų) for Yb₅Rh₆Sn₁₈

Atom	Site	x	y	z	U eq/Uiso
200 K
Yb1	2b	3/4	1/4	1/4	21.0(4)
Yb2	8g	1/4	0.0255(1)	0.38263(8)	19.4(2)
Rh1	4c	3/4	1/4	0.5060(2)	15.8(6)
Rh2	8f	0.4953(1)	−x + 1	1/4	17.2(4)
Sn1	4d	1/4	1/4	0.0389(2)	27.5(1)
Sn2	8g	1/4	0.0999(1)	0.7388(2)	21.6(4)
Sn3	8g	1/4	0.5787(2)	0.1634(1)	19.6(3)
Sn4	16h	0.5758(2)	0.0743(1)	0.0737(1)	22.9(3)
293 K
Yb1	2b	3/4	1/4	1/4	29(6)
Yb2	8g	1/4	0.0229(9)	0.3826(7)	31(4)
Rh1	4c	1/4	1/4	0.5069(9)	26(9)
Rh2	8f	0.4970(9)	−x + 1	1/4	26(6)
Sn1	4d	1/4	1/4	0.0354(9)	39(7)
Sn2	8g	1/4	0.0991(9)	0.7404(9)	20(6)
Sn3	8g	1/4	0.5743(9)	0.1643(9)	29(5)
Sn4	16h	0.5737(9)	0.0744(9)	0.0739(9)	26(3)

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Table 3 Anisotropic thermal displacement parameters (U_ij × 10⁻³ Ų) for Yb₅Rh₆Sn₁₈

Atom	U 11	U 22	U 33	U 12	U 13	U 23
Yb1	20.2(7)	U 11	23(1)	0	0	0
Yb2	18.4(5)	19.9(5)	U 22	0	−0.1(4)	0
Rh1	14(1)	16(1)	18(2)	0	0	0
Rh2	16.9(6)	U 11	18(1)	−0.4(4)	U 12	0.0(8)
Sn1	24(1)	41(1)	18(1)	0	0	0
Sn2	15.5(9)	17.6(9)	31.6(9)	−0.9(6)	0	0
Sn3	16.4(7)	21.1(7)	21.5(8)	3.2(6)	0	0
Sn4	26.0(6)	23.1(6)	19.6(8)	1.9(4)	−0.4(4)	−8.9(5)

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Interatomic distances in the crystal structure of Yb₅Rh₆Sn₁₈ are presented in Table S2.† As for the majority of phases crystallizing in the Remeika type and derived arrangements,²⁷ there are no Yb–Yb and Rh–Rh bonding contacts (i.e., these distances exceed the limit, where r_Yb = 1.94 Å, r_Rh = 1.34 Å and r_Sn = 1.41 Å (ref. 57) are atomic radii of the elements) in the herein studied compound. Remarkable shortenings occur in Yb₅Rh₆Sn₁₈ for Yb2–2Rh2 (8.0%) and Yb2–1Sn1 (8.5%) distances, whereas the Rh–Sn and Sn–Sn contacts are either slightly shrunk (e.g., Rh1–4Sn4 by 4.0%) or even longer than the corresponding sum (e.g., Sn4–1Sn4 by −2.6%).

Having a stoichiometry 5 : 6 : 18, the compound studied here is expected to be structurally related to the series of M₅Rh₆Sn₁₈ (M = Y, Tb, Dy, Ho, Er, and Tm) stannides crystallizing with the (Sn_1−xTb_x)Tb₄Rh₆Sn₁₈ type (SG I4₁/acd, a_tetr ≈ 13.8 Å, c_tetr ≈ 27.5 Å),⁵⁸ which is considered to be a derivative of the Yb₃Rh₄Sn₁₃ Remeika prototype (SG Pm3̄m, a_cub ≈ 9.7 Å). Indeed, comparing the unit cell parameters from Table 1 with the abovementioned observations, clear relations become visible: and . Such a close relationship between UCPs is also reflected in the structural arrangements. As one can see from Fig. 2a, b and c, in all three types [for better visualization, we present Sc₅Rh₆Sn₁₈ in Fig. 2c, which is an ordered variant of the (Sn_1−xTb_x)Tb₄Rh₆Sn₁₈ type], an array of corner sharing [RhSn₆] trigonal prisms (tan) with [YbSn₁₂] cuboctahedra (gray) incorporated into the free space in-between occurs. In the case of the cubic Yb₃Rh₄Sn₁₃ prototype, trigonal faces of prisms and cuboctahedra are shared with those of [SnSn₁₂] icosahedra (cyan) (Fig. 2a), whereas for the tetragonally distorted 5 : 6 : 18 variants (Fig. 2b and c), the 16-vertices distorted Frank–Kasper [SnR₂Sn₁₄] polyhedra share these with only cuboctahedra. Additionally, the latter units in Sc₅Rh₆Sn₁₈ are distorted in a different way due to diverse distributions of M atoms therein (Fig. 2c), which results in the shifted positions of centering Sn atoms (black connected with red zigzag). On the other hand, in the structures of Yb₃Rh₄Sn₁₃ (Fig. 2a) and Yb₅Rh₆Sn₁₈ (Fig. 2b), the Sn atoms centering the similar voids are along a straight line (red).

