Introduction

Understanding the ground states that result from hybridization between local moments and conduction electrons remains a major goal of the condensed matter community. In a purely magnetic picture, localized f magnetic moments can hybridize with the itinerant metallic spd electrons in the valence band, forming spin singlets and resulting in a screening of the f magnetic moments that reduces the tendency toward magnetic order. Within this picture, introducing a Kondo coupling \({{{{{{{\mathcal{J}}}}}}}}\) between the local f moment and the conduction electron c, one traverses the well-known Doniach phase diagram between an RKKY-mediated magnetically ordered ground state (\(\sim {{{{{{{{\mathcal{J}}}}}}}}}^{2}\)) and a nonmagnetic heavy fermi liquid (\(\sim {e}^{-1/{{{{{{{\mathcal{J}}}}}}}}}\)), separated by a quantum critical point (QCP). Microscopically, the hybridization proceeds as the formation of a virtual bound state (Abrikosov-Suhl resonance) between the mobile c carriers at the Fermi energy EF, and the local f electrons, with the bare interaction potential Vcf enhanced based on how far the f levels happen to be from EF. For nearby f states, the resonant enhancement \({{{{{{{\mathcal{J}}}}}}}}={V}_{cf}/({E}_{F}-{E}_{{{\mbox{4f}}}})\) leads to an increased density of heavy fermion states at EF1,2,3. It is the strength of the Kondo coupling which determines whether the f electrons can form a nonmagnetic Fermi liquid with heavy bands containing the f electrons, which can host unconventional superconductivity. Experimental evidence for a lattice of heavy Kondo bands is observed in Ce-based materials such as CeCoIn54,5,6, CeIrIn57, and CeRh2Si28.

The virtual bound state of the Kondo lattice is generally considered to be a singlet state of the magnetic spins, but cf-hybridization between conducting carriers c and local f moments in general also allows for fluctuating charge degrees of freedom. When f electrons are strongly localized in a metallic host, the charges may be conceptualized as a kind of Wigner crystal with high electronic repulsion between neighboring f states, which prevents them from moving, similar to the physics of a Mott insulator2,9. When the primary valence configuration becomes unstable, however, charge fluctuations and ultimately delocalization of the f electrons may also occur, which result in a superposition of fn andfn±1 valence configurations of the localized ion9. If both f valence states lie within the bandwidth of the metallic spd electrons, one of the f configurations may then be found close enough to the Fermi energy to form an Abrikosov-Suhl resonance, an effect that has been studied with a high degree of detail in systems such as YbRh2Si210. In intermediate valence systems, the Kondo model is not necessarily able to capture the essential physics due to the existence of the charge fluctuation degree of freedom.

Distinguishing whether charge fluctuations play a role in the mechanism of unconventional superconductivity is generally difficult, but the shape of the superconducting dome has historically provided some insight. Specifically, small superconducting domes may be magnetically driven, whereas large superconducting domes extending far from the QCP and exhibiting unusual shapes are assumed to include charge fluctuations11. In CeCoIn5, a tetragonal material, the microscopic mechanism for superconductivity was often believed to be spin-dominated due to its much larger critical temperature Tc compared to CeIn3, a scenario consistent with the expectations of spin-fluctuation driven superconductivity in a system with reduced dimensionality12. Nevertheless, a very recent study argued that CeCoIn5 also exhibits a simultaneous charge delocalization QCP, which was suggested as evidence for the separation of spin and charge sectors on a more equitable footing13, a conclusion that agrees with an earlier review of critical valence fluctuations14. Thus, the microscopic details of the delocalization mechanism from the point of view of the charge sector may deserve more attention in “115” materials. From this perspective, materials with multiple f electrons are highly suitable for such an investigation because both valence states will have a nonzero number of f electrons.

The microscopic mechanism of f electron delocalization is a topic of ongoing debate, and which has invited a variety of perspectives on its origin and appropriate theoretical treatment. In the case of a single f electron or hole, such as in Ce and Yb, materials with an intermediate-valence scenario are often viewed as the natural consequence of a “Kondo interaction with high Kondo temperature,” which progressively brings the f state closer to the delocalized limit as the temperature is lowered. However, in the multi-f-electron case and sometimes including Yb, it has been recognized that this picture misses some additional complexity2,15. Starting with basic experimental observations, the f delocalization mechanism seems to be enhanced in ions which already have a tendency toward an f valence instability, including Ce, Sm, Yb, U, and Pu3. In 5f intermediate-valence materials with U and Pu, theoretical efforts have been made to identify an integer number of delocalized f electrons based on stability conditions16,17,18, while for 4f materials with Ce and Yb, as well as Sm-based materials like SmB6, the valence is often well-established to be a fraction of a single f electron19,20. In YbCoIn5, it was argued that the intermediate-valence state does not change with temperature, meaning Yb acts as a magnetic impurity similar to the role of Nd in Ce1−xNdxCoIn521 and suggesting a non-Kondo origin for the intermediate-valence character19. A particularly intriguing mystery was found by comparing different Yb-based materials, where delocalization through a intermediate valence mechanism occurs at a fixed temperature scale, even when the Kondo temperatures differ by over 4 orders of magnitude22. For Sm-based compounds, modern experiments have shown that they often exhibit heavy fermion and/or intermediate valence character9,23,24,25,26,27,28,29,30,31,32,33,34. Of these cases, a temperature-induced Kondo effect was most clearly observed in (La,Sm)Sn3, with a Kondo temperature near 125 K in the dilute case, and a Kondo lattice in the dense (Sm-rich) limit at a similar temperature scale25,35,36. In SmOs4Sb12, on the other hand, an exotic low-temperature heavy fermion state was suggested to be non-magnetically driven, and valence fluctuations are also observed, suggesting a novel type of heavy fermion mass enhancement for the multi-f-electron case27,34. A key observation about the origin of heavy masses, pointed out by Higashinaka et al.37, is that Sm-based systems do not follow the expected mass enhancement factor m*/m = (1 + Δnf)/(2Δnf). This factor, calculated to be the dominant scale factor in the renormalized quasiparticle mass, reflects the decrease in effective mass when the itinerancy of the f states is large, and has been empirically shown to work well for systems with a single f electron in the strongly correlated limit14. However, Sm-based systems tend to show a large mass enhancement of 102–103 when they exhibit a valence of vSm ≈ 2.8 (Δnf = 0.2), whereas the formula predicts only m*/m = 3. Therefore, since the phenomenology of mass enhancement in Sm-based materials is very different from expectations, it is important to show how f electrons in Sm-based materials can exhibit temperature-dependent delocalization in order to understand how Kondo-like behavior arises within the many-body intermediate valence scenario.

