Introduction

Valence fluctuations in the f-electron based materials near the localization threshold attract significant attention in the condensed matter physics. The intermediate valence has been considered originally to describe some of the rare-earth compounds with Ce, Sm, Eu, Tm and Yb elements. The original idea was that the single-particle “promotion energy” from 4f to 5d states changes the sign in these systems1,2. Soon, it was realized that the situation is different for the special case of Ce. In mixed-valence Ce compounds there is a partial delocalization of 4f electrons due to direct overlap of their wave functions (4f band formation), rather than their promotion to 5d band3. Later it was suggested that similar physics is relevant for 5f electrons in Pu4.

A careful examination of various intermediate valence systems uncovers many differences between them. At first, what are the properties of competing configurations? For Ce, this is f  0 and f 1; for Yb (like in YbB122 or elemental Yb under pressure5) this is f  13 and f  14. In both these cases one of those configurations is trivial in a many-body sense (completely empty or completely occupied 4f shell). For Sm the competing configurations are f  5 and f  6 and for Eu - f  6 and f  7. In this situation, the atomic f  n spin-orbital coupling (SOC) and term effects are essential. Rather than to assume the promotion between single-particle f- and d-states, one needs to consider the competition of ground-state multiplets corresponding to those configurations. Roughly speaking, this is the case when the Hubbard bands originated from these multiplets are well separated. Namely, one of the sub-bands has a well-pronounced multiplet structure in solids and for another part of the spectrum, the multiplets are merged into a single quasiparticle sub-band6. This picture bears close similarities to the case of δ-Pu, which was called “Racah metal”7.

Here we apply this concept to another 4f and 5f systems, using SmB6 and PuB6 as examples of the “Racah materials”. Recently, these materials were proposed as candidates to 3D topological insulators8,9,10, as well as ytterbium borides11. We primarily focus not on the topological properties of electronic bands in SmB6 and PuB6, but on the physics of valence fluctuations and multiplet transitions in these systems. The formation of mixed valence singlet non-magnetic states in effective Anderson impurity model for these compounds crucially depends on hybridization parameters with the ligand bath orbitals and is not the universal property of such “Kondo insulators”. Empirically, all known mixed valence Sm and Eu compounds are nonmagnetic, similar to Yb mixed-valence compounds and contrary to Tm ones1,2; the case of Tm is special in a sense that the ground-state multiplets for both competing configurations, f  12 and f  13 are magnetic. One can speculate that there is a general reason that mixed valence systems cannot be magnetically ordered if one of the competing ground states are nonmagnetic. We show that this is, rather, a “play of numbers”; and requires the optimal hybridisation strength. In particular, we have demonstrated that a typical energy of magnetic excitations is an order of magnitude smaller than a typical energy of valence fluctuations.

Although PuB6 has lately attracted the theoretical attention, very little is known about its properties. The CaB6 structure type corresponds to the cubic CsCl-type lattice in which the B6 octahedra occupy the Cl site. In this structure, the B6 octahedra are linked together in all six orthogonal directions and the Pu-Pu contact distance of 4.11 Å is essentially non-bonding. The paper12 mentions only a weak temperature dependence of magnetic susceptibility. This would suggest that the 5f occupancy should be at least 5.2 or higher, as Pu systems with lower 5f count are known to be magnetic13. It is interesting that the suggestion that PuB6 has a valency lower than 3+ appeared already in the work of Smith and Fisk12 on the basis of volume and color and that the Kondo effect was considered to be responsible for the lack of magnetic moments. SmB6 belongs to canonical valence fluctuation materials (valence estimated as 2.5–2.6) with the Fermi level in a hybridization gap14. Careful photoemission experiments15,16,17 clearly support the complicated mixed valence nature of this “topological insulator”.

Our aim is to apply the state-of-the-art many-body method to develop a complete quantitative theory of electronic structure in SmB6 and PuB6. We follow the “LDA++” methodology18 and consider the multi-band Hubbard Hamiltonian H = H0 + Hint, , where i, j label lattice sites and γ = (lmσ) mark spinorbitals {ϕγ}, is the one-particle Hamiltonian found from ab initio electronic structure calculations of a periodic crystal; Hint is the on-site Coulomb interaction18 describing the f- electron correlation. The effects of the interaction Hamiltonian Hint on the electronic structure are described by a k-independent one-particle self energy, Σ(z) (where z is a (complex) energy), which is constructed with the aid of an auxiliary impurity model describing the complete seven-orbital 5f shell. This multi-orbital impurity model includes the full spherically symmetric Coulomb interaction, the spin-orbit coupling (SOC) and the crystal field (CF). The corresponding Hamiltonian can be written as19

where creates an electron in the 5f shell and creates an electron in the “bath” that consists of those host-band states that hybridize with the impurity 5f shell. The energy position of the impurity level and the bath energies are measured from the chemical potential μ. The parameters ξ and ΔCF specify the strength of the SOC and the magnitude of the crystal field (CF) at the impurity. The parameter matrices Vk describe the hybridization between the f states and the bath orbitals at energy .