Fig. 2 Packing of [RhSn₆] trigonal prisms (tan), [YbSn₁₂] cuboctahedra together with [SnSn₁₂] icosahedra (gray) and 16-vertices distorted Frank–Kasper [SnM₂Sn₁₄] (M = Sc and Yb) polyhedra (cyan) in the structures of Yb₃Rh₄Sn₁₃ (a), Yb₅Rh₆Sn₁₈ (b) and Sc₅Rh₆Sn₁₈ (c). [Yb₄] tetrahedra together with tetragonal fragments [Sn₄] within the icosahedra (d) and [M₂Sn₂] within the Frank–Kasper polyhedra (e and f) are given in red and thick blue lines, respectively. Yb and Sn atoms at the vertices of polyhedra are depicted as magenta and white balls, whereas Sn atoms centering them are depicted as black balls.

In Fig. 2d, e and f, we demonstrate the packing of [Yb₄] tetrahedra (red) together with tetrahedral fragments (marked with thick blue lines) within the [SnSn₁₂] icosahedra and 16-vertices distorted Frank–Kasper [SnM₂Sn₁₄] polyhedra (both units are given in cyan). These fragments consist of only Sn atoms in the case of Yb₃Rh₄Sn₁₃ (Fig. 2d), whereas for Yb₅Rh₆Sn₁₈ (Fig. 2e) and Sc₅Rh₆Sn₁₈ (Fig. 2f), they contain 2Sn and 2M. Again, being nearly equally oriented in Yb-containing stannides, they reveal different distributions in the Sc₅Rh₆Sn₁₈ type.

Similarly to Yb₅Rh₆Sn₁₈, Yb_6.6Ir₆Sn₁₆ crystallizes with the same space group, close unit cell parameters and even identical Wyckoff positions.²⁹ The only difference between these structural arrangements is the appearance of an 8g site occupied by Yb (≈40%) in the Ir-containing stannide instead of Sn1 at the 4d position (Table 2), which results in various compositions.

All these let us to conclude that Yb₅Rh₆Sn₁₈ crystallizes with a unique structural arrangement, which belongs to the family of derivatives of the Yb₃Rh₄Sn₁₃ Remeika prototype.

3.2 X-ray absorption spectroscopy

The X-ray absorption spectra (XAS) of Yb₅Rh₆Sn₁₈ measured at the YbL_III edge are presented in Fig. 3. They reveal two white lines, which are attributed to Yb-4f¹⁴ (≈8940.5 eV) and Yb-4f¹³ (≈8946 eV) configurations, respectively. Interestingly, the XAS measured here at room temperature are nearly identical to those reported for a (Yb_xSn_1−x)Yb₄Rh₆Sn₁₈ compound, which was believed to crystallize with the face-centered cubic structure (SG Fm3̄m, a = 13.74 Å).⁴⁴

Fig. 3 Temperature-dependent X-ray absorption spectra of Yb₅Rh₆Sn₁₈ at the YbL_III edge in comparison with the literature data for (Yb_xSn_1−x)Yb₄Rh₆Sn₁₈ ⁴⁴ and the theoretically calculated Yb-5d density of states.

A clear reduction of the intensity of the low-energy line (≈8940.5 eV) in the XAS of Yb₅Rh₆Sn₁₈ with increasing temperature is observed. This is an indication of the fluctuating valence [i.e. intermediate valence state (IVS)] of ytterbium. Indeed, by deconvoluting the measured spectra, we obtain Yb^2.59(1)+ and Yb^2.62(1)+ at 100 K and 293 K, respectively. Interestingly, similar valence evolution is reported for the Yb₃Co₄Ge₁₃ modulated Remeika phase⁴⁶ and YbPtGe₂: a multivalent charge-ordered system with an unusual spin pseudogap.⁵⁹ The IVS in Yb₅Rh₆Sn₁₈ is in contrast with the prevailing temperature-independent state of Yb-4f¹⁴ (i.e., Yb²⁺) observed in the weakly Pauli paramagnetic and superconducting Yb₃Rh₄Sn₁₃ compound with a close composition.^44,47

Furthermore, we compared the Yb-5d electronic density of states (DOS) obtained from the theoretical DFT calculations with the measured XAS. They are expected to reproduce the lower-energy feature in the spectra and therefore became shifted to ≈8940.5 eV. As one can see from Fig. 3, the Yb1-5d orbitals are mainly unoccupied (they are above the Fermi level E_F) and nicely coincide with the position of the corresponding absorption maximum. However, unexpectedly, the 5d states of the Yb2 atom are found to be partially occupied. Also, these orbitals contribute remarkably to the higher-energy absorption maximum at ≈8946 eV corresponding to the Yb-4f¹³ configuration, which would assume an underestimation of the Yb²⁺ fraction in the valences obtained from the deconvolution of XAS. Since the temperature dependence of the Yb2-5d DOS cannot be simulated, its influence on the IVS behavior in Yb₅Rh₆Sn₁₈ remains rather unclear.

