Spin-orbit materials, i.e., systems in which physical properties are strongly affected by the spin-orbit coupling (SOC), undoubtedly became one of the central subjects in modern condensed matter physics1,2. In particular, it is essential for various topological effects and anisotropic exchange interaction in magnetic materials, which may result, e.g., in a mysterious Kitaev quantum spin liquid3,4. Since the SOC constant is large for heavy elements5, the investigations were initially concentrated on the 4d and 5d transition-metal compounds such as α − RuCl3 and (Li,Na)2IrO34,6,7. However, it was very recently shown that more conventional 3d oxides also can demonstrate similar behavior with the ground state characterized by an effective moment Jeff = 1/2 (New is often well overlooked old.)8,9.

Indeed, the spin-orbital entangled state can be realized, for example, in the case of Co2+ ions having octahedral coordination or Cu2+ ions surrounded by a ligand tetrahedron. Intensive ongoing studies of layered honeycomb cobaltites have demonstrated that anisotropic and bond-dependent exchange coupling can be sufficiently strong and may result in an elusive quantum spin liquid state10,11,12,13. Strong exchange anisotropy was also predicted for Cu2+ ions occupying A sites in spinels such as CuAl2O4, if the ground state is characterized by Jeff = 1/214,15. However, whether this situation realizes accurate 3d materials is an open question.

Resonant inelastic X-ray scattering (RIXS) was shown to be a powerful technique for studying Kitaev materials based on 4d and 5d transition metals. It can be used both to estimate physical parameters of a system, such as the crystal-field splitting or the SOC constant, and also anomalous magnetic excitation spectra of Kitaev systems, see, e.g.,16,17,18,19,20. In the present work we demonstrate that RIXS turns out to be a sensitive probe in the case of 3d transition-metal compounds and can discriminate between conventional S = 1/2 and spin-orbit Jeff = 1/2 states, which may lead to anisotropic exchange interactions. Applying this technique to CuAl2O4, we show that the vibronic coupling suppresses the formation of the Jeff = 1/2 state in this material.

CuAl2O4 belongs to the A-site spinel system, in which the Cu2+ ions site at the center of the tetrahedral A-sites and the nonmagnetic Al3+ ions are located at the center of octahedral B-sites. The 3d states of tetrahedral Cu2+ are split into t2 and e states because of the Td crystal field of the four oxygen ions, as illustrated in Fig. 1. An atomic \({t}_{2}^{5}\) configuration is therefore realized with a single hole in the upper t2 manifold, making CuAl2O4 a possible candidate of the spin-orbit Mott insulator because the on-site Coulomb interaction amplifies the effect of relativistic SOC14,15,21. The t2 degeneracy can be lifted through the following two channels. In the presence of SOC, the triply degenerate t2 states are split into states of effective total angular momenta Jeff = 1/2 and 3/2 through the approximation of effective orbital angular momentum Leff = 1 for t2 states. Hence the ground state would be a spin-orbital entangled state in the SOC limit. On the other hand, the degeneracy can be lifted due to the Jahn-Teller distortion, which lowers the Td symmetry of crystal field, resulting in a spin-half ground state with a quenched orbital angular momentum.

Recent theoretical studies based on first-principles calculations conclude that CuAl2O4 is a spin-orbital-entangled Jeff = 1/2 Mott insulator14,15. X-ray and neutron diffraction results also show that the crystal structure of CuAl2O4 at ambient pressure is in cubic phase without evidence of tetragonal distortion22. However, the breakdown of local symmetry induced by Jahn-Teller distortion can not be ruled out. Moreover, the diffraction data have shown a finite site-disorder in CuAl2O4, where about 30% of Cu2+ ions occupy the octahedral sites23,24. To unravel the ground state of CuAl2O4, we used Cu L-edge RIXS to investigate the electronic structure as RIXS is an element- and site-selective probe. In combining with multiplet calculations, our RIXS results demonstrate the existence of local Jahn-Teller distortion in the tetrahedral sites, in contrast to the scenario of spin-orbital entanglement.


