Relationship between specific heat, valence and effective magnetic moment of Sm ions in strongly correlated Sm compounds Open Access

Recent studies on strongly-correlated electron states emerging in cubic Sm-based compounds, e.g., SmOs₄Sb₁₂¹ and Sm Tr₂Al₂₀ (Tr: transition metal elements),^2–4 have revealed that these compounds form a unique class of heavy fermion (HF) systems. The resistivity of Sm Tr₂Al₂₀ shows a clear −log T dependence, suggesting the occurrence of Kondo effect.^2–4 The −log T dependence of the resistivity remains even in a La-diluted compound Sm_0.01La_0.99Ta₂Al₂₀. This finding provides clear evidence that it is caused by a local single-ion Kondo effect.⁵ Large values of the electronic specific heat coefficient γ, e.g., 820 mJ/mol K² for SmOs₄Sb₁₂ and 150, 720, 1000, and 3200 mJ/mol K² for Sm Tr₂Al₂₀ with Ti, V, Cr, and Ta, respectively, reflect the formation of heavy quasiparticles at low temperatures. Surprisingly, the −log T dependences of the resistivity and the enhanced γ values are insensitive to applied magnetic fields in many of the cubic Sm-based compounds. In addition, it has been found that Sm ions in these compounds are in strongly mixed valence states. The Sm valence $v$_Sm determined by x-ray absorption spectroscopy (XAS) in SmOs₄Sb₁₂ (⁠$v$_Sm = +2.84 at RT) shows a log T dependence, indicating that the valence change is caused by an unconventional Kondo effect (the associated Kondo temperature is 56 K).^6,7 The log T dependent behavior remains down to the Sm concentration x = 0.2 in (Sm_xLa_1−x)Os₄Sb₁₂. This finding suggests that the Kondo effect has a local single-site nature. Sm Tr₂Al₂₀ compounds also have intermediate valence states although they show no clear temperature dependence.^5,8 These behaviors, which are in marked contrast to those of Ce-based HF systems, suggest strongly that the formation of heavy quasiparticles in Sm-based compounds involves unconventional nonmagnetic mechanisms probably associated with Sm 4f-electron charge degrees of freedom.

The electronic level schemes of Sm²⁺ and Sm³⁺ ions in cubic point symmetry are schematically shown in Fig. 1 in the LS coupling regime. Sm²⁺ has a nonmagnetic ground state (the total angular momentum J = 0) and a magnetic first excited state (J = 1) at the excitation energy of ∼420 K, which is comparable with the thermal energy at RT and therefore leads to Van Vleck paramagnetism. Sm³⁺ has a magnetic ground state (J = 5/2), which splits into a Γ₇ doublet and a Γ₈ quartet due to the cubic crystalline-electric-field (CEF) effect. The Γ₇ state has a magnetic dipole and the Γ₈ state has a magnetic dipole, electric quadrupoles and magnetic octupoles.⁹ Therefore, not only charge degrees of freedom (the Sm-ion valence) but also multipole degrees of freedom are expected to play essential roles for the realization of the unconventional Kondo effect and the field-insensitive HF states in Sm-based compounds (the difference in the effective ionic radius between Sm³⁺ and Sm²⁺ suggests possible involvement of phonon degrees of freedom). In this paper, we compare some of the key physical parameters of the cubic Sm-based compounds to extract characteristic features of the Sm-based strongly-correlated electron states.

In Table I, we summarize some of the key physical parameters of cubic Sm-based intermetallic compounds, i.e., the lattice constant a, Néel temperature T_N or Curie temperature T_C of magnetic orderings, Weiss temperature θ_CW and the effective magnetic moment μ_eff along with the temperature range for the Curie-Weiss (CW) fitting, the electronic specific-heat coefficient γ, Sm valence $v$_Sm determined by XAS, CEF splitting energy, and CEF ground state (GS).^{1–8,11–57} For compounds that show magnetic orderings, γ values were estimated from C_4f/T vs T data (C_4f: 4 f electron contribution to the specific heat) at temperatures lower than 1/10 of their magnetic transition temperatures.

In Ce-based HF compounds, γ is inversely proportional to the Kondo temperature T_K and θ_CW provides a rough measure of T_K, i.e., T_K ∼|θ_CW/2|,⁵⁸ leading to the relation γ ∝|1/θ_CW|. At low temperatures (T < T_K), the magnetic susceptibility χ(T) flattens out into a largely-enhanced T-independent Pauli paramagnetic contribution χ_Pauli ∝ 1/T_K. One comparison of the relative enhancements of γ and χ_Pauli is the dimensionless Wilson ratio R_W (∝ χ_Pauli/γ), which is roughly unity (∼ 2) for many of the HF compounds.⁵⁹ This fact evidences the formation of Fermi liquid state at low temperatures with strongly mass-enhanced quasiparticles. In Sm-based HF compounds (see Table I), however, there seem to be no clear correlations expected from these relations among the physical quantities. In fact, Sm Tr₂Al₂₀, which have largely enhanced γ values, do not show any pronounced χ_Pauli contribution at low temperatures. Instead, the CW behaviors remain down to the ordering temperatures. One of the most anomalous points is the extremely small μ_eff (∼ 0.1 μ_B), which is quite rare in Ce-based HF materials. This fact suggests that the HF state in Sm-based materials has a mechanism that is markedly different from that of Ce-based materials.