The band Lanczos method20 is employed to find the lowest-lying eigenstates of the many-body Hamiltonian Himp and to calculate the one-particle Green’s function in the subspace of the f orbitals at low temperature (kBT = 1/500 eV). The selfenergy is then obtained from the inverse of the Green’s function matrix [Gimp].

Once the selfenergy is known, the local Green’s function G(z) for the electrons in the solid,

is calculated in a single-site approximation as given in21. Then, with the aid of the local Green’s function G(z), we evaluate the occupation matrix . The matrix is used to construct an effective LDA + U potential VU, which is inserted into Kohn–Sham-like equations:

These equations are iteratively solved until self-consistency over the charge density is reached. In each iteration, a new Green’s function GLDA(z) (which corresponds to G(z) from Eq. (2) with the self energy Σ set to zero) and a new value of the 5f-shell occupation are obtained from the solution of Eq. (3). Subsequently, a new self energy Σ(z) corresponding to the updated f-shell occupation is constructed. Finally, the next iteration is started by evaluating the new local Green’s function, Eq. (2).

SmB6 and PuB6 crystalize in the CaB6-structure with the space group Pn3m (221), as shown in Fig. S1 (supplementary information). The experimental lattice constants of 4.1333 Å for SmB6 and 4.1132 Å for PuB6 are used. In the calculations we used an in-house implementation22,23 of the FP-LAPW method that includes both scalar-relativistic and spin-orbit coupling effects. For SmB6, the Slater integrals were chosen as F0 = 6.87 eV and F2 = 9.06 eV, F4 = 6.05 eV and F6 = 4.48 eV24. They corresponds to commonly accepted values for Coulomb U = 6.87 eV and Hund exchange J = 0.76 eV and are in the ballpark of the parameters commonly used in the calculations of the rare-earth materials25. For PuB6, the Slater integrals F0 = 4.0 eV and F2 = 7.76 eV, F4 = 5.05 eV and F6 = 3.07 eV were chosen26. They corresponds to commonly accepted values for Coulomb U = 4.0 eV and exchange J = 0.64 eV. The SOC parameters ξ = 0.16 eV for SmB6 and 0.29 eV for PuB6 were determined from LDA calculations. CF effects were neglected and ΔCF was set to zero. For the double-counting term entering the definition of the LDA + U potential, VU, we have adopted the fully-localized (or atomic-like) limit (FLL) Vdc = U(nf − 1/2) − J(nf − 1)/2. Furthermore, we set the radii of the atomic spheres to 2.85 a.u. (Sm), 3.0 a.u. (Pu), 1.53 a.u. (B). The parameter RSm × Kmax = 9.98 determined the basis set size and the Brillouin zone (BZ) sampling was performed with 1331 k points. The self-consistent procedure defined by Eqs (1, 2, 3) was repeated until the convergence of the f-manifold occupation nf was better than 0.01.

In order to determine the bath parameters Vk and , we assume that the LDA represents the non-interacting model. We then associate the LDA Green’s function GLDA(z) with the Hamiltonian of Eq. (1) when the coefficients of the Coulomb interaction matrix are set to zero (Ummm′′m′′′ = 0). The hybridization function is then estimated as . The curve obtained for is shown in Fig. 1, together with the LDA density of states (total and j = 5/2, 7/2-projected). The results show that the hybridization matrix is, to a good approximation, diagonal in the {j, jz} representation. Thus, we assume the first and fourth terms in the impurity model, Eq. (1), to be diagonal in {j, jz}, so that we only need to specify one bath state (six orbitals) with and and another bath state (eight orbitals) with and . Assuming that the most important hybridization is the one occurring in the vicinity of EF, as suggested by the curve shown in Fig. 1, the numerical values of the bath parameters are found from the relation27 averaged over the energy interval, EF − 0.5 eV  eV, with Nf = 6 for j = 5/2 and Nf = 8 for j = 7/2. The bath-state energies shown in Table 1 are adjusted to approximately reproduce the LDA f-states occupations and .

