Introduction

Studied for more than 50-years, the compound SmB61,2,3,4 has recently become a strong candidate for a 3-dimensional topological Kondo insulator (TKI) with a robust bulk insulating gap5,6,7,8,9,10,11,12,13. The insulating gap in SmB6 is supposed to be created via the Kondo hybridization of localized Sm-4f and itinerant Sm-5d electrons, with the Fermi level residing in the hybridization gap. This situation is in contrast to that of a conventional band insulator13. The energy value of the hybridization gap depends on the experimental probe being approximately 20 meV (=232 K). However, some experiments point to the existence of in-(hybridization)gap states14,15,16,17 due to the observation of some lower-energy excitations. A 14-meV bulk collective mode observed within the hybridization gap by inelastic neutron scattering (INS) by Alekseev et al.17 has been recently interpreted as a spin exciton18,19.

Difficulty in understanding electronic and magnetic properties has led to a conclusion regarding SmB6 as a mixed-valence system in which the Sm ions rapidly fluctuate between non-magnetic Sm2+ (4f 6) and magnetic Sm3+ (4f 5 5d1) electronic configurations, resulting in an average intermediate valence of 2.5–2.7. This valence can additionally change with temperature20. Recently, the temperature dependence of the specific heat c(T) of SmB6 was re-measured in two different laboratories on very good quality single crystals: in the USA21 and in Slovakia22.

In this contribution, we analyse the temperature dependence of the specific heat c(T) of SmB6 in a wide temperature range from 2 to 300 K, presenting, for the first time, a consistent explanation for the c(T) of SmB6 and the samarium contribution cSm(T) to its specific heat.

Results and Discussion

The experimentally derived specific heat of SmB6 exhibits a large extra heat compared with that of isostructural, nonmagnetic LaB6; see Fig. 1, which was redrawn from refs21,22. This excess heat in SmB6 is enormous because the entropy related to this excess, up to 300 K, amounts to 19–23 J/K mol f.u.21. It corresponds to 2.3–2.9 R (the gas constant R = 8.314 J/(K mol f.u.)). This extra entropy would point to a number of involved localized energy states from 10 to 16. These are large numbers totally not expected for the Sm2+ and Sm3+ ions. This problem was noticed 50 years ago1,2,3,4, and it is still under debate. In 2014, Phelan et al.21 wrote that “Some of the excess entropy will be due to the localization of the conduction electrons when the hybridization gap forms, but more likely explanation for the observed large value of ΔS is an additional phonon (lattice) contribution.” This additional phonon (lattice) contribution was not specified. In their recent 2017-year paper22, Orendac et al. mentioned only the existence of undefined in-(hybridization)gap states.

Figure 1
figure 1

Experimental temperature dependence of the specific heat of SmB6 and LaB6, read from refs21,22, showing a very large excess specific heat of SmB6 compared to that of LaB6.

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As has been restated by Sudermann et al.23, the CEF level scheme of the Sm3+ ion, which is a quantum system 4f5 with L = 3 and S = 5/2, in SmB6 is quite similar to that of the Ce3+ (4f1) ion, L = 3 and S = 1/2, in the isostructural CeB624. In both cases, the spin-orbit interaction splits the 4f states into J = 5/2 and J = 7/2 multiplets. The J = 5/2 multiplet is further split by the cubic crystalline field into a Γ7 doublet and a ground-state Γ8 quartet. The energy difference between the Γ8 quartet and the excited Γ7 doublet is suggested to be approximately 15 meV for Sm3+ in SmB6 and 46 meV in CeB6, as shown in Fig. 1 of ref.23. The fine electronic structure for the Sm2+ (4f6) ion is even simpler. It is governed by the spin-orbit interactions, customarily assumed by a value of the spin-orbit coupling λs−o of 35 meV (=420 K), and due to the compensation of the orbital and spin momenta, the resultant ground multiplet is a singlet (J = 0). Considering that a λs−o value of 35 meV excited 6 multiplets, characterized by J values from 1 to 6, and with a (2J + 1) number of states, are at 35 meV (J = 1, triplet) and 105 meV (J = 2, quintuplet), other multiplets are at much higher energies.