Figure 2 displays a log-log plot of γ versus μ_eff of cubic Sm-based compounds. In the LS coupling regime, μ_eff of a Sm³⁺ ion is 0.845, 0.665, and 0.412 μ_B for free, Γ₈, and Γ₇ states, respectively. In the region of μ_eff > 0.4 μ_B, Sm Tr₂Zn₂₀, Sm Tr₂Cd₂₀, and filled skutterudites are located. 4 f electrons of Sm ions in these compounds are expected to be in well localized states (hybridization with conduction electrons are weak). In compounds with μ_eff > 0.845 μ_B, magnetic moments of d electrons of Tr ions probably contribute to the CW behaviors. Sm ions in most of the compounds with μ_eff < 0.4 μ_B have intermediate valence states. This fact suggests that, in these compounds, the Sm 4 f electrons are strongly hybridized with conduction electrons. All the Sm Tr₂Al₂₀ compounds belong to this group, and they tend to have large γ values (>10² mJ/K²mol). Most striking one is SmTa₂Al₂₀, which has γ = 3200 mJ/K²mol and μ_eff = 0.09 μ_B. Interestingly, as Fig. 2 displays, most of the compounds lie in the zone expressed by $γ=α/μeff2$ with α = 5 ∼ 2 × 10² mJ$μB2$/K²mol. Interpretation of this trend is not trivial.

Figure 3 displays a log-log plot of γ versus the average Sm valence $v$_Sm determined by XAS of cubic Sm-based compounds. Largely enhanced γ is observed in compounds for $v$_Sm slightly lower than 3+. This fact suggests that the Sm valence fluctuation is associated with the heavy quasiparticle formation. As shown in Fig. 1, Sm³⁺ ion has a degenerate ground state with several types of degrees of freedom, which can lead to strongly-correlated electron states via hybridization with conduction electrons. On the other hand, Sm²⁺ ion has a singlet ground state with no degrees of freedom. Therefore, the decrease in γ approaching $v$_Smto2 + is reasonable. In the ordinary Ce-based HF compounds, γ ≥ 10³ mJ/K²mol suggests that it is in the strongly correlated limit. In this limit, the mass-enhancement factor m^*/m and the shift in the 4 f-electron number Δn_f caused by the Kondo effect are expected to satisfy m^*/m = (1 + Δn_f)/(2Δn_f).⁶⁰ If this approximate formula is tentatively applied to the Sm-based compounds, $v$_Sm ∼ + 2.8 (Δn_f ∼ 0.2) leads to m^*/m ∼ 3, which is a couple of orders of magnitude smaller than the observed values. This fact clearly demonstrates that the HF states in the Sm-based compounds are quite extraordinary.

As shown above, the Sm-based HF states have the remarkable characteristics, which are quite different from the Ce-based ones. In Sm-based systems, valence fluctuations occur between Sm³⁺(4f⁵) and Sm²⁺(4f⁶) states, while they occur between Ce³⁺(4f¹) and Ce⁴⁺(4f⁰) states in the Ce-based ones. It seems that involvement of multiple 4 f elections of a Sm ion in the formation of HF states plays a key role for the above-mentioned characteristics.

A theoretical model has been proposed recently by Siina using a two-orbital impurity Anderson model.^61,62 This model suggests the occurrence of a crossover between two extremal ground states, i.e., a local many-body singlet phase and a global Kondo singlet phase. Around the crossover, there appears a HF state, where Sm ions have an intermediate valence and R_W becomes zero.⁶² The crossover may correspond to $v$_Sm ∼ 2.8 in Fig. 3. However, it seems that the two characteristic features, i.e., the absence of χ_Pauli at low temperatures and extremely small μ_eff (∼ 0.1 μ_B) cannot be accounted for in this model.

In summary, we have compared the key physical parameters of cubic Sm-based intermetallic compounds. The results revealed that the Sm-based heavy fermion states have some significant characteristics, which are quite different from the Ce-based ones. The main remarkable features are the following; (i) there is an inverse relation between the electronic specific heat coefficient γ and the effective magnetic moment μ_eff (⁠$γ=α/μeff2$ with α = 5 ∼ 2 × 10² mJ$μB2$/K²mol), and (ii) largely enhanced γ appears in the intermediate Sm valance states around $v$_Sm ∼ +2.8. Other remarkable features are the absence of enhanced Pauli paramagnetism at low temperatures and largely suppressed μ_eff (down to ∼ 0.1 μ_B). These are quite different from those of Ce-based heavy-fermion compounds, pointing to an unconventional mechanism for the formation of Sm-based heavy quasiparticles, possibly caused by the involvement of multiple 4 f elections of a Sm ion in the hybridization process.

We gratefully appreciate Prof. Ryousuke Shiina for fruitful discussions. This work was financially supported by JSPS KAKENHI Grant Numbers 15H03693, 15H05884, 15J07600, and 15K05178.