Turning now to SmCoIn5, we present evidence for an f electron delocalization mechanism at temperatures below Tv ≈ 60 K, which we identify as a valence crossover temperature. In Fig. 1, we draw a cartoon schematic that showcases our understanding of the valence change that is coincident with a change in the shape of the Sm 4f wavefunctions. Figure 1a shows one layer of Sm3+ ions in SmCoIn5 at high temperature, where all of the levels in the ground spin-orbit multiplet are thermally populated, and the total 4f wavefunction at each site resembles a sphere. At temperatures below Tv, the Sm3+ wavefunctions become thermally isolated in the ground state doublet, which have a box-like shape, while delocalized states corresponding to the Sm2+ valence configuration are enhanced at the Fermi energy EF, as depicted in Fig. 1b. Evidence for this scenario comes primarily from our x-ray absorption spectroscopy (XAS) experiments, combined with full multiplet calculations, which reveal the microscopic configuration of f electrons in the Sm3+ valence state of SmCoIn5 and determine the energy level scheme (Figs. 2 and 3). Further evidence for a valence crossover Tv associated with this crystal field scheme is observed with a variety of experimental probes (Fig. 4), including the onset of magnetic fluctuations in bulk susceptibility, the appearance of magnetism in the Co d electron orbitals, a crossover in the lattice parameter aspect ratio a/c, a departure from T-linear resistivity and inflection point in carrier density, and most significantly, a redistribution of f spectral weight in the valence band structure favoring a Sm2+ component around the Fermi energy EF at low temperature, which represents a delocalized hole in the conduction volume (Fig. 5). These results indicate that Tv is related to a change in the microscopic shape of the Sm wavefunctions that allows electrons in the Fermi sea to spend more time on the Sm sites, resulting in a larger Sm2+ signature. In addition, our full multiplet calculations (Figs. 2 and 3) show that the mutual Coulomb repulsion between f electrons is responsible for separating the Sm single-ion energy eigenstates over more than 16 eV in the valence band. Such a wide distribution of multiplet levels, containing wavefunctions with many different shapes and symmetries, may point to the importance of one of these levels serving as a virtual intermediate state in the formation of an Abrikosov-Suhl resonance and heavy fermion character. The appearance of Tv ≈ 60 K in bulk probes (magnetization, lattice constant, resistivity, and carrier density) also shows that the crossover is a bulk property, with possible consequences for the eventual magnetically ordered ground state below TN ≈ 11 K38. Therefore, our work shows that SmCoIn5 exhibits a delocalization mechanism driven by the crystal field that resembles the Kondo effect in CeCoIn5 generalized to the multi-f-electron case, and shows that charge fluctuations are an important phenomenon found widely in the ground state of “115” materials.

Fig. 1: Cartoon schematic of the f delocalization process in SmCoIn5.
figure 1

a Lattice of Sm3+ ions at high temperature above the valence transition Tv ≈ 60 K, with a graph showing the position of Sm3+ in the density of states (d.o.s.) with respect to the Fermi energy EF. b Lattice of Sm3+ ions at low temperature below Tv, but above the antiferromagnetically ordered phase at TN ≈ 11 K, containing screening clouds around each site representing the delocalized Sm2+ states that appear in the d.o.s. near EF.