3.3 Magnetic susceptibility

Temperature dependence of magnetic susceptibility [χ(T)] for Yb₅Rh₆Sn₁₈ is depicted in Fig. 4. It obeys the Curie–Weiss (CW) law for 180 K < T < 350 K (inset in Fig. 4). The effective magnetic moment μ_eff = 2.94(5)μ_B deduced from such a fit is approximately by a factor of 3/2 smaller than 4.54μ_B – a value theoretically calculated for the Yb-4f¹³ configuration. A similar situation was observed in YbPtGe₂, where the Yb atom in one of the crystallographic positions was shown to be non-magnetic (i.e., in the Yb-4f¹⁴ state),^59,60 which resulted in the reduction of μ_eff by a factor of . Also, the performed CW fit indicated a large negative Weiss temperature of Θ_P = −175(1) K, which can be explained by an enhanced characteristic energy (≈47 K) of valence fluctuations.

Fig. 4 Temperature dependence of magnetic susceptibility (μ₀H = 7 T) for Yb₅Rh₆Sn₁₈ together with the predictions of the Coqblin–Schrieffer (CS) and Bickers–Cox–Wilkins (BCW) models for the J = 3/2 multiplet. Insets: temperature dependence of the reciprocal magnetic susceptibility of Yb₅Rh₆Sn₁₈ together with the Curie–Weiss (CW) fit.

Below 180 K, the χ(T) of Yb₅Rh₆Sn₁₈ deviates from the CW law, overcomes a maximum centered at T^χ_max ≈ 47 K and finally reveals an upturn for T < 25 K (Fig. 4) due to a minor (invisible in XRD) paramagnetic impurity. Such a behavior of magnetic susceptibility is in line with the IVS observed in XAS. Noteworthily, T^χ_max varies in a broad range for Yb-containing intermetallics, revealing the IVS [e.g., 25 K for YbPd₂Si₂,²⁰ 30 K for YbCuAl,¹⁸ 150 K for Yb₃Co₄Ge₁₃,⁴⁶ 240 K for Yb₄Sb₃,⁶¹ 250 K for YbNi_0.8Al_4.2,⁶² 250 K for YbPtGe₂,⁵⁹ 400 K for YbNiAl₄,⁶³ 500 K for Yb₃Ge₅,⁶⁴etc.]. Such a behavior is expected to be universally described by a Coqblin–Schrieffer (CS) model proposed by Rajan for the whole Yb³⁺ multiplet J = 7/2 (i.e., degeneracy N = 2J + 1 = 8).¹³ However, since this model does not account for crystal electric field (CEF) effects, the low temperature lattice problem and the so-called Kondo limit (i.e., valence variation, 0.5 < n_f < 1), it frequently fails in the description of the IVS, as it is the case for e.g., YbPd₂Si₂,²⁰ YbCuAl,¹⁸ Yb₃Co₄Ge₁₃,⁴⁶etc.

A similar observation is made for the herein studied Yb₅Rh₆Sn₁₈, where a good description for T > 100 K is obtained with J = 3/2 (N = 4) and the characteristic spin-fluctuation temperature T₀ = 201 K. The situation can be somewhat improved while switching to the Anderson model developed for the same multiplet and n_f = 1 by Bickers, Cox and Wilkins (BCW)⁹ (Fig. 4). From such a fit, we obtained a similar characteristic temperature T₀ = 208 K. Using the Rajan prediction¹³ for the Sommerfeld coefficient of electronic specific heat, we calculated γ = 2Jπk_BN_A/6T₀ ≈ 63 mJ mol⁻¹ K⁻² (with k_B = 1.38 × 10⁻²³ J K⁻¹ and N_A = 6.022 × 10⁻²³ mol⁻¹ as Boltzmann and Avogadro constants, respectively) – a value that seems to be degeneracy-independent and would even be in fair agreement with the corresponding value obtained from c_p-measurements (see below). However, implying the obtained T₀ to be correct, we calculated a Kondo temperature: T_K = 2πT₀W_J(2J + 1)⁻¹ ≈ 191 K (where W_3/2 = 0.5843 is the Wilson number),⁶⁵ which is in complete disagreement with the measured electrical resistivity, revealing no signature of any Kondo-like behavior at this T_K.

In view of the complicated Yb valency situation (i.e., Yb2-5d states contribute to the Yb³⁺ fraction), it is not surprising that the classical CS and BCW models do not work well. Additionally, as it is shown in section 1.1 and Fig. S3 of the ESI,† the simplified two-level interconfigurational fluctuation model (ICF)⁶⁶ fails in the description of the valence fluctuations in Yb₅Rh₆Sn₁₈ as well, thus confirming the conclusion above.