Table 1 f-states occupations and and bath state parameters (eV), (eV) for Sm and Pu-atoms in SmB6 and PuB6 from LDA calculations.
Figure 1
figure 1

LDA j = 5/2, 7/2 projected DOS and LDA hybridization function for SmB6 (a) and PuB6(b).

The magnitude of Δ(EF) (Δv) is a characteristic energy of the valence fluctuations, in a sense that for the time scale the system behaves as a homogeneous with the physical properties which are intermediate between those for Sm2+ and Sm3+ whereas for t < τfl it is a random configuration of “frozen” Sm2+ and Sm3+ ions1,2,5. Lattice parameter and core-level X-ray spectra serve as examples of the properties of the first and the second kind.

SmB6

First, we focus on SmB6 and discuss the solution of Eq. (1). The ground state of the cluster formed by the 4f shell and the bath is given by a non-magnetic singlet with all angular moments of the 5f-bath cluster equal to zero (S = L = J = 0). For the 4f shell alone, the 〈nf〉 = 5.63 and the 〈nbath〉 = 6.37 bath states. Note that 〈nf〉 slightly exceeds its LDA value of 5.54. The expectation values of the spin Sf, orbital Lf and total Jf angular moments can be calculated as (Xf = Sf, Lf, Jf), giving Sf = 2.77, Lf = 3.80 and Jf = 1.88. The ground state is separated from the first excited state by the gap Δm = 2.6 meV. Surprisingly, this value is in a very good agreement with the experimental activation gap value of 3 meV28. This gap should show itself in the magnetic susceptibility, which is anticipated to behave as 1/[T + Tm] at high temperatures, with saturation below Tm temperature ~Δm, in qualitative agreement with the experimental data14 and other experiments which measure the two-particle excitations. This excitation in two-particle spectrum can be contrasted with first single-particle photoemission peak around 20 meV15. It is important to mention that formation of mixed-valance multi-orbital singlet in effective Anderson model is very sensitive to hybridization parameters (Table 1) and with relative small changes the magnetic ground states is formed in ED calculations. It is also important that this magnetic exciton energy is an order of magnitude smaller than the energy of the valence fluctuations Δv ≈ 70 meV. This means that the nonmagnetic character of the ground state is not directly related to the valence fluctuations: the system possesses local magnetic moments in the energy (and temperature) range between Δm and Δv, that is, within the homogeneous intermediate valence regime.

The f-orbital density of states (DOS) obtained from Eq. (2) for SmB6 is shown in Fig. 2(a). The f-DOS is in agreement with the experimental x-ray photoelectron spectra (XPS)29 and previously reported Hubbard-I calculations30. The many-body resonances near the Fermi energy are produced by f  6 → f  5 multiplet transitions, they are in a way analogues to the Racah peaks, specific transitions between Racah multiplets31 of f  n and f  n±1.

Figure 2
figure 2

f-electron density of states (fDOS and j = 5/2, 7/2 projected) for the Sm atom in SmB6 (a) and the Pu atom in PuB6 (b).

Also comparison with the experimental XPS spectra is given for SmB6.

Figure S2(a) (supplementary information) shows the LDA band structure together with the band structure calculated from the solutions of Eq. (3), which represents an extended LDA + U band structure with the 5f-states occupation matrix obtained from the local impurity Greens function Eq. (2) (LDMA). Note that the LDA band structures are very similar to previously reported results of WIEN2K for SmB69.

A more detailed look at the band structure is shown in Fig. 3(a) SmB6 is close to a very narrow band insulator already in LDA. There is a tiny amount of holes in the vicinity of the X-point (similar to ref. 9) and a direct gap of ~30 meV right above. When the Coulomb interaction is added, it becomes an indirect band insulator with the gap of ~60 meV. Note that the band-gap value exceeds somewhat the experimental gap of around 20 meV. Incorporating the dynamical self-energy effects into the LDMA band structure, as described in the supplemental material Fig. S3, we obtain that the indirect band gap is somewhat reduced to ~30 meV becoming closer to the experimental value of 20 meV.

Figure 3
figure 3

SmB6 (a) and PuB6 (b) LDA and LDMA band structure on the small energy scale.

The circles indicate the f-character of the electronic states.