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Fig. 2: Single-ion properties of Ce3+ and Sm3+ in the “115” materials.
figure 2

a–g Level scheme of the Ce3+ and Sm3+ ions in CeCoIn5 and SmCoIn5, calculated using the software package QUANTY. The ground manifold in each case consists of three Kramers doublets with J ≈ 5/2, separated from the 7/2 manifold by spin-orbit coupling as indicated by the zigzag lines. The CeCoIn5 calculations in a–c were performed for the 4f1 configuration using the crystal field parameters reported by Willers et al.39 and spin-orbit interaction ξSO = 87 meV, which leads to a splitting between the J = 5/2 and J = 7/2 spin-orbit manifolds of Δ ≈ 300 meV. The SmCoIn5 calculations in d–g were performed using the parameters found in this work and with ξSO ≈ 102 meV, which leads to a lower Δ ≈ 180 meV. h Stevens A parameters for the crystal field potential used in the calculations, in meV: (−4, 170, 50) for SmCoIn5 and (2, 240, 80) for SmIrIn5, which were found via an exhaustive search in the space of \({A}_{2}^{0}\), \({A}_{4}^{0}\), and \({A}_{4}^{4}\). For SmCoIn5, the error bars are chosen by hand to account for reasonable variation in the parameters that still remains a close match to the experimental data (XLD and magnetic susceptibility) and reproduce the temperature (Tv = 60 K) at which the XLD pattern inverts. For SmIrIn5, the error bars were chosen to match INS and magnetic susceptibility. i–j Ground state wavefunctions (one from each Kramers doublet) of CeCoIn5 and SmCoIn5, plotted with QUANTY. The radius corresponds to the magnitude of the charge density, with magnetization density overlayed as the surface color from red to blue. For CeCoIn5 and CeIrIn5, the values were converted from the Stevens B parameters reported by Willers et al.39 using the conversion factors \((-35,1260,18\sqrt{70})\)72.

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Fig. 3: XAS spectra and model calculations for SmCoIn5.
figure 3

a XAS at the Sm M4,5 edges, measured at X-Treme in grazing incidence using the total electron yield (TEY) method. b XLD as a function of temperature, and corresponding QUANTY calculations for the Sm3+ ions. The XLD data is normalized to the XAS signal, with a vertical scale of 2% shown with horizontal lines around each curve, as indicated by double-headed arrows. The overall agreement between the data and model calculations at all temperatures is sufficient to uniquely determine the Stevens parameters presented in Fig. 2.

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Fig. 4: Evidence for a temperature scale Tv ≈ 60 K in SmCoIn5.
figure 4

a Magnetic susceptibility of SmCoIn5 measured in a vibrating sample magnetometer with H = 5000 Oe, shown along with calculations of the magnetization using the crystal field model in QUANTY, and with a Curie–Weiss fit. The magnetic ordering temperature is visible at TN = 11 K. b (inset) State populations of the three multiplet levels of SmCoIn5, showing an inflection point in the ground state population at T = 56 K. The equivalent temperature for CeCoIn5 is found to be T = 36 K. c Difference between the experimentally observed magnetization and calculated magnetization for different field directions. The overlayed points are the total magnetic moment of the conduction electrons at H = 6.8 T with H∥101, which we obtained from sum rule analysis of x-ray magnetic circular dichroism (XMCD) data measured at the Co L-edge assuming a Co2+ valence (see Supplementary Note 4 and Supplementary Figs. 5–7). d–f Temperature dependence of the a and c lattice parameters, and corresponding fits to the high temperature data with a hyperbolic form \({a}_{i}={a}_{i,0}+b\sqrt{{(T/{T}^{* })}^{2}+1}\), where b and T* are fitted parameters with error bars smaller than the point size in the figure. The departure from the expected form occurs near Tv. g Direct current electrical resistance showing a departure from linear behavior at Tv. The solid line shows a fit to a hyperbolic form. Inset: resistance continues in a linear fashion up to T = 300 K. h Net carrier density from Hall resistance measurements with current applied along the a axis direction and field along c. Error bars represent fitted errors.

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Fig. 5: Evidence for intermediate valency in SmCoIn5 below Tv ≈ 60 K.
figure 5

a Temperature dependence of the momentum-resolved Res-ARPES spectra along the Γ − X − Γ high-symmetry cut at the Sm M4 edge (hν = 1106 eV), showing an enhancement of the spectral weight near −2 < EB < −0.5 eV that is associated with the appearance of the Sm2+ configuration (see Supplementary Note 2 and Supplementary Fig. 2). The emission patterns are consistent with the multiplet spectrum of Sm final states in the resonant ARPES process52,53. b Momentum-integrated cuts along Γ − X − Γ as a function of binding energy, at the Sm M4,5 resonant edges, for several temperatures below Tv ≈ 60 K. The peak structure near −2 < EB < −0.5 eV associated with Sm2+ loses spectral weight as the temperature is increased. A nonresonant ARPES measurement carried out at T = 12 K with hν = 605 eV and integrated over the same momentum cut is shown offset at the bottom, which also contains Co and In bands. c X-ray photoemission spectroscopy at T = 12 K, showing an intermediate valence scenario.

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Results

Crystal field ground state and multiplet structure of SmCoIn5 and SmIrIn5

The Hamiltonian of the localized 4f electrons may be written as H = HCoul + HSO + HCEF, which contains the Coulomb interaction, spin-orbit coupling, and crystal electric field with D4h site symmetry. In CeCoIn5, the crystal field was fully determined from measurements of the x-ray linear dichroism (XLD) in x-ray absorption spectroscopy (XAS) experiments at the Ce M4,5 edges, combined with inelastic neutron scattering (INS)39,40,41. We determined the single-ion properties of Sm in SmCoIn5 by combining M4,5-edge XLD with bulk magnetization measurements. In order to model the XLD results, we used the full-multiplet software package QUANTY42 to diagonalize the full Hamiltonian for the 4f ground state. With D4h site symmetry and 5 electrons in the f shell, there are three independent parameters in the crystal field potential expansion written in Stevens operator formalism, specifically, a dipole term \({A}_{2}^{0}\) which describes the ionic potential along the long tetragonal axis, and two quadrupolar terms \({A}_{4}^{0}\) and \({A}_{4}^{4}\).