3.4 Specific heat

The temperature dependence of specific heat capacity c_p(T) of Yb₅Rh₆Sn₁₈ reveals no anomalies for 3 K < T < 200 K (inset in Fig. 5) and reaches the Dulong–Petit limit c_p = 3nR ≈ 723 J mol⁻¹ K⁻¹ (R = 8.31 J K⁻¹ mol⁻¹ is the gas constant) at ≈183 K. However, it reveals an anomaly at T_N = 2.3 K (Fig. S4†). This is due to a minor (not detectable with XRD or EDX) Yb₂O₃ impurity ordering antiferromagnetically.⁶⁷ The influence of such an impurity on the physical properties of the similar Yb₃Pt₄Ge₁₃ is discussed in detail in ref. 22. Noteworthily, the cusp at the transition in the current case is by a factor of ≈3 smaller than that in ref. 22. Therefore, we further consider the specific heat of Yb₅Rh₆Sn₁₈ in the c_p/T(T²) presentation and assume it to follow an ansatz of c_p = γT + βT³ in a temperature range of ≈4.5–6.1 K (Fig. S4†). This would indicate γ ≈ 55(5) mJ mol⁻¹ K⁻² and Θ_D = 148(2) K – values that are very similar to those reported for non-magnetic Sc₅Rh₆Sn₁₈ ³⁴ and Lu₅Rh₆Sn₁₈.⁶⁸

Fig. 5 Specific heat for Yb₅Rh₆Sn₁₈ in the c_p/T³(T) presentation together with a fit to eqn (1) and Debye (Θ_D1 + Θ_D2), Einstein (Θ_E) and electronic (γ) contributions. Inset: specific heat for Yb₅Rh₆Sn₁₈.

For T > 6 K, the c_p(T) of Yb₅Rh₆Sn₁₈ deviates from the above given ansatz, which is confirmed by a well-pronounced maximum centered at ≈8 K observed for the c_p/T³(T) presentation (Fig. 5). It signals a contribution of low energy Einstein optical modes to the measured specific heat. One of the possible reasons of their appearance could be the so-called ‘rattling’ motion⁶⁹ of Sn1 atoms within the enlarged 16-vertices distorted Frank–Kasper polyhedra (Fig. 2b). Such an effect is discussed for some Remeika phases.²⁷ To describe it, a combined Debye–Einstein model is frequently used.^70,71 It assumes as a first approximation⁷² the separation of phononic contributions originating from different types of sublattices (i.e., covalently and ionically bonded) and is given as follows:

where the Debye contribution is given as:(with x = ħω/k_BT) and the Einstein one as:

By fitting the c_p(T) of Yb₅Rh₆Sn₁₈ to eqn (1), the characteristic Debye and Einstein temperatures and number of modes were found to be Θ_D1 = 300(2) K, Θ_D2 = 134(2) K, and Θ_E1 = 47(1) K and N_D1 = 59(1), N_D2 = 30(1), and N_E1 = 4(1), respectively. Comparing this result with the Yb₅Rh₆Sn₁₇Sn₁ (Sn1 should remain for a possible ‘rattling’) stoichiometry, N_D1 could be assumed to originate mainly from the covalently bonded Sn₁₇ framework (i.e., theoretically expected number of Debye modes is 51). N_D2 is then due to 5Yb and 6Rh atoms (theoretical value of 33) and finally N_E1 is from 1Sn1 in the enlarged [Sn1M₂Sn₁₄] polyhedron (should be 3). In total, the abovementioned stoichiometry indicates 87 possible modes, whereas our fit results in N_tot = N_Di + N_Ej = 93(1). The mismatches between the theoretical expectations and the experiment are not a rare case (here we would like to refer to ref. 47, 71, 73 and 74) and reflect the shortcomings of the simple model applied to the complex phononic spectra of compounds.

Interestingly, taking the thermal displacement parameter B_iso = 8π²U_iso = 3.1(1) Ų of the ‘rattling’ Sn1 atom refined from the powder XRD data at room temperature (Table 2) and implementing it into eqn (4), we obtained the Einstein temperature Θ^B_E = 55(1) K, which fairly agrees with the specific heat data. This is a rather unexpected result. As it has been shown for such well-established ‘rattling’ systems as filled skutterudites^75,76 and the recently studied Yb₃Rh₄Sn₁₃,⁴⁷ if the atomic masses of the ‘rattling’ systems are comparable with those of the atoms forming the framework, Θ_E and Θ^B_E differ strongly.

(where ħ = 1.055 × 10⁻³⁴ J s is a Dirac constant).

The fit to eqn (1) also resulted in an enhanced Sommerfeld coefficient of the electronic specific heat γ = 196(1) mJ mol⁻¹ K⁻², which is by a factor of ≈4 larger than the values observed from the classical ansatz as well as for the non-magnetic Sc₅Rh₆Sn₁₈ ³⁴ or Lu₅Rh₆Sn₁₈.⁶⁸ This is again due to the enhancement of low-temperature c_p(T) caused by the Yb₂O₃ impurity.

3.5 Electrical transport

The temperature dependence of electrical resistivity ρ(T) for Yb₅Rh₆Sn₁₈ is presented in Fig. 6. In contrast to that for Lu₅Rh₆Sn₁₈ – a phonon reference compound⁷⁷ – as well as Y₅Rh₆Sn₁₈,⁷⁸ Sc₅Rh₆Sn₁₈,^34,35 Y₅Ir₆Sn₁₈,³⁰ analogous intermetallics, etc., it increases with increasing temperature and is of the same order of magnitude as that of a metal.⁷⁹ Furthermore, in the temperature range of ≈4–12 K, ρ(T) of Yb₅Rh₆Sn₁₈ follows the Fermi-liquid behavior and fits well to ρ₀ + A_FLT², with ρ₀ = 61(1) μΩ cm and A_FL = 0.13(1) μΩ cm K⁻². However, this ρ(T) cannot be described by a Bloch–Grüneisen model (typical of metals)⁸⁰ in the whole temperature range (for more details, see the discussions in section 1.2 of the ESI†), which is most probably due to the change of the scattering mechanism at T ≈ 100 K. Similar temperature dependencies of electrical resistivity were reported for such unusual IVS as CePt₄Ge₁₂ ⁸¹ and YbPtGe₂.⁵⁹

