Single crystals of (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} were characterized by magnetization, specific heat, and electrical resistivity measurements. Doping Yb into the non-magnetic Lu_{3}Rh_{4}Ge_{13} compound tunes this cubic system’s properties from a superconductor with disordered metal normal state (x < 0.05) to a Kondo for 0.05 ≤ x ≤0.2 and intermediate valence at the highest Yb concentrations. The evidence for intermediate Yb valence comes from a broad maximum in the magnetic susceptibility and X-ray photoelectron spectroscopy. Furthermore, the resistivity displays a local maximum at finite temperatures at intermediate compositions x, followed by apparent metallic behavior closest to the Yb end compound in the series.

The hybridization of transition metal or rare earth localized *d* or *f*-electrons with conduction electrons often gives rise to exotic electronic properties such as intermediate valence (IV) states, Kondo insulator, pseudogap, superconductivity, heavy fermion (HF), and quantum criticality.^{1–5} Intermediate valence states generally form with electronically unstable rare earths like Ce, Yb, Eu, and Sm.^{1,6–8} The *f* electron levels in such systems are close to the s-d band favoring an intermediate valence state. Hirst^{9} first pointed out that rare earth ions in this state fluctuate between two 4*f* electronic configurations, one with n electrons and the other with n − 1 electrons, competing for stability.^{10} In Yb based compounds, these are the nearly degenerate 4*f*^{13} (magnetic) and 4*f*^{14} (non-magnetic) states, a necessary condition for the IV state. This happens when the difference in Fermi energy E_{F} and 4*f* energy level E_{f} is comparable to the width of the hybridized 4*f* level.

By the application of pressure or doping, the Coulomb interaction (charge fluctuations) and hybridization strength of *f*-electrons with the conduction electrons change, and hence, different behaviors can be induced. Two competing energy scales, the Ruderman-Kittel-Kasuya-Yosida (RKKY) *T _{RKKY}* ∝ (

*JN*(

*E*))

_{F}^{2}and the Kondo energy

*T*∝

_{K}*e*

^{−1/JN(EF)}(where J is the spin coupling constant and N(

*E*) is the density of states at the Fermi level), give rise to either Kondo, HF, or HF superconducting behavior when

_{F}*T*≫

_{K}*T*, or magnetic when

_{RKKY}*T*≫

_{RKKY}*T*.

_{K}^{12–16}A crossover from Kondo lattice or HF to an intermediate valence state has been observed in a number of Kondo lattice or HF compounds, such as doped CeCu

_{2}Si

_{2},

^{17}CePtSi,

^{18}or under pressure in Y bCu

_{4}Ag,

^{19}CeCu

_{6},

^{20}and CeBe

_{13}.

^{21}

In this study, a continuous solid solution (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} displays a continuous evolution from superconductivity in a disordered metal close to x = 0 (Ref. 5) to Kondo to intermediate valence regimes for low (x ≤ 0.3) and high (0.3 ≤ x ≤ 1) Yb compositions.

Single crystals of (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} (x = 0, 0.05, 0.1, 0.15, 0.2, 0.4, 0.6, 0.8, and 1) were prepared using a self-flux method as described elsewhere.^{5} Room temperature powder X-ray diffraction data were collected in a Rigaku D/Max X-ray diffractometer using Cu K*α* radiation with wavelength *λ* = 1.5406 Å. Rietveld analysis was performed using the GSAS software package.^{22} *DC* magnetic susceptibility was measured in a Quantum Design (QD) Magnetic Properties Measurement System (MPMS) with a magnetic field *H* ∥ a. Specific heat measurements were performed in a QD Physical Properties Measurement System (PPMS) with a ^{3}He insert using a thermal relaxation method. The temperature-dependent *AC* resistivity of bar-shaped crystals was collected in the QD PPMS, with i ∥ a, *i* = 1 mA, *f* = 17.77 Hz. Room temperature X-ray photoelectron spectroscopy (XPS) spectra were taken in the PHI Quantera XPS, Physical Electronics instrument with a X-ray source Al anode at 25 W with a pass energy of 26.00 eV, 45° take-off angle, and a 100 *μ*m beam size.

(Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} (*x* = 0, 0.05, 0.1, 0.15, 0.2, 0.4, 0.6, 0.8, and 1) single crystals form with a cubic structure $Pm 3 \u0304 n$. Fig. 1 displays the *x* = 1 powder X-ray diffraction pattern, with the inset showing the shift of the (320) and (321) peaks with x. The change of the unit cell volume with x (Fig. 1(b), full symbols, left-axis) is consistent with Vegard’s law, as the unit cell expands for bigger Yb substituting for Lu ions. However, the unit cell increase is faster at first (for x < 0.2) when only Yb^{+2} replaces some of the Lu^{3+} ions. For increasing Yb amounts (0.2 ≤ x ≤ 1) a mixture of Yb^{+2} and Yb^{+3} (with the radius r for the trivalent ion being smaller than that of the divalent one) in place of Lu^{3+} results in a lower increase in a. The R–R bond distances d_{R−R} calculated from the Rietveld refinements (Fig. 1(b), open symbols, right-axis) further support the observed peak shift and hence the unit cell expansion. The inset in Fig. 1(b) displays the cubic crystal structure (space group $Pm 3 \u0304 n$) consisting of corner-sharing Ge(2) (green) trigonal prisms with Rh ions (blue) at the center, while the Ge(1) ions (pink) form a FCC structure, with two R ions (orange) on each face of the cube.

The (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} magnetic susceptibility M(T)/H measured in a field H = 0.1 T (Fig. 2) is dominated by the Pauli contribution for 0.05 ≤ x ≤ 0.15 (Fig. 2(a)), with small low temperature increase due to the small amounts of magnetic Yb^{3+} ion contribution. When no Yb ions are present (x = 0, inset Fig. 2(a)), no magnetic contribution is registered; instead, a superconducting state is present below ∼2 K (H = 5 Oe), which seems to rapidly disappear with x, and is no longer observed for x = 0.05. However, as x increases to around x = 0.20 (Fig. 2(b)), the susceptibility is nearly Curie-Weiss, as H/M (right axis) becomes linear for most of the measured temperature range. A fit of H/M above ∼20 K yields an effective moment p_{eff} ≈ 2.5, much smaller then the theoretical value for Yb^{3+} p$ eff theory =4.54$. Upon further increasing x, a broad maximum develops at high temperatures on top of the Curie-Weiss contribution (Figs. 2(c) and 2(d)), indicative of fluctuating valence behavior.^{23,24} The observed trend in the magnetization is consistent with the room temperature Yb valence changing from 2 + for x = 0 to mixed 2 + and 3 + for x = 1, while the broad maximum for Y b_{3}Rh_{4}Ge_{13} (Fig. 2(d)) is associated with temperature valence fluctuations in this series. Similar magnetization behavior was observed in Ce(Pt_{1−x}Ir_{x})_{2}Si_{2}, where the Ce valence changed with Pt-Ir substitution.^{25} The Y b_{3}Rh_{4}Ge_{13} susceptibility can be fit to the interconfiguration fluctuation (ICF) model,^{8,10}

where C and *θ* are the Curie constant and the Weiss temperature associated with a fraction y of magnetic Yb^{3+} ions with p_{eff} = 4.54, E_{ex} is the energy separation between the two 4f^{n} and 4f^{n−1} levels (with n = 14 for Yb), and T_{sf} is a characteristic spin fluctuation temperature related to the rate of valence fluctuation between n and n − 1. The ICF model fit (dashed line in Fig. 2(d)) corresponds to a small (y = 0.03) fraction of trivalent Yb ions, with a non-magnetic contribution χ(0) ∼ 10^{−4} emu/mol_{Yb}, Weiss temperature *θ* = 50 K, Curie constant C = 0.08 emu K/mol_{Yb}, and E_{ex} = 810(1) K and T_{sf} = 208(1) K. It should be noted that the Curie constant C scaled to the number y of Yb^{3+} becomes C = 2.67 emu K/mol_{Yb3+}, which is equivalent to an effective moment *μ _{eff}* ∼ 4.6

*μ*very close to the theoretical value for Yb

_{B}^{3+}of 4.54

*μ*. All of the parameters obtained from this fit are comparable with those for other intermediate valence Yb systems, such as YbCuGa,

_{B}^{8}Y b(Cu

_{y}Si

_{1−y})

_{2−x},

^{26}and YbPtGe

_{2}.

