Electron configuration of uranium in UBi 2 and USb 2

Unlike the case with stronger ligands such as oxygen and chlorine, there is no unambiguously favored effective valence picture for uranium pnictides. Density functional theory suggests that the charge and spin density on uranium are significantly modified by itinerancy effects14,15 (see also Supplementary Note 1), as we will discuss in the analysis below, making it difficult to address this question from secondary characteristics such as the local or ordered moment. However, analyses in 2014–2016 have shown that resonant fine structure at the O-edge (5d→5 f transition) provides a distinctive fingerprint for identifying the nominal valence state and electronic multiplet symmetry on uranium16,17,18,19. X-ray absorption spectra (XAS) of UBi 2 and USb 2 were measured by the total electron yield (TEY) method, revealing curves that are superficially similar but quantitatively quite different (Fig. 2a). Both curves have prominent resonance features at hυ~100 and ~113 eV that are easily recognized as the ‘R1’ and ‘R2’ resonances split by the G-series Slater integrals16. Within models, these resonances are narrowest and most distinct for 5 f0 systems, and merge as 5f electron number increases, becoming difficult to distinguish beyond 5f2 (see Fig. 2a (bottom) simulations). The USb 2 sample shows absorption features that closely match the absorption curve of URu 2 Si 2 16, and are associated with the J = 4 ground states of a 5f2 multiplet. This correspondence can be drawn with little ambiguity by noting a one-to-one feature correspondence with the fine structure present in a second derivative analysis (SDI, see Fig. 2b).

Fig. 2 XAS fine structure and valence of UBi 2 and USb 2 . a The x-ray absorption of UBi 2 and USb 2 on the O-edge of uranium is compared with (bottom) multiplets simulations for 5 f1 (U5+), 5f2 (U4+), and 5f3 (U3+). b A negative second derivative (SDI) of the XAS data and simulated curves, with drop-lines showing feature correspondence. Noise in the SDI has an amplitude comparable to the plotted line thickness, and all features identified with drop-lines were consistently reproducible when moving the beam spot. Prominent absorption features are labeled peak-A (UBi 2 , hυ = 99.2 eV), peak-B (USb 2 , hυ = 98.2 eV), and peak-C (USb 2 , hυ = 100.8 eV). Source data are provided as a Source Data file Full size image

The R1 and R2 resonances of UBi 2 are more broadly separated than in USb 2 , and the lower energy R1 feature of UBi 2 is missing the prominent leading edge peak at hυ~98.2 eV (peak-B), which is a characteristic feature of 5f2 uranium16,17. The UBi 2 spectrum shows relatively little intensity between R1 and R2, and the higher energy R2 resonance has a much sharper intensity onset. All of these features are closely consistent with expectations for a 5f1 multiplet, and the SDI curve in Fig. 2b reveals that the R1 fine structure of UBi 2 is a one-to-one match for the 5f1 multiplet. We note that a close analysis is not performed for R2 as it is influenced by strong Fano interference (see Supplementary Note 2). The lack of prominent 5f2 multiplet features suggests that the 5f1 multiplet state is quite pure, and the measurement penetration depth of several nanometers (see Methods) makes it unlikely that this distinction between UHV-cleaved UBi 2 and USb 2 originates from surface effects. However, the picture for UBi 2 is complicated by a very rough cleaved surface, which our STM measurements (see Supplementary Note 3) find to incorporate at least two non-parallel cleavage planes. Surface oxidation in similar compounds is generally associated with the formation of UO 2 (5f2) and does not directly explain the observation of a 5f1 state.

We note that even with a clean attribution of multiplet symmetries, it is not at all clear how different the f-orbital occupancy will be for these materials, or what magnetic moment should be expected when the single-site multiplet picture is modified by band-structure-like itinerancy10,11 (see also Supplementary Note 1). The effective multiplet states identified by shallow-core-level spectroscopy represent the coherent multiplet (or angular moment) state on the scattering site and its surrounding ligands, but are relatively insensitive to the degree of charge transfer from the ligands20.