The results of our analysis are summarized in Fig. 2, which shows both Sm compounds exhibit a Γ7 ground state wavefunction, similar to the results previously found for the Ce-based 115s. We find the crystal field scheme for Sm is very close to the limit of cubic Oh symmetry, where the tetragonal terms are constrained to a fixed ratio and \({A}_{2}^{0}\) = 043,44. Upon the lowering of the point group symmetry from cubic to tetragonal, the Γ7 doublet survives while the Γ8 cubic quartet separates into Γ6 and Γ7 doublets45. In SmCoIn5, we find a very small dipolar \({A}_{2}^{0}\) term, as well as a ratio close to the cubic limit for the quadrupolar terms, which implies that the tetragonal anisotropy of the lattice has a weak effect on the single-ion properties of Sm. Within this crystal field scheme, the Sm3+ level spacing and wavefunctions are significantly affected by the Coulomb repulsion between mutual f electrons (see Supplementary Note 1 and Supplementary Fig. 1), a situation not encountered in the single-electron case. Moreover, we find that the Coulomb repulsion admixes the ground state wavefunction with other multiplets, resulting in a situation where the combination of irreducible representations contained in the ground state wavefunction is not perfectly described by mixing the Jz = 5/2 and 3/2 states, as it is in the single-electron Ce3+46. In Table 1, we compute the expectation values of quantum angular momentum operators of the ground state, as well as the projection into the irreducible symmetries of the D4h point symmetry group, under different combinations of the terms in the Hamiltonian. Introducing spin-orbit coupling (SOC), the crystal field potential, and Coulomb repulsion have different effects on the quantum numbers, indicating that the ground state of Sm is not in the limit of a pure LS multiplet state.

Table 1 Angular momentum quantum numbers of the crystal field ground states (g.s.) of Sm3+ in SmCoIn5 calculated using QUANTY.
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In Fig. 3, we present the x-ray linear dichroism (XLD) raw data, displayed as the c-axis component of the XAS measurement minus the a-axis component (see Methods). By examining the XLD data above and below a temperature near Tv ≈ 60 K, the shape of the dichroism reverses sign. Since the XAS spectra include transition processes that span several eV, and each transition depends on the Boltzmann factor of its initial state, the most obvious qualitative crossover of the overall XLD lineshape will occur when the system transitions between an ensemble of populated crystal field states to a pure doublet state at low temperature, which is precisely the crossover temperature Tv ≈ 60 K that we find. Specifically, we found that the dipole \({A}_{2}^{0}\) parameter is coupled strongly to the temperature of this sign reversal; this allows us to calculate \({A}_{2}^{0}\) in a way that is highly robust against small errors in the raw data, giving us confidence in our analysis. The model calculations in Fig. 3 are generated with QUANTY and are calculated for the Sm3+ configuration. For the Sm2+ configuration, the 4f6 configuration exhibits J = 0 with no low-lying crystal field states, and is a non-Kramers ion; therefore, we ignore the Sm2+ states in the analysis of the XAS data and magnetization.

Since our crystals of SmCoIn5 are small, only around 10 mg, we were unable to perform inelastic neutron scattering (INS) to confirm the crystal field level spacing. Thus, in order to roughly confirm the results for SmCoIn5, we also synthesized SmIrIn5, obtaining large crystals which we prepared in a powder sample for INS with a total mass of 1.8 g (see Methods). We conducted constant-Q scans at six positions between Q = 1.79 and 4.00 Å−1, which all show a broad peak that is consistent with two overlapping resolution-limited peaks near E = 17.5 and E = 20 meV, suggesting that the two crystal field doublets from Sm are at these positions (see Supplementary Note 3 and Supplementary Fig. 3). This level scheme for SmIrIn5 is also consistent with the magnetization data we collected on a 63 mg single crystal of SmIrIn5 at high temperature (see Supplementary Note 4 and Supplementary Fig. 4), which shows that the crystal field in SmIrIn5 is close to the cubic limit such that \({A}_{2}^{0}\approx 0\), as we found with SmCoIn5. Therefore, by combining magnetization data with with inelastic neutron scattering for the SmIrIn5 compound, as a complementary technique to combining magnetization with XAS experiments as we did for SmCoIn5, and finding a similar level scheme in the ground multiplet for both Sm-based materials, gives us confidence that the crystal field ground state of SmCoIn5 and SmIrIn5 are accurately determined.