Fig. 6 Temperature dependence of electrical resistivity ρ(T) for Yb₅Rh₆Sn₁₈ together with the fit to the Fermi-liquid model. Inset: temperature dependence of the Hall coefficient R_H(T) for Yb₅Rh₆Sn₁₈ in different magnetic fields. The red line is a guide for the eye only.

Taking the values obtained from the Fermi-liquid fit, we estimated the residual resistance ratio RRR = ρ₍₃₀₀₎/ρ₀ ≈ 9.3, which indicates a fair quality of the studied sample. From the quadratic term, Kadowaki–Woods ratios⁸² (R_KW = A_FL/γ²) of 4.2 × 10⁻⁵ μΩ cm (mol K mJ⁻¹)² [for γ ≈ 55(5) mJ mol⁻¹ K⁻²] and 3.4 × 10⁻⁶ μΩ cm (mol K mJ⁻¹)² [for γ ≈ 196(1) mJ mol⁻¹ K⁻²] were obtained. Both the obtained values differ strongly from the R_KW = 1.0 × 10⁻⁵ μΩ cm (mol K mJ⁻¹)² expected for a heavy fermion compound, which is in line with the IVS of Yb₅Rh₆Sn₁₈.

Temperature dependence of the Hall coefficient R_H(T) for Yb₅Rh₆Sn₁₈ in different magnetic fields is presented in the inset of Fig. 6. Being nearly field independent, R_H(T) mimics the behavior of the Seebeck coefficient (see below, Fig. 7), revealing a well-pronounced minimum at and changing the sign from negative to positive at T ≈ 120 K. This latter observation confirms the switching of the conduction mechanism from electron- to hole-like type. By applying the one-band model [R_H = (n|e|)⁻¹ and μ = R_H/ρ], we found that their charge carrier concentration n and mobility μ vary (with the exception of singularity at R_H = 0) in the narrow ranges of 7.6–9.4 × 10⁻²¹ cm⁻³ and 1.1–2.1 cm² V⁻¹ s⁻¹, respectively. Weak temperature dependence of both these characteristics and their order of magnitudes indicate Yb₅Rh₆Sn₁₈ to be a metal.⁷⁹ By further applying the free electron gas model, we calculated the Fermi energy E_F = (ħ²/2m)(3π²n)^2/3e⁻¹ = 1.52(9) eV and the DOS at the Fermi level D(E_F) = 3nV/2E_F = 10.9(4) states eV⁻¹ f.u.⁻¹. The latter value is strongly underestimated in comparison with the theoretically calculated one, which is in line with a complex electronic band structure of Yb₅Rh₆Sn₁₈ (see the discussion below).

Fig. 7 Temperature dependence of the Seebeck coefficient S(T) for Yb₅Rh₆Sn₁₈ on a logarithmic scale together with the fit to eqn (5). Inset: temperature dependence of the Seebeck coefficient S(T) for Yb₅Rh₆Sn₁₈ on a linear scale together with the linear fit as expected for a metallic system.

Temperature dependence of the Seebeck coefficient of thermopower S(T) for Yb₅Rh₆Sn₁₈ in logarithmic and linear scales is shown in Fig. 7 and the inset therein, respectively. It is negative in the LT range and decreases with increasing temperature, passing through a minimum centered at T^S_min ≈ 41 K (in fair agreement with T^χ_max). Furthermore, S(T) starts to increase steeply, becoming almost linear, and changes the sign to positive at T ≈ 220 K, which indicates the switch from n- to p-type conductivity.

Having two crystallographic positions for Yb atoms in the structure of Yb₅Rh₆Sn₁₈, we assumed that the conduction electrons scattered on two independent 4f quasiparticle bands. This prompted us to try to describe S(T) using the model proposed by Bando et al.:⁸³

withwhere Δ_N,W = ε_N,W − ε_F and Γ_N,W are the position relative to the Fermi level and half width of the band, respectively. Hence, the corresponding Lorentzian density of states is expressed as:Here, W = Q⁺z + Q⁻(1 − z) is the integrated weight approximated for the intermediate valence regime with Q⁺ = N (i.e., the degeneracy of Yb ions), Q⁻ = 1 and the number of holes in the 4f-shell is estimated as n_h = 1 − z.⁸⁴ The latter value can also be derived from:

The parameters obtained from the fit of S(T) of Yb₅Rh₆Sn₁₈ to eqn (5) and the theory related to it are presented in Table 4. Knowing that the Yb valence is given as ν = 3 − z, we can state that from these calculations, it is ≈2.9 for both bands if one assumes the whole J = 7/2 (N = 8) multiplet for Yb ions. These values strongly deviate from those observed experimentally from XAS and can become somewhat improved (i.e., to ≈2.8) while taking into consideration the J = 3/2 (N = 4) multiplet applied for the description of temperature dependence of magnetic susceptibility. Such an inconsistency can be explained by the fact that the applied model is much too simple, which stresses again the fact that the Yb valence fluctuation in the studied stannide is of a very complex mechanism. Interestingly, the Yb-valences obtained here are in good agreement with those deduced from the ICF fit to the magnetic susceptibility of Yb₅Rh₆Sn₁₈. With due caution, we thus conclude that the simplifications in both theories (i.e., electron scattering on two quasiparticle bands and valence fluctuations between two energetic levels, respectively) result in comparable values.