Nonetheless, the 5f1 and 5f2 nominal valence scenarios have very different physical implications. A 5f1 nominal valence state does not incorporate multi-electron Hund’s rule physics21,22 (same-site multi-electron spin alignment), and must be magnetically polarizable with non-zero pseudospin in the paramagnetic state due to Kramer’s degeneracy (pseudospin ½ for the UBi 2 crystal structure). By contrast in the 5f2 case one expects to have a Hund’s metal with strong alignment of the 2-electron moment (see dynamical mean field theory (DFT + DMFT) simulation below), and the relatively low symmetry of the 9-fold ligand coordination around uranium strongly favors a non-magnetic singlet crystal electric field (CEF) ground state with Γ 1 symmetry, gapped from other CEF states by roughly 1/3 the total spread of state energies in the CEF basis (see Table 1). The Γ 1 state contains equal components of diametrically opposed large-moment |m J = + 4 > and |m J = -4 > states, and is poised with no net moment by the combination of spin-orbit and CEF interactions. This unusual scenario in which magnetic phenomena emerge in spite of a non-magnetic singlet ground state has been considered in the context of mean-field models23,24,25,26, and appears to be realized at quite low temperatures (typically T < ~10 K) in a handful of rare earth compounds. The resulting magnetic phases are achieved by partially occupying low-lying magnetic excited states, and have been characterized as spin exciton condensates23.

Table 1 The CEF energy hierarchy in USb 2 Full size table

Multiplet symmetry from XLD versus temperature

To address the role of low-lying spin excitations, it is useful to investigate the interplay between magnetism and the occupied multiplet symmetries by measuring the polarization-resolved XAS spectrum as a function of temperature beneath the magnetic transition. Measurements were performed with linear polarization set to horizontal (LH, near z-axis) and vertical (LV, a–b plane) configurations. In the case of UBi 2 , the XAS spectrum shows little change as a function of temperature from 15 to 210K (Fig. 3a, b), and temperature dependence in the dichroic difference (XLD, Fig. 3b) between these linear polarizations is inconclusive, being dominated by noise from the data normalization process (see Methods and Supplementary Note 4). This lack of temperature dependent XLD is consistent with conventional magnetism from a doublet ground state. The XLD matrix elements do not distinguish between the up- and down-moment states of a Kramers doublet, and so strong XLD is only expected if the magnetic phase incorporates higher energy multiplet symmetries associated with excitations in the paramagnetic state.

Fig. 3 Temperature dependence of occupied f-electron symmetries. a The R1 XAS spectrum of UBi 2 is shown for linear horizontal (LH) and vertical (LV) polarizations. b The dichroic difference (LH-LV) is shown with temperature distinguished by a rainbow color order (15K (purple), 40K (blue), 80K (green), 120K (yellow), and 210K (red)). c, d Analogous spectra are shown for USb 2 . Arrows in d show the monotonic trend direction on the peak-B and peak-C resonances as temperature increases. e, f Simulations for 5f2 with mean-field magnetic interactions. g A summary of the linear dichroic difference on the primary XAS resonances of USb 2 , as a percentage of total XAS intensity at the indicated resonance energy (hυ = 98.2 eV for peak-B, and hυ = 100.8 eV for peak-C). Error bars represent a rough upper bound on the error introduced by curve normalization. h The linear dichroic difference trends from the mean field model. Source data are provided as a Source Data file. Shading in g, h indicates the onset of a magnetic ordered moment Full size image

By contrast, the temperature dependence of USb 2 shows a large monotonic progression (Fig. 3c, d), suggesting that the atomic symmetry changes significantly in the magnetic phase. The primary absorption peak (hυ~98.2 eV, peak-B) is more pronounced under the LH-polarization at low-temperature, and gradually flattens as temperature increases. The LV polarized spectrum shows the opposite trend, with a sharper peak-B feature visible at high temperature, and a less leading edge intensity at low temperature. This contrasting trend is visible in the temperature dependent XLD in Fig. 3d, as is a monotonic progression with the opposite sign at peak-C (hυ~100.8 eV).