X-ray absorption

L-edge X-ray absorption spectroscopy (XAS) is an effective tool to investigate the SOC in the ground state of transition-metal compounds because it probes the dipole transitions from 2p electrons to unoccupied d states. If the Cu2+ is in the pure Jeff = 1/2 ground state, the L2 edge is forbidden due to the dipole selection rule, i.e., the transition from 2p1/2 to Jeff = 1/2 is not allowed25,26. Figures 2(a) plots Cu L-edge XAS of CuAl2O4. Consistent with recent XAS results of single-crystal CuAl2O422, our data show that the L3-edge XAS contains two distinct features; they arise from the transition to the unoccupied 3d states of tetrahedral and octahedral Cu2+. In addition, we observed non-vanishing Cu L2-edge XAS intensity, implying the existence of octahedral Cu2+ or tetrahedral Cu2+ which has a ground state with a Jahn-Teller distortion. Although previous XAS study concluded that the L2 XAS intensity solely originates from the octahedral Cu2+ site22, the measured XAS can also be explained by the scenario of the coexistence of octahedral Cu2+ and tetrahedral Cu2+ of a spin-half ground state. In other words, whether CuAl2O4 is a Jeff = 1/2 Mott insulator remains an open question. To resolve this issue, we resort to RIXS measurements to separate the contribution of octahedral Cu2+ to the L2 absorption from that of tetrahedral sites by examining the incident-energy dependence of Cu L2-edge RIXS.

Fig. 1
figure 1

Schematic illustration on the t2 splitting of tetrahedral Cu2+ due to the Jahn-Teller effect (left) and spin-orbit coupling (right).

Fig. 2: RIXS intensity distribution maps of CuAl2O4 in the plane of energy loss vs. incident photon energy.
figure 2

a Cu L-edge XAS spectrum measured with total electron yield. b & c RIXS intensity maps measured around Cu L3 and L2 edges, respectively. The vertical red dashed lines indicate RIXS features from tetrahedral and octahedral Cu2+, corresponding to the absorption energies of the XAS spectrum shown in (a), i.e., 930 & 949.6 eV and 930.8 & 950.2 eV, respectively. All spectra were recorded at room temperature with π-polarized incident X-ray.

Resonant inelastic X-ray scattering

RIXS has been proved to be a powerful probe of crystal-field excitation from different site symmetry27. By selecting the incident photon to particular absorption energy, RIXS measures electronic excitations of d electrons with the site-specific orbital degree of freedom. Figure 2(b),(c) show the RIXS intensity maps of CuAl2O4 measured about Cu L3 and L2 edges, respectively. The scattering angle was fixed at 90. The L3-edge RIXS map shows two sets of distinct excitation structures which resonate at incident energies of 930 eV and 930.8 eV, respectively, indicating that these excitations arise from two Cu sites with different crystal field symmetries. Two excitations with energy loss centered at 0.3 eV and 0.82 eV were observed at the first excitation energy, resulting from the dd excitations of tetrahedral Cu2+. The excitation at the other incident energy shows only one broad peak at 1.52 eV and is best explained as the dd excitations of octahedral Cu2+. Interestingly, similar features occur at Cu L2-edge RIXS map as well. The Cu L2-edge RIXS intensity map displays two sets of excitations at incident energies of 949.6 eV and 950.2 eV, respectively. These experimental observations indicate that the Cu L2-edge XAS spectrum is composed of the 2p − to − 3d transition of two Cu sites with different crystal field symmetries, like those in the Cu L3-edge XAS spectrum.

To understand the electronic structure of CuAl2O4, we analyzed RIXS data through crystal-field multiplet calculations. The RIXS spectra measured at the L3 resonance energy of Td Cu2+, i.e., 930 eV, were compared with the atomic multiplet calculations of a single Cu2+ ion in the crystal field produced by four ligands O2−, as shown in Fig. 3(a). With the Hartree