Temperature scale near Tv ≈ 60 K

In Fig. 4, we show the evidence for a crossover in the electronic character of SmCoIn5 near Tv ≈ 60 K. Magnetization measurements of SmCoIn5 are shown in Fig. 4a, along with the results of the calculated magnetic susceptibility within our crystal field model of Sm3+, for magnetic field of 5000 Oe applied in different directions, within the ab plane and parallel to the tetragonal c axis. The magnetization contains no contribution from the J = 0 Sm2+ states. Near Tv, we find that the calculated magnetization exhibits a hump as a function of temperature, and below Tv begins to deviate from the experimental results. To better understand the origin of the temperature Tv = 60 K, in the inset labeled Fig. 4b we plot the Boltzmann population factors for the levels found from the XLD data shown in Fig. 3. The inflection point in the population factor of the ground state is calculated to be 56 K, which is an excellent match for the observed Tv in all experimental probes and strongly suggests that the crystal field is the driving mechanism behind the observation of Tv. Moreover, the experimental magnetization data show the f magnetic moments in SmCoIn5 do not follow the expected magnetic susceptibility from the crystal field model, but instead follow almost perfect Curie–Weiss behavior with Weiss temperature Θ ≈ −24 K and J ≈ 2.04. Below TN = 11 K, the susceptibility exhibits a sharp change due to the onset of long-range antiferromagnetic order. The departure of the magnetic susceptibility from the crystal field model of Sm3+ suggests that the magnetic degree of freedom is not strongly affected by the valence crossover and change of shape of the 4f atomic wavefunction.

To better understand the meaning of the departure from the crystal field model below Tv, we measured x-ray magnetic circular dichroism (XMCD) at the Co L-edge to understand whether the magnetic susceptibility arises from the conduction electrons. In previous experiments, it was determined that the magnetic moment specifically tied to the conduction electrons could be measured with XMCD at the ligand ions of UAs, UGa3, and UGe247,48 Here, we perform the same analysis, while also considering that the Co 3d states may have a small intrinsic moment. Using the XMCD sum rules49, we determine that below Tv, the conduction electron moment is increasing in proportion with the departure of the bulk magnetic susceptibility from the crystal field model, gaining approximately 0.3 μB between Tv and TN = 11 K. In Fig. 4c, we overlay the total magnetic moment of Co onto the difference curve between bulk magnetization and our crystal field model from part (a). The agreement between the bulk and local probe experiments demonstrates that itinerant electrons carry part of the f electron magnetic moment below Tv, a signature of delocalization.

To further demonstrate a change of physical properties across Tv, we used high-resolution powder x-ray diffraction experiments to search for changes in the structure of the lattice. In Fig. 4d–f, we show that the a and c tetragonal lattice parameters both change qualitatively below Tv with respect to the temperature dependence of the total unit cell volume a2/c. The total volume follows the form of a hyperbola, which smoothly interpolates between constant and linear thermal expansion as the temperature is increased. However, the individual lattice parameters both deviate from a hyperbolic form below Tv, with the a lattice parameter even beginning to increase as the temperature is lowered further, indicating an unexpected in-plane expansion of the unit cell. This result shows phenomenologically that the electronic structure undergoes a crossover transition at Tv, and proves that the transition occurs in the bulk of SmCoIn5. Moreover, the direct cause of the change in the aspect ratio a/c may be connected to magnetostrictive effects, due to the partial transfer of the magnetic moments onto the itinerant sites (Fig. 4a, b). In this way, the lattice parameter measurements show that the valence crossover also affects the bulk magnetic properties and leads to structural changes, while the overall tetragonal symmetry is preserved. It is also possible that Tv is associated with the development of an intrinsic Co moment, which may also be related to the overall change in electronic structure connected to the valence transition at Tv.

Finally, in Fig. 4g, h we show a qualitative change in the electrical resistance and carrier density which occur below Tv. In the temperature-dependent resistance R(T) at temperatures above Tv (inset of Fig. 4g), the resistivity is clearly linear in T, a phenomenon previously associated with the existence of critical valence fluctuations14. Below Tv, the resistance begins an “upturn,” which keeps the value higher than the T-linear model predicts as the temperature is lowered. This observation is consistent with opening more scattering channels for the conduction electrons and holes due to the increase in the number of Sm2+ states at EF50. A similar “upturn” in resistivity measurements of SmSn3 was taken as a sign of the Kondo effect25.

To further show that the upturn in resistivity is connected to a qualitative change in the metallic properties of SmCoIn5, we carried out measurements of the Hall effect as a function of temperature, and extracted the net total carrier density from a model fitted to data between H = −9 and +9 T. With current along the a-axis direction and magnetic field of H = 9 T applied along the c-axis, we measured the carrier density as a function of temperature between T = 2 and 200 K. The result shows that the carrier density also exhibits a qualitative decrease below Tv ≈ 60 K, consistent with the introduction of more hole-like states in the carrier volume. This simultaneous increase in hole carriers and qualitative change in R(T) shows that the resistivity “upturn” in Fig. 4g is most likely associated with an increased number of scattering channels below Tv due to the presence of more Sm2+ states. Therefore, bulk resistivity measurements provide evidence for a valence crossover in SmCoIn5 at Tv.