Augmenting the atomic multiplet model for 5f2 uranium with mean-field magnetic exchange (AM + MF) aligned to match the T N ~203K phase transition (see Methods) results in the temperature dependent XAS trends shown in Fig. 3e. The temperature dependent changes in peak-B and peak-C in each linear dichroic curve match the sign of the trends seen in the experimental data, but occur with roughly twice the amplitude, as can be seen in Fig. 3d, f. No attempt is made to precisely match the T > 200K linear dichroism, as this is influenced by itinerant and Fano physics not considered in the model. The theoretical amplitude could easily be reduced by adding greater broadening on the energy loss axis or by fine tuning of the model (which has been avoided – see Methods). However, it is difficult to compensate for a factor of two, and the discrepancy is likely to represent a fundamental limitation of the non-itinerant mean field atomic multiplet model. Indeed, when the competition between local moment physics and electronic itinerancy is evaluated for USb 2 with dynamical mean field theory (DFT + DMFT), we find that the uranium site shows a non-negligible ~25% admixture of 5f1 and 5f3 configurations (Fig. 4a).

Fig. 4 Electronic symmetry convergence in USb 2 . a The partial multiplet state occupancy on uranium in USb 2 from DFT + DMFT numerics, with Hund-aligned symmetries highlighted in bold (3H 4 and 4I 9/2 ). b Temperature dependence of the partial occupancy of different multiplet states within a 5f2 mean field model. In spite of a magnetic transition above 200K, roughly 1/3rd of the ground state convergence occurs in the range from 30–100K. The labeled CEF symmetries are only fully accurate in the high temperature paramagnetic state. Beneath the Néel temperature, the Γ 1 ground state is magnetically polarized by admixture with Γ 2 . Shading indicates the onset of a magnetic ordered moment. c The ordered magnetic moment of (red circles) USb 2 and (black circles) UBi 2 from elastic neutron scattering. The mean field multiplet model for USb 2 is shown as a solid blue curve, and critical exponent trends near the phase transition are traced with dashed black lines representing m(T) = m max (1-T/T N )β. The USb 2 data are overlaid with a steep critical exponent trend of β = 0.19 indicating strong fluctuations, and the UBi 2 data are overlaid with the conventional 3D Ising critical exponent (β = 0.327). d The Néel temperature as a function of doping level in U 1-x Th x Sb 2 (red circles), and the simulated ordered moment in Bohr magnetons (renormalized to 62% as described in Methods; red-hot shading). Source data for all curves are provided as a Source Data file Full size image

Magnetic ordered moment and the nature of fluctuations

Compared with conventional magnetism, the singlet ground state provides a far richer environment for low temperature physics within the magnetic phase. In a conventional magnetic system, the energy gap between the ground state and next excited state grows monotonically as temperature is decreased beneath the transition, giving an increasingly inert many-body environment. However, in the case of singlet ground state magnetism, the ground state is difficult to magnetically polarize, causing the energy gap between the ground state and easily polarized excited states to shrink as temperature is lowered and the magnetic order parameter becomes stronger. Consequently, within the AM + MF model, many states keep significant partial occupancy down to T < 100K, and the first excited state (derived from the Γ 5 doublet) actually grows in partial occupancy beneath the phase transition (see Fig. 4b). Of the low energy CEF symmetries (tracked in Fig. 4b), Γ 5 and Γ 2 are of particular importance, as Γ 5 is a magnetically polarizable Ising doublet, and Γ 2 is a singlet state that can partner coherently with the Γ 1 ground state to yield a z-axis magnetic moment (see Supplementary Note 5). These non-ground-state crystal field symmetries retain a roughly 1/3rd of the total occupancy at T = 100K, suggesting that a heat capacity peak similar to a Schottky anomaly should appear at low temperature, as has been observed at T < ~50K in experiments (see the supplementary material of ref. 10). Alternatively, when intersite exchange effects are factored in, the shrinking energy gap between the Γ 1 and Γ 5 CEF states at low temperature will enable Kondo-like resonance physics and coherent exchange effects that are forbidden in conventional magnets.