It is obvious from this generally accepted description of the Sm3+ and Sm2+ ions that there is not so many CEF states below, say, 25 meV, as the experimental extra specific heat needs.

In our description, we also base our considerations on the CEF theory, thinking about the existence in SmB6 of samarium ions with an integer number of electrons of only 5 or 6, which are denoted as Sm3+ (4f5) and Sm2+(4f6) ions, respectively. However, we pay attention to the fact that the spin-orbit coupling in the case of the Sm2+ ion could be weaker than previously considered. The multiplet structure results from the very large spin-orbit coupling, a situation realized in most rare-earth compounds. Computer programs, in which the spin-orbit coupling can be given a finite value, are used in the description of 3d ions in, for instance, NiO and CoO oxides25,26, yielding the fine electronic structure and the orbital magnetic moment. This last outcome is important for describing the 3d magnetism. The Ni2+ and Co2+ ions are 3d8 and 3d7 quantum systems, respectively, and are both characterized by L = 3, similar to the 4f6 (Sm2+) quantum system.

In this contribution, we performed calculations of the fine electronic structure of the Sm2+ (4f6) ion in SmB6 within the spin-orbital |LSLzSz〉 space for the 7F term (L = 3 and S = 3) given by the two Hund’s rules. This fine electronic structure results from the combined action of the cubic crystal-field and the intra-atomic spin-orbit interactions and was calculated with the sole aim of describing the temperature dependence of the 4f contribution to the specific heat of SmB6.

The spin-orbital space of the Sm2+ ion is much bigger, i.e., 49 by 49, than the generally accepted ground state singlet multiplet J = 0 and the excited triplet multiplet J = 1 at 35 meV23.

The highly-correlated atomic-like 4f+ electronic system has the 7F ground term given by two Hund’s rules yielding S = 3 and L = 3. Its 49-fold degeneracy, 7 associated with the orbital degeneration times a 7-fold spin degeneracy, is lifted by intra-atomic spin-orbit interactions and, in a solid, by crystal-field interactions. The cubic crystal-field splits the 49 7F states into 7 orbital states denoted as Γ2 (orbital singlet), Γ4 (orbital triplet) and Γ5 (orbital triplet)27. These Γ states have 7, 21, and 21 degeneracy, as shown in Fig. 2(left). On the other hand, the spin-orbit interactions split the 49 7F states into 7 multiples denoted by J = 0, …, 6, as shown in Fig. 2(right). The lowest multiplet is a singlet J = 0, whereas excited multiplets are (2J + 1)-fold degenerated.

Figure 2
figure 2

The energy states of the highly-correlated 4f6 electronic system associated with the 49-fold degenerated 7F term given by two Hund’s rules: S = 3 and L = 3, as the effect of the cubic crystal field with an exemplary six-order cubic CEF parameter B6 = +0.1 K (left) and as the sole effect of the spin-orbit coupling with an exemplary value of λs−o = +200 K (right). The spin-orbit multiplet structure has been shifted up by 1536 K to obtain the same energy as that of the lowest CEF level to show the involved possible excitation energies. The CEF Γ energies are −864 K, −360 K and +648 K. The multiplet structure is formed in the free ion, i.e., in the absence of CEF interactions.

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In Fig. 2, the six-order cubic CEF parameter, \({B}_{6}^{0}\) (and \({B}_{6}^{4}\) = −21 \({B}_{6}^{0}\) in the exact cubic symmetry), has been taken as +0.1 K, keeping the zero B4 term. A value of λs−o in Fig. 2 has been taken as +200 K, instead of the value of 420 K presented in the literature23, to show, for a better comparison, more multiplets resulting from the spin-orbit interactions.

From an inspection of Fig. 2, one sees that in the case of the Sm2+ ion, (a) the CEF and spin-orbit interactions are of comparable energies and thus must be treated on the same footing and, second, that (b) there is strong competition between them about the ground state and the number of states at the lowest energies.