Table 4 Positions (Δ) and half-widths (Γ) of 4f quasiparticle bands in Yb₅Rh₆Sn₁₈ derived from the temperature dependence of the Seebeck coefficient and assuming different degeneracies (N) for Yb ions

i	Δ i (meV)	Γ i (meV)	z (N = 8)	z (N = 4)
N	−0.86(5)	7.30(5)	0.09(1)	0.15(1)
W	−0.02(1)	0.56(1)	0.10(1)	0.19(1)

---

In the temperature range of 150–250 K, the Seebeck coefficient of Yb₅Rh₆Sn₁₈ fits well to S(T) = A_ST + B (the inset in Fig. 7) with the carriers’ diffusion component A_S = π²k_B²/2eE_F = 0.15(1) μV K⁻² and the term due to the contribution of electronic interaction B = −32(1) μV K⁻².^85,86 Having these fit parameters, we calculated the Fermi energy E_F = 1.64(9) eV, which is in good agreement with the value obtained from Hall effect measurements.

3.6 Thermal conductivity

The temperature dependence of the thermal conductivity [κ_tot(T)] of Yb₅Rh₆Sn₁₈ is depicted in Fig. 8. It is very low and thus comparable with those of such well-established ‘rattling’ systems as intermetallic clathrates^87–89 and filled skutterudites.⁹⁰ To shed light on this behavior, we decomposed κ_tot(T) into its electronic (κ_el = [L(T)/ρ(T)]T) and phononic (κ_ph = κ_tot − κ_el) parts. Knowing that the Lorenz number L₀ = 2.44 × 10⁻⁸ W Ω K⁻² is temperature dependent, we corrected it as proposed in:⁹¹[in this equation, |S(T)| is given in μV K⁻¹ and L(T) is given in 10⁻⁸ W Ω K⁻²].

Fig. 8 Temperature dependence of thermal conductivity and its electronic κ_el and phononic κ_ph contributions for Yb₅Rh₆Sn₁₈ together with the fit to eqn (10). For T > 250 K, κ_tot(T) and κ_ph reveal an upturn following the ∝T³ law due to radiation heat losses. Inset: κ_ph together with the fit to eqn (10) on a logarithmic scale for the better visualization of a minimum at T_min ≈ 40 K.

As one can see from Fig. 8, the phononic contribution to the thermal conductivity of Yb₅Rh₆Sn₁₈ is larger than the electronic one, which is again similar to the situation in the abovementioned ‘rattling’ systems. Furthermore, the obtained κ_ph(T) reveals a well-pronounced minimum at T_min ≈ 40 K (inset in Fig. 8), which could be a signature of the so-called phonon resonance (PR).⁹² There are two known reasons for the appearance of such a phenomenon: (i) resonant interaction occurring between phonons and non-paramagnetic lattice point defects (reported for mixed halides^93,94 and recently for natural galena⁹⁵) and/or (ii) anticipated scattering of the phonons by the vibrations of a weakly bound cation incorporated inside an enlarged anionic void (i.e., ‘rattling’ effect).⁹⁶ The modified Debye–Callaway model⁹⁷ [eqn (10)] allows accounting for different scattering mechanisms of phonons (including PR) and to describe κ_ph(T).

In eqn (10), we used Θ_D = 148(2) K, the average sound velocity v_S = k_BΘ_D/h(6π²n_at)^1/3 = 1435(1) m s⁻¹ (with the concentration of atoms per unit cell volume n_at = 4.48 × 10²⁸ m⁻³) deduced from XRD refinement and x = ħω/k_BT. The total relaxation time τ_tot⁻¹ in the current case includes the following contributions: phonon scattering on (i) grain boundaries (τ_B⁻¹) and (ii) point defects (τ_PD⁻¹), (iii) normal (τ_N⁻¹) and (iv) umklapp (τ_U⁻¹) phonon processes and (v) τ_PR⁻¹ due to phonon resonance:

τ_tot⁻¹ = τ_B⁻¹ + τ_PD⁻¹ + τ_N⁻¹ + τ_U⁻¹ + τ_PR⁻¹ (11)

with

Parameters obtained from the fit to eqn (10) are presented in Table 5. Interestingly, the observed C₁ value is by 1–2 orders of magnitude smaller than those reported for the intermetallic ErPdSb⁹⁸ or semiconducting PbS⁹⁵ and In_1−y@_yIn₂S₄ ⁷⁴ thermoelectric materials. This means that phonon scattering on the grain boundaries in Yb₅Rh₆Sn₁₈ is rather weak. On the other hand, parameter C₂, which is due to the point defects, is (for the stannide studied here) by a factor of 10–100 larger than the analogues in the abovementioned compounds. Thus, this scattering process is a dominating one in the low temperature regime. For T > 50 K, the dominance of normal and umklapp phonon–phonon processes is observed, which is reflected in the remarkable C₃ and C₄ parameters. Such a clear separation of both contributions is frequently not the case, especially for intermetallic systems. For instance, the normal processes seem to be completely screened by the umklapp ones in ErPdSb⁹⁸ and as a result, no C₃ parameter is obtained from the fit to the Debye–Callaway model. In concluding this discussion, we emphasize that the C_1–4 parameters obtained for Yb₅Rh₆Sn₁₈ are quite conventional and do not provide any insights into the ultra-low values of κ(T). Herein, it should be also noted that the applied model does not distinguish between three- and four-phonon decay mechanisms in umklapp processes, and as is known, the latter can lead to extremely low κ(T), as it is the case for PbCuSbS₃.⁷³ Obviously, to shed more light on this problem, inelastic neutron scattering on single-crystalline Yb₅Rh₆Sn₁₈ combined with theoretical lattice dynamics studies would be required.

Table 5 Relaxation time constants (C_n) and the characteristic PR frequency (ω_PR) deduced from the fit to the modified Debye–Callaway model

τ ^−1	Equation^97	C n
τ B ^−1	v S/C1	4.0(1) × 10^−7 m
τ PD ^−1	C 2 x ^4 T ^4	4.0(1) × 10^5 s^−1 K^−4
τ N ^−1	C 3 x ^3 T ^4	1.5(1) × 10^3 s^−1 K^−4
τ U ^−1	C 4 x ^2 T ^2 e ^(−ΘD/3T)	8.7(1) × 10^5 s^−1 K^−2
τ PR ^−1	C 5	5.1(1) × 10^35 s^−3
	C 6	2.1(1)
	ω PR	5.8(1) THz

---

The possible occurrence of phonon resonance in Yb₅Rh₆Sn₁₈ could be the main reason for its ultra-low thermal conductivity. This phenomenon is usually described by a simple mechanical oscillator model given by eqn (12), where the half-width of the energy resonance curve is defined as Γ = ω_PRC₆/2 and a proportionality factor containing the concentration of oscillators is A = C₅/Γ.⁹² Since a resonance interaction with a relaxation time τ_PR⁻¹ is believed to occur between phonons and defects, the enhanced C₂ parameter would nicely agree with this expectation. One of the central questions for such systems is the nature of the resonator. In early studies of mixed halides, this role was ascribed to the impurity non-magnetic ions,^92–94 whereas in the cage compounds to ‘rattling’ atoms.⁹⁶ However, the concept of inelastic resonance associated with energy dissipation of isolated ‘rattling’ atoms has been recently challenged^99–101 by inelastic neutron scattering and lattice dynamics studies for clathrate- and skutterudite-based cage compounds. Since the value of the characteristic PR frequency ω_PR, deduced for Yb₅Rh₆Sn₁₈ from our fit, is comparable with that reported for the Sr₈Ga₁₆Ge₃₀ clathrate,⁹⁶ one could conclude on the same nature of the resonator (i.e., ‘rattling’ atoms) in both substances. Despite being less understood and its applicability to thermal properties of cage compounds is debated,^72,102–104 PR would clearly contribute to a drastic reduction in κ(T). For instance, as it has been shown in ref. 95, its appearance in PbS leads to a decrease of κ_ph(T) by ≈70% at a certain temperature. Thus, since the lattice contribution is a dominant one in the case of Yb₅Rh₆Sn₁₈, its suppression plays a central role in the minimization of κ_tot(T).

Due to their ultra-low thermal conductivity, clathrates and filled skutterudites are known to be promising thermoelectric (TE) materials, which is reflected in their enhanced dimensionless figure of merit zT = S²T/ρκ > 1.^87,90 In the case of Yb₅Rh₆Sn₁₈, the highest zT = 5 × 10⁻³ is observed at 40 K. The poor TE performance of this stannide is explained by the negligibly small Seebeck coefficient due to its predominantly metallic behavior.⁷⁹

3.7 Electronic band structure

The atomic- and orbital-resolved electronic densities of states (DOS) of Yb₅Rh₆Sn₁₈ are presented in Fig. 9a and b, respectively. They show similar features to those reported earlier for the Yb₃Rh₄Sn₁₃ Remeika prototype⁴⁷ and the idealized structural model of Sc₅Rh₆Sn₁₈.³⁵ These are: (i) a low-lying energy band in the ≈−10.8 – ≈−5.8 eV range formed mainly by Sn-5s states, followed by (ii) an energy gap of ≈1 eV and (iii) a broad valence band extending from ∼−4.5 eV to the Fermi level E_F dominated mainly by Rh-4d and Sn-5p electrons.

Fig. 9 (a) Total and atomic resolved electronic density of states (DOS) for Yb₅Rh₆Sn₁₈. Inset: DOS near the Fermi level E_F. (b) Orbital resolved DOSs. E_F is marked as a green dashed line.