It is known that the d-f Coulomb interaction G (Falicov-Kimball interaction) plays a role for the intermediate valence1,32,33. This interaction leads to the excitonic renormalization of the effective hybridization. The effective hybridization Veff between d and f states with the many-body renormalization can be calculated using the electronic structure expression32 which for zero temperature reads:

In this Eq. (4), is the total DOS without the f-projected contribution and V is the LDA hybridization from the Table 1. Importantly, the renormalized hybridization turns out to be quite strongly temperature dependent32.

The parameter G can be determined as the derivative of the center of the 5d band with respect to the number nf of 4f electrons5). In practice, we have varied nf by changing the double-counting term from the FLL (nf = 5.63) to the “around-mean-field” (AMF, nf = 5.68) and obtained the Falicov interaction parameter of 3.8 eV. Solution of the Eq. (4) yields the Veff/V renormalization of 1.77.

Thus, the d-f excitonic effects enhance the hybridization making the hybridization gap larger and therefore favoring the topological insulator behavior. We performed the calculations with this renormalised Veff in Eq. (1) and obtained again the singlet ground state. The 〈nf〉 = 5.61 has decreased slightly. This numerical stability of the Sm singlet ground state with respect to a hybridization strength is important since experiments34 show a strong temperature dependence of the energy gap in SmB6 which cannot be explained in a purely hybridization model; they were explained in ref. 32 via excitonic effects. Recently, a strong decrease of the hybridization gap with the temperature increase in SmB6 was found in ARPES16. This can be also considered as a confirmation of strong many-body (excitonic) renormalization of the hybridization.

To estimate the temperature dependence of the hybridisation due to Falicov-Kimball interaction we use the theory32 for the finite temperatures, that is, Eq. (4) with the replacement,

where and is the Fermi function. The results are shown in Fig. 4.

Figure 4
figure 4

The temperature dependence of the hybridization gap (indirect), Δv(T)/Δv [LDA] = (Veff(T)/V)2 calculated in the Falicov-Kimball model Eqs (4) and (5)

.

The presence of the non-magnetic f  6 multiplet is crucially important for the non-magnetic singlet ground state of SmB6. For instance, in the intermediate valence TmSe (competition of f  12 and f  13 configurations) the ground state is magnetic since both configurations are magnetic. At the same time, there is no “theorem” that for the non-magnetic ground state of one of the competing configurations the system cannot be magnetic and the specific values of the relevant parameters are important. As we have seen, even typical energy scales for the magnetic (Δm) and valence (Δv) fluctuations are different.

For the f-shell occupation nf of 5.6, we show in Fig. 5 the energy difference between the first excited eigenstate for given number of particles (N = nbath + nf) and the ground state of the Eq. (1) for different values of hybridization: those calculated in LDA and given in Table 1, reduced by a factor of 2 and renormalised by the Falicov-Kimball model, as it was described above. In all those calculations, the ground state is a non-magnetic singlet with N = 12. For the LDA hybridization, the lowest excited state belongs to the same N = 12 and is lying 3 meV above the ground state. The excited magnetic N = 11 and N = 13 states are shifted upwards in the energy by 70 meV and 47 meV respectively. When the hybridization is reduced (twice smaller than its LDA value), a non-magnetic ground state singlet with N = 12 is by 6 meV lower than almost degenerate N = 11 and N = 12 magnetic excited states. The N = 13 excitation is substantially (by 70 meV) higher in the energy. At the same time, for the hybridization renormalised by the Falicov-Kimball model Eq. (4), the situation is inverse: the lowest magnetic excited state of 4 meV belongs to N = 13, next (9 meV) has the same N = 12 and the N = 11 excitation exceeds the singlet ground state by 139 meV. Further increase of the hybridization, say by a factor of 2 with respect to the LDA value, leads to occurrence of the magnetic N = 13 ground state.

Figure 5
figure 5

The energy difference between the first excited eigenstate for given number of particles (N = nbath + nf) and the ground state of the Eq.

(1) for different values of hybridization. (i) VLDA/2 reduced by a factor of 2 from those calculated in LDA and given in Table 1, (ii) VLDA from Table 1; (iii) renormalised by the Falicov-Kimball model Eq. (4).

In this sense, while it is possible to call the situation “Kondo singlet with high Kondo temperature” (which simply means a formation of singlet from the states of localized and itinerant electrons) one should keep in mind that microscopically some effects beyond the Kondo or Andreson model, such as Falicov-Kimball interactions can contribute significantly. There is an essential difference with various Ce- and Yb-based systems where multiplets are not important and the situation is indeed closer to the Kondo lattice with high Kondo temperature.