^{27}

The susceptibility behavior with x is also reflected in the composition dependence of the low temperature (T = 1.8 K) magnetization isotherms M(H) shown in Fig. 3(a): the initial Yb doping in Lu_{3}Rh_{4}Ge_{13} results in an increase in the overall M(H) up to x = 0.15, followed by a rapid decrease with higher Yb amounts. However, as the values at the highest measured field H = 7 T reflect (Fig. 3(b)), the magnetization reaches only 0.2 *μ _{B}*/ Yb, compared to

*μ*= 4

_{sat}*μ*/

_{B}*Yb*

^{3+}. The moment is quenched because of Yb valence fluctuations.

^{24}Thus, the moment is comparable to the moment for either intermediate valence or Kondo lattice systems.

^{28–30}Above

*x*= 0.15, the M(1.8 K;7 T) scales almost linearly with the Yb concentration, a behavior commonly seen in the valence change or Kondo lattice systems where this behavior emerges upon replacing the magnetic rare earth by a non-magnetic rare earth (e.g., Y b

_{x}Y

_{1−x}CuAl.

^{2,31})

The H = 0 specific heat is plotted in Fig. 4 in the form C_{p}/T *vs.* T^{2}. For Lu_{3}Rh_{4}Ge_{13}, the curve in Fig. 4 corresponds to H = 3 T, since this field value was needed to suppress the superconducting temperature to below 0.4 K.^{5} The electronic specific heat coefficient *γ*, estimated at the lowest measured temperature (T = 0.4 K) as *γ* = C_{p}/T, increases for x ≤ 0.15 and decreases for higher x, following same trend as M(7 T). The maximum *γ*(0.4 K) = 126 mJ/mol_{R}K^{2} for *x* = 0.15 is likely an underestimate, given the nearly divergent C_{p}/*T* at the lowest measured temperatures. Kasuya and Saso^{11} established that materials at the boundary between the Kondo and intermediate valence regime could have *γ* ≈ 100 mJ/mol_{R}K^{2}, confirming that the *x* = 0.15 and 0.2 (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} compounds are in this crossover regime, with enhanced mass electrons.

This observation is further supported by the H = 0 electrical resistivity shown in Fig. 5. The x = 0 superconducting state below T = 2.1 K (Ref. 5) is preceded by a semiconducting-*like* resistivity, shown to be due to the crystallographic disorder and enhanced atomic displacement parameter ratios. With Yb doping, the superconducting state is suppressed below 0.4 K, and the low T resistivity increases up to three times from the normal state *ρ* for x = 0. A broad maximum is observed for *x* = 0.1–0.4 (Figs. 5(c)-5(f)), followed by a ln*T* behavior at higher temperatures (dashed line, Figs. 5(k)-5(n)). Such trends in resistivity are anticipated when the Kondo effect coexists with the crystalline electric fields (CEFs).^{32,33} The Kondo temperature *T _{K}* is determined from the specific heat coefficient

*γ*using

^{26}

As is shown below, T_{K} increases with increasing of *x* for 0.1 ≤ x ≤ 0.3, due to the change of the Yb valence, similar to the case of Ce_{1−x}Y _{x}Al_{2}.^{34} At *x* = 0.6, the peak shifts close to 300 K, as in Y bRh_{2}Ga and Y bAl_{3}.^{7,35} For *x* ≥ 0.8, the system becomes metallic, however with large resistivity values ρ(300 K) 1 mΩ cm (Figs. 5(h) and 5(i) and small residual resistivity ratio RRR = ρ(300 K)/ρ(0.4 K), RRR ∼ 1.5. A possible explanation could come from the crystallographic disorder persisting throughout the whole (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} series. The intermediate valence systems can have *T _{K}* up to 2000 K, much larger than CEF splitting and comparable to spin-orbit coupling.

^{11}It is therefore plausible that the Kondo peak exists above 300 K for

*x*= 0.8 and 1.0, which would provide an alternative explanation for the large resistivity values in these samples. However, the normalized resistivity ρ(

*T*)/ρ(300 K) (Fig. 5(j)) decreases systematically with increasing x in the range

*x*= 0.1–1. Such behavior could be consistent with rare earth valence fluctuations, or a structural phase transition, or both.