Valence crossover observed with resonant ARPES

To further show that Tv ≈ 60 K is associated with hybridization of the Sm f electrons with conduction states at the Fermi level, we measured the position and intensity of the Sm3+ and Sm2+ configurations in the valence band using resonant angle-resolved photoemission (Res-ARPES) at the Sm M4,5 edges. In the resonant process, photons may excite electrons from the Sm d-shell into the vacuum through an intermediate state in the valence band. Since the intermediate state is short lived, its energy width is large by the uncertainty principle, and therefore any available Sm f state around the valence band may be temporarily occupied. This means that Res-ARPES observes resonant emission at the positions of any multiplet satisfying the appropriate photon selection rules, regardless of whether it is thermodynamically occupied, generating a “fingerprint” of different multiplet bands that can be used to identify the binding energy of different valence states. Analysis using the Res-ARPES spectrum has been carried out previously in several studies of multi-f-electron systems, including YbCuIn451 and SmB623,51,52.

The Res-ARPES raw data at different temperatures is shown in Fig. 5a, with momentum-integrated cuts shown in Fig. 5b as a function of binding energy EB. The valence states of Sm are Sm3+ (4f5) and Sm2+ (4f6), which after emitting a photoelectron are in the 4f4 and 4f5 final state configurations, respectively, which are the states observed in the experiment. The multiplet spectrum from each valence shows the position of the initial state, because the ground multiplet of the final state is energetically degenerate with the initial state. In our DFT calculations of SmCoIn5, we find excellent agreement for the position of the Sm3+ states at EB ≈ 6 eV (see Supplementary Note 2 and Supplementary Fig. 2), with the Sm2+ states appearing close to the Fermi energy, which is in agreement with the Res-ARPES data. The intensity and “fingerprint” of multiplet emission lines for each valence state were tabulated by Gerken53 and also agree with our observations. Moreover, the resonant spectrum in SmCoIn5 at low temperature appears to be highly similar to that of intermediate-valence SmB652. In SmCoIn5, we find that the Sm3+ band maintains similar intensity as a function of temperature between T = 12 and 60 K, whereas the Sm2+ component gains intensity at low temperature, consistent with a change in valence state across the region below Tv ≈ 60 K. To further confirm the intermediate valence scenario at low temperature, in Fig. 5c we show x-ray photoemission spectroscopy measurements at T = 12 K, which also show unambiguously the existence of the Sm2+ state using an incident energy of 1400 eV, similar to the analysis of SmOs4Sb12 data carried out by Yamasaki et al.29.

The energy difference observed between the Sm2+ component and the Fermi energy EF observed in our Res-ARPES spectra raises the interesting possibility that some of the atomic multiplets of Sm play a role in the efficiency of the microscopic f electron delocalization mechanism in 115 materials. While the appearance of the Sm2+ level near EB = 0.4 eV, marked by a dashed green line in Fig. 5b, might be explained by a surface-core shift similar to SmSn331 and SmRh2Si233, we also cannot discount the possibility that a multiplet level of Sm that is not the ground state participates in the Abrikosov-Suhl resonance process. In the Kondo effect, the width of the Kondo resonance is an indicator of the separation between the Fermi energy and the localized magnetic state. In CeCoIn5, CeRhIn5, and CeIrIn5, the width of the resonance indicates the Ce states to be less than 5 meV from EF39, while in SmB6, the Sm2+ states were shown to be 16 meV below EF54. These results suggest that the width of the Kondo resonance in rare earth intermetallics is of order meV, far smaller than the separation of ~6 eV in SmCoIn5 would allow for hybridization to proceed through the ground state of Sm3+. However, the Coulomb repulsion between mutual f electrons in Sm3+ distribute the \(\left(\begin{array}{c}14\\ 5\end{array}\right)\) = 2002 f levels over more than 16 eV, as diagrammed in Fig. 2d, and the distribution of \(\left(\begin{array}{c}14\\ 6\end{array}\right)\) = 3003 Sm2+ multiplet states is similarly large. This fact suggests that a multiplet level of Sm could be found within 10–15 meV of EF and therefore have a much larger Kondo hybridization matrix element compared to the actual crystal field ground state wavefunction found in this work. This opens the intriguing possibility of multiplet structure participating in the Kondo effect at EF.

Multiplet-selective hybridization between rare earths and metallic bands has been observed in other materials. In EuNi2P2, multiplets of the final-state 4f6 configuration were directly observed to hybridize selectively with the valence band structure55. In SmB6, many-body correlations can explain the formation of the indirect band gap that leads to a topological insulating state, with evidence for formation of isolated multiplet levels in one part of the spectral function and a single quasiparticle band in a different part15. In SmSn3, high-resolution Res-ARPES combined with DFT predicted a shift of 0.13 eV of the ground multiplet with respect to EF that positions a higher multiplet directly at EF31; however, this result was also dependent on some particular details of the calculations33. Perhaps most similarly to our work, a recent inelastic x-ray scattering experiment, combined with DMFT, suggested that the Kondo interaction proceeds in CePd3 by shifting part of the rare earth spectral weight across the Fermi energy56. Therefore, multiplet hybridization effects may be important for the electronic structure and Kondo interaction, and our observation of a shift of the Sm2+ state in SmCoIn5 suggests that this possibility should be further explored. Given the large number of Sm-based compounds with heavy fermion and/or intermediate valence characteristics37, it is likely that multiplet effects play a more frequent role than has been previously reported.