As one can see from the inset in Fig. 9a and b, the majority of the states at E_F originate from Yb2-4f (≈41%), Sn-5p (≈29%), Yb1-4f (≈8%) and Rh-4d (≈9%) (the small admixture of Yb1,2-5d, Rh-5p, and Sn-5s together contributing ≈9% is not shown in the figures). Thus, in total, we obtain at the Fermi level a DOS of D(E_F) = 18.8 states eV⁻¹ f.u.⁻¹, which can be recalculated in a Sommerfeld coefficient of the electronic specific heat γ_theor = 44.5 mJ mol⁻¹ K⁻¹. This value would be in acceptable agreement with those approximately estimated from the experiment. Taking into account that four electronic bands, with a curvature revealing no similarities to a parabola (Fig. S6†), cross E_F, one cannot expect that the simple free electron gas model would satisfactorily work in the case of Yb₅Rh₆Sn₁₈. Indeed, as it has been shown above, the electrical conduction mechanisms are found here to be of a complex nature.

4 Conclusions

Yb₅Rh₆Sn₁₈ crystallizes with a unique primitive tetragonal structural arrangement, which is closely related with the Yb₃Rh₄Sn₁₃ Remeika and (Sn_1−xTb_x)Tb₄Rh₆Sn₁₈ types. Such a relationship is reflected in the unit cell parameters which are derived from a ≈ 9.5 Å as well as in the common [RhSn₆] corner sharing trigonal prismatic arrays with cuboctahedra incorporated into the free space in between. An important feature of the Yb₅Rh₆Sn₁₈ crystal structure is that the relatively small Sn1 atoms reside inside enlarged 16-vertices distorted Frank–Kasper polyhedra thus possessing potential for a ‘rattling’ motion. This expectation is confirmed by the fitting of the temperature dependence of specific heat capacity to the combined Debye–Einstein model. The Einstein temperature deduced from the fit is in good agreement with the value obtained from the thermal displacement parameter of the Sn1 atom refined from the powder XRD data.

The two white lines with temperature-dependent intensities in the X-ray absorption spectra (XAS) of Yb₅Rh₆Sn₁₈ indicate that the Yb ions are in the intermediate valence state (IVS) switching between the 4f¹³ (Yb³⁺) and 4f¹⁴ (Yb²⁺) configurations. The latter one was simulated by quantum mechanical density functional theory (DFT) calculations, which showed that the Yb2-5d states are unexpectedly strong, contributing to the Yb³⁺ white line. This finding can explain why the known theories fail to describe the IVS in the new stannide.

Indeed, the well-established Coqblin–Schrieffer model fails, whereas the Bickers–Cox–Wilkins one fairly describes the maximum in the temperature dependence of the magnetic susceptibility of Yb₅Rh₆Sn₁₈ only on the assumption of the J = 3/2 multiplet (degeneracy N = 4) which, however, is atypical for Yb atoms. Furthermore, the Kondo temperature of 191 K, obtained from such a fit, is much too low, which is not confirmed in further electrical transport studies.

Electrical resistivity, which increases with increasing temperature and fits well to the quadratic Fermi-liquid approach in the low-T regime, together with the observed small Seebeck coefficients as well as high charge carrier concentrations and their mobilities hints towards the metallic properties of Yb₅Rh₆Sn₁₈. However, application of the free electron gas model failed in the description of electrical transport of this stannide. This fact is further ascribed to its complex electronic structure revealing four bands at the Fermi level (E_F) as well as to the temperature-dependent domination of the type of charge carrier (i.e., electrons below ≈200 K and holes above).

A well-pronounced minimum in the T-dependent Seebeck coefficient is a feature indicating IVS. By applying a two-band theory, we found that the mean Yb valence is ν ≈ 2.8, if one assumes smaller degeneracy (N = 4), and it increases up to ≈2.9 for N = 8. Although both ν-values differ strongly from ν = 2.59 to 2.62 (deduced from XAS), the observed tendency is in line with those found for magnetic susceptibility (i.e., better description with models for smaller N).

Yb₅Rh₆Sn₁₈ shows an ultra-low thermal conductivity dominated by phonons in the low-T range. The lattice contribution shows also a clear minimum, which (at the level of this investigation) most likely can be attributed to the appearance of phonon resonance due to the ‘rattling’ motion of Sn1 atoms in the structure of this new stannide. Indeed, by applying the Debye–Callaway model, we deduced the characteristic resonance frequency [5.81 THz], which is comparable with the values for such cage compounds as intermetallic clathrates. Therefore, it is argued that the phonon resonance can explain why the thermal conductivity of Yb₅Rh₆Sn₁₈ becomes ultra-low.

Data availability

The data supporting this article are included as part of the ESI.†

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

The DynaCool-12 system was acquired within the DFG project 422219907. VL is grateful to DFG for financial support within the project 467257848. The authors thank H. Borrmann for performing the powder XRD analysis. We are also grateful to U. Burkhardt, S. Kostmann and P. Scheppan for the microstructural and EDX studies.

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Footnote

† Electronic supplementary information (ESI) available. CCDC 2383845. For ESI and crystallographic data in CIF or other electronic format see DOI: https://doi.org/10.1039/d4dt02759k

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