^{36}The lack of any anomalies in the magnetization render the structural transition scenario unlikely, but it remains to be unambiguously verified with temperature-dependent X-ray diffraction measurements. Systematic changes in the residual resistivity as function of pressure, similar to the trend induced by doping in (Lu

_{1−x}Y b

_{x})

_{3}Rh

_{4}Ge

_{13}(Fig. 5(j)), were also reported in other compounds, which also showed a Yb valence transition and Kondo behavior.

^{36,37}

Indirectly, intermediate valence behavior can be inferred from lattice parameters, transport properties, susceptibility, and Mössbauer spectra. However, fluctuations in the electronic configuration can be probed directly using XPS. The valence fluctuation time of most intermediate valence systems falls in a range between the measuring time of Mössbauer spectroscopy (10^{−11} s) and XPS (10^{−17} s).^{38} Therefore, XPS can capture both valence states in the valence-fluctuating compounds, while Mössbauer spectroscopy yields an average valence state in most cases.^{38} Room temperature XPS measurements on (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} (Fig. 6) show the Yb^{+2} peak at 181.4 eV for all compositions. The Yb^{+3} at 185.3 eV starts developing for x ≥ 0.2. Both states of Yb were also observed in the XPS study of other intermediate valence systems, such as Y bInCu_{4}^{39} or YbGaGe.^{40} The indirect observations of Yb valence changes and intermediate valence from magnetic susceptibility and resistivity measurements are confirmed directly by XPS.

As the composition changes, the lattice constant also changes. This lattice constant change is sensitive to the hybridization.^{31} The hybridization parameter Δ is related to the Kondo temperature T_{K} through

where *D* is an effective bandwidth, ε_{f} is the binding energy of *f* electrons or holes, and N_{f} is the degeneracy of the *f* electrons. As can be seen in the Fig. 1(b), the lattice parameters and the unit cell volume change monotonously as the Yb concentration increases, implying that the Δ is also changing. This is consistent with the T_{K} (open circles) rapid increase with x, given that a small change in the hybridization parameter Δ would result, according to Eq. (2), in a substantive change in T_{K}.

For *x* = 0.15 and 0.2, the rapid increase in *M*(*T*)/*H* (Fig. 2) and *γ*(*T*) (Fig. 4) at low temperatures points to the strong hybridization of the 4*f* states.^{42} The Lu_{3}Rh_{4}Ge_{13} superconductivity disappears in the doped samples. The suppression in the superconducting transition temperature can be either due to changing conduction electron density or the exchange interaction between the conduction electrons and *f*-shell spins.^{41} Since 5% Yb doping in Lu_{3}Rh_{4}Ge_{13} suppresses T_{c}, the drastic change in conduction electron can be eliminated. Here, superconductivity may disappear with a small amount (*x* = 0.05) of magnetic Yb ion due to the role of the exchange interaction of spin of the conduction electrons with the spin of magnetic Yb atoms.

In summary, the (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} series displays a disordered metal state for x < 0.05, with a superconducting state at very low temperatures (T_{c} < 2 K). In the Kondo regime (0.05 ≤ x ≤0.2), both the electronic specific heat coefficient *γ* and the magnetization M[7 T;1.8 K] are maximum around x = 0.15. It should be noted that the determination of *γ* (Fig. 4) is ambiguous for x between 0.05 and 0.1, and therefore, this region is represented by the dashed area in the phase diagram (Fig. 7). The XPS measurements (Fig. 6) together with the enhanced *γ* and very small saturated moment values provide evidence for the valence fluctuations in (Lu_{1−x}Y b_{x})_{3}Rh_{4}Ge_{13} above x = 0.05.

This work was supported by the Robert A. Welch Foundation. The authors thank Jiakui K. Wang, E. Svanidze, J. Santiago, and A. Subramanian for useful discussions and critical reading of the manuscript.

## REFERENCES

_{3}T

_{4}Ge

_{13−x}(T = Co, Rh, Os) and Y

_{3}T

_{4}Ge

_{13−x}(T = Ir, Rh, Os)