Conclusions

In summary, we identified thermal isolation of the ground state crystal field doublet as a driving mechanism for f electron delocalization in SmCoIn5, a close relative of the unconventional superconductor CeCoIn5. At low temperatures, the superposition of Sm valence states resembles a Kondo scenario that also hosts charge fluctuations between the Sm3+ and Sm2+ configurations. The delocalization begins when the temperature is low enough that the shape of the single-ion Sm wavefunction becomes more box-like compared to a filled shell containing thermally populated excited states. At the same temperature, we begin to see magnetism appear in the bulk properties, suggesting a form of the hybridization that involves a magnetic degree of freedom. In this way, we connect delocalized itinerant band states to the single-ion properties of Sm3+ through a generalization of the Kondo effect to the multi-f-electron systems. Finally, since the single-ion properties and crystal field ground state of CeCoIn5 are similar to those we found in SmCoIn5, similar charge fluctuations may also be present in the superconducting dome of CeCoIn5. However, in SmCoIn5, the ground state is antiferromagnetically ordered38, indicating that the Kondo screening is not sufficient to form a Kondo lattice, and we do not observe any signatures of Kondo coherence. Our results therefore show that SmCoIn5 is in a hybrid region close to the quantum critical point containing aspects from both sides, where the microscopic ingredients for f delocalization are present, but the magnetic RKKY exchange nevertheless becomes dominant.

Methods

Sample growth

Single crystals of SmCoIn5 and SmIrIn5 were grown by a molten In flux method. For SmCoIn5, the starting materials were Sm (99%, Goodfellow Cambridge Ltd.), Co (99.9+%, Alfa Aesar), and In (99.9999%, Alfa Aesar). Sm (1.078 g, 0.717 mmol) and Co (0.4223 g, 0.717 mmol) were reacted together by means of arc melting method with negligible weight loss (less than 0.15%). In a helium-filled glovebox, the resulting materials were placed into a 5 ml alumina Canfield crucible57 and mixed with In (31.999 g, 27.869 mmol). The crucible set was placed in a quartz ampule, evacuated, backfilled with ~100 mbar of Ar, and sealed in a quartz ampoule. The specimen was heated up to 1150 °C, with a rate 400 °C/h, and annealed at that temperature for 1 h. Annealing at 1150 °C was assisted by motorized rotation of the ampoule for better homogenization. Subsequently, the sample was cooled down to 800 °C with a rate of 400 °C/h, and further cooled down to 350 °C with a rate of 1 °C/h. After the growth step, the excess In flux was separated from the single crystals using a centrifuge. For SmIrIn5, the starting materials were 0.820 g (0.545 mmol) of Sm rod; 1.048 g (0.545 mmol) of Ir powder (2N5+; chemPUR), and 33.531 g (29.204 mmol) of In. The starting materials were placed directly into the crucible without the arc melting step. During the growth, the specimen was held at 1150 °C for 3 h, and cooled to 450 °C with a rate of 5 °C/h for the centrifugation. For both materials, the remaining In flux was dissolved in aqueous (37%) HCl. The sample composition was confirmed by x-ray fluorescence spectroscopy and powder x-ray diffraction. The SmCoIn5 samples exhibited a plate-like habit corresponding to the basal plane of the crystal, while SmIrIn5 crystals appeared to be more block-like.

XAS measurements

X-ray absorption spectroscopy (XAS) experiments were carried out at the X-Treme beamline at the Swiss Light Source, PSI, Switzerland58. Samples were mounted on a copper plate using silver paint, including drops of silver paint connecting the copper to the top surface of the sample, ensuring excellent electrical contract with the area exposed to the beam. To ensure consistent sample preparation, we used three methods and tested all of them for differences: first, we prepared as-grown samples with no surface preparation; second, we cleaved samples before introducing the samples into the vacuum chamber; third, we cleaved the samples in a helium glove box before transferring them into the experimental chamber. We collected XAS scans on all three samples and searched for changes in the spectra under these conditions, or a reduction of the intensity, but we found that all three were equivalent; therefore, we rule out any surface-specific effects in our data. The samples were mounted vertically on a rotating cold finger with the horizontal plane corresponding to the (100) × (001) plane of the crystal axes. To measure the x-ray linear dichroism (XLD), the samples were rotated into grazing incidence so that the beam direction changed by 60 degrees from the (001) axis toward the (100) axis, which coincidentally is very close (about 1 degree) to the (101) direction in reciprocal space. A diagram of the experimental configuration of the XAS experiments is shown in the Supplementary Fig. 5. In this way, horizontally polarized light probes a component in the (001) direction which is the sine of the rotation angle. Additional experiments of the x-ray magnetic circular dichroism (XMCD) were carried out at the same grazing incidence, with the magnetic field of up to 6.8 T applied in the direction of the beam. To analyze the XAS experimental data, the scans were normalized such that the height of the largest edge (M5 or L3), averaged between the polarizations, was 1 relative to the value of the scan before the pre-edge.

Single-ion multiplet calculations

Full multiplet calculations of Sm, including magnetic susceptibility and XLD patterns at the Sm M4,5 edges, were performed with the software package QUANTY42, similar to the calculations presented by Sundermann et al.59. The atomic parameters were taken from the Crispy interface for QUANTY60 which reproduces the values from the Cowan code, and the values were reduced to 80% in order to reproduce our experimental results, which is a typical value59. The values of spin-orbit coupling were found to be 180 meV for the Sm 4f states, 200 meV for the Sm 4f states in the core-hole state, and 10.51 eV for the Sm 3d states. Gaussian broadening of 0.05 eV was applied to the spectra to simulate the experimental conditions.