PuB6

Now we turn to the case of PuB6. In this case, the hybridization strength is substantially increases (see Table 1). The hybridized ground state of the Pu atom in PuB6, the solution of Eq. (1), is a non-magnetic singlet with all angular moments of the 5f-bath cluster equal to zero (S = L = J = 0). It consists of 〈nf〉 = 5.49 f states and 〈nbath〉 = 8.51 bath states. As in the case of SmB6, the magnetic moment of the 5f shell (Sf = 2.23, Lf = 3.68, Jf = 1.94) is completely compensated by the moment carried by the electrons in the conduction band. As the value of the 5f magnetic moment fluctuates in time, because of the intermediate valence electronic configuration, this compensation must be understood as dynamical in nature. The same situation is realized in δ-Pu (Sf = 2.11, Lf = 4.21, Jf = 2.62), whose ground state is found to be a non-magnetic singlet with 〈nf〉 = 5.21 and 〈nbath〉 = 8.797.

The f-orbital density of states (DOS) obtained from Eq. (2) for PuB6 is shown in Fig. 2(b). No experimental photoelectron spectra available in this case. As in δ-Pu, there are three many-body resonances near the Fermi energy which are produced by f  6 → f  5 Racah multiplet transitions.

The LDA band structure is very similar to previously reported results of WIEN2K for PuB610 as shown in Fig. S2(b) (supplementary information), and, in more details, in Fig. 3(b). Already in the LDA, PuB6, is close to an insulator with a small amount of holes near the X-point and the indirect band gap of ~60 meV. In the LDMA, PuB6 becomes almost an insulator, with the tiny fraction of holes near the X-point and direct band gap of ~60 meV (see Fig. S2(b) and Fig. 3(b)).

As to PuB6, we have very little material for comparison with experiment, as there is much less data not only comparing to rare earth borides but also with respect to other Pu compounds. A group of analogous compounds with an energy gap and non-magnetic behaviour are Pu chalcogenides PuX, with X = S, Se, Te. Photoelectron spectra35,36 reveal a pronounced fingerprint of the final-state 5f  5 multiplet close to the Fermi level, which implies that the 5f  6 state must contribute to the ground state. The Pu chalcogenides have also qualitatively similar non-metallic conductivity explained by hopping37, qualitatively analogous not only to SmB6, but also to Sm chalcogenides.

Conclusions

The electronic structure calculations are performed within the density functional plus dynamical mean-field theory (“LDA++”18) approach combining the local density approximation (LDA) with an exact diagonalization (ED) of the Anderson impurity model for SmB6 and PuB6. The intermediate valence singlet ground states are found for these materials. When the Coulomb f − f (Hubbard) correlations are included, SmB6 becomes an indirect band gap insulator, while PuB6 is a direct band gap insulator. A combined effect of specific Racah multiplet structure with intermediate valence behavior of these compounds results in complicated excitation spectrum clearly seen in different photoemission experiments. Formation of singlet ground state in the ED impurity calculations is not universal and crucially depends on structure of two mixed valance multiplets and parameters of effective Anderson model. The Coulomb f − d (Falicov-Kimball) interactions increase essentially the effective hybridization influencing additionally the singlet state. Their role may be essential in explanation of recently observed temperature-dependent electronic structure of SmB616. The calculations illustrate that many-body effects are relevant to form the indirect band gap. In PuB6 we have found also a mixed-valent singlet ground state with basically the same multiplet physics as was discussed earlier for δ-Pu7.

To emphasize the role of multiplet effects in competing valence states for this class of mixed valence systems, we suggest the term “Racah materials”. The distinguishing feature for these materials is that part of electron excitation spectrum originated from one the valence configurations is more atomic like (with well-pronounced multiplets) whereas for the other valence configuration it is more itinerant-like. The consept of “Racah materials” is somewhat related to the idea of “quasiparticle multiplets”38. Those are represented by atomic-like multiplet transitions f  6f  5 near the Fermi edge. In addition, there is a second part at the lower energy (f  5f  4) which are more itinerant-like and merged into the quasi-particle subband6. Co-existence of these two types of the Hubbard bands in SmB6 and PuB6 defines them as Racah materials.

Additional Information

How to cite this article: Shick, A. B. et al. Racah materials: role of atomic multiplets in intermediate valence systems. Sci. Rep. 5, 15429; doi: 10.1038/srep15429 (2015).