The energy levels resulting from the combined action of the cubic CEF and spin-orbit interactions can be calculated by the standard single-ion CEF-like calculations25,26,28,29,30. The resulting fine electronic structure depends on three parameters: the fourth-order cubic CEF parameter B4, the six-order cubic CEF parameter B6 and the strength of the spin-orbit coupling λs−o. In Fig. 3, we show the dependence of the fine electronic structure resulting from only the sixth-order B6 parameter, assuming that the fourth-order B4 parameter is zero, as a function of λs−o. We have chosen the B4 parameter only, with a value of +0.1 K, from a didactic point of view to clearly show the idea of our approach and the resulting energy states. In Fig. 3, the change in the fine electronic structure from the CEF electronic structure with the Γ2 ground state to the J = 0 and J = 1 multiplet structure is clearly visible.

Figure 3
figure 3

The calculated CEF +s−o energy states as a function of the spin-orbit coupling λs−o for six-order CEF interactions of the cubic symmetry B6 = +0.1 K with a zero B4 term. The change from the CEF electronic structure with the Γ2 ground state to the J = 0 and J = 1 multiplet structures is clearly visible. For λs−o = +110 K (10 meV), the lowest energy states are at 0 (singlet), 91 K (triplet), 221 K (triplet), 350 K (doublet), 505 K (triplet), 521 K (triplet), 610 K (singlet), ….. These energy states well reproduce the experimental temperature dependence of the samarium contribution to the specific heat of SmB6, as shown in Fig. 4.

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We have found preliminary values of λs−o = +110 K with cubic CEF \({B}_{6}^{0}\) = 0.1 K and \({B}_{6}^{4}\) = −2.1 K interactions (with B4 = 0) that well reproduce the experimental temperature dependence of the samarium contribution to the specific heat of SmB6. We obtained the lowest fine electronic structure with energy states at 0 K (singlet), 91 K (triplet), 221 K (triplet), 350 K (doublet), 505 K (triplet), 521 K (triplet), 610 K (singlet)….. It is clear that the 221 K triplet and the 350 K doublet will form the J = 2 multiplet for the large s − o coupling.

The temperature dependence of the resulting 4f contribution to the specific heat of SmB6 is shown in Fig. 4, together with experimental data read from Fig. 1 in ref.22. We conclude that our calculations, which can still be improved, very well describe the overall temperature dependence of the specific heat of SmB6. This indicates a quite surprising result: practically all Sm ions are in the divalent state. Leaving the problem of understanding of other than specific-heat phenomena for future studies, we conclude that our theoretical approach, with the spin-orbit coupling assumed to have a finite, relatively weak value, reveals a fine electronic structure with a quite large number of low-energy states.

Figure 4
figure 4

The calculated temperature dependence of the samarium 4f contribution to the specific heat of SmB6 associated with only Sm2+ ions. The combined action of exemplary cubic CEF (B4 = 0 K, B6 = +0.1 K) and s−o (λs−o = +110 K) interactions produce the lowest energy states at 0 (singlet), 91 K (triplet), 221 K (triplet), 350 K (doublet), 505 K (triplet), 521 K (triplet), 610 K (singlet), ….. Points are experimental results read from Fig. 1 in ref.22.

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Conclusions

We have, for the first time, described the temperature dependence of the samarium 4f contribution to the specific heat of SmB6, explaining the large extra specific heat of SmB6 compared to that of LaB6, with a maximum at 40 K, which is in good agreement with experimental observations. We calculated the fine electronic structure of the Sm2+ (4f6) ion, finding 15 spin-orbital states within the lowest 50 meV, which are responsible for the large extra specific heat of SmB6. In our calculations, the finite, relatively weak strength of the relativistic spin-orbit coupling plays a fundamentally important role, indicating the breakdown of the strong multiplet description of the Sm2+ ion in SmB6. Our approach has theoretically confirmed the existence of the suggested in-(hybridization)gap states and given them physical meaning.