Magnetization measurements

Bulk magnetization measurements were carried out in a Quantum Design MPMS3 vibrating sample magnetometer (VSM). For measurements with the field in the basal plane of the single crystal, the sample was attached to a quartz sample paddle with GE varnish and wrapped with teflon tape. For measurements with the field along the (001) axis, the samples were mounted between two quartz cylinders inside a brass straw.

Inelastic neutron scattering measurements

Inelastic neutron scattering was carried out with a 1.8 g powder sample of SmIrIn5 at the EIGER thermal triple-axis spectrometer, PSI, Switzerland. Because of the extremely strong thermal neutron absorption resonance in natural Sm, we could not mount the samples in a standard powder can. Therefore, we mixed the powder with several drops of Cytop hydrogen-free glue and spread it as evenly as possible on a 9 cm × 4 cm aluminum plate of thickness 0.5 mm. The aluminum plate was mounted in the spectrometer with its long axis at a fixed angle to the incoming beam, as the outgoing beam angle was varied in the process of conducting energy scans at constant scattering wavevector Q. For each energy scan at constant Q, we fixed the sample angle at a single optimal position for the entire scan, to avoid having the long direction of the plate coincident with either the incoming and outgoing beam, which would lead to very large neutron absorption.

X-ray diffraction measurements

Temperature-dependent lattice parameter measurements were carried out in the cryostat at the MS beamline at the Swiss Light Source, PSI, Switzerland. Powdered SmCoIn5 was prepared and diluted with cornstarch by about 50% by volume in order to ensure constant illumination, then placed into a quartz capillary of diameter 0.3 mm. We used a beam energy E = 22 keV (λ = 0.563564 Å), which is below the In absorption edge near 27 keV. The wavelength was calibrated using a silicon standard, while starting u,v,w parameters used in the refinements were determined using the LaB6 NIST660A standard. The temperature in the cryostat was varied between 4 and 300 K. Refinements were carried out using the Fullprof software package and found to agree with the expected lattice structure. In Fullprof, an absorption correction μR = 0.9 was applied, and we used X = 0.47 and Y = 0.00017. We found evidence for a secondary phase of Sm2CoIn8 which was 2.9% by volume, which we assume comes from small crystallites on the edges of the SmCoIn5 single crystals (see Supplementary Note 5 and Supplementary Fig. 8).

ARPES measurements

Angle-resolved photoemission spectroscopy (ARPES) experiments were carried out at the soft-x-ray endstation61 of the ADRESS beamline62 at the Swiss Light Source, PSI, Switzerland. Single crystals were mounted using the natural basal plane on copper plates using high-strength H20E silver epoxy. Posts made of small stainless steel screws were attached onto the top surfaces of the samples with Torr-Seal epoxy. The samples were cleaved in ultrahigh vacuum of about 10−10 mbar. We used circularly polarized light with a beam spot of about 60 × 100 μm on the sample. We determined the coupling of beam energy to the out-of-plane kz direction by scanning the beam energy over hν = 480–700 eV, and we identified hν = 602 eV as a scan through the Γ point and hν = 564 eV as a scan through Z. Maps were then collected at these energies to determine the Fermi surface and valence band structure, where we found broad similarity to the published results of CeCoIn55. We also conducted XPS measurements (Fig. 4c) as well as resonant partial spectra at the Sm M4,5 edges (Fig. 5b).

DFT calculations

All the density functional theory calculations were carried out using the Quantum ESPRESSO package63,64,65. Exchange and correlation effects were modeled using the PBEsol functional66, augmented by a Hubbard U term to better describe the physics of the localized Sm 4f electrons. The scalar-relativistic pseudopotentials are taken from the SSSP library67. To sample the Brillouin zone, we used a 5 × 5 × 5 Monkhorst-Pack grid; a grid twice as fine was used for the calculation of the projected density of states. The wave-functions (charge densities and potentials) were expanded using a kinetic energy cutoff of 50 Ry (400 Ry). We used the experimental geometry66,67 throughout this work. The values of U used in this work (U = 6.03 eV and U = 6.26 eV for Sm3+ and Sm2+ calculations, respectively) were computed fully ab-initio by using the linear response approach68 as implemented by Timrov et al.69,70. In order to force the self-consistent-field calculations to converge to the Sm3+ solution, we constrained the number of electrons in the Sm f shell to be 5. The data used to produce the simulation results presented in this work are available at the Materials Cloud Archive71.

Transport measurements

Transport measurements were performed in a Quantum Design PPMS equipped with a 9 T magnet and a sample rotator. The sample was mounted on a sapphire plate with GE varnish, and leads were painted on by hand in a Hall bar geometry with current always flowing along the (100) axis. Aluminum wires were then attached to the paint leads by wire bonding. For the Hall effect measurements, we oriented the magnetic field along the (001) direction, we used a spear-shaped single crystal which was longer along the (100) axis compared to the (010) axis, ensuring a nearly Hall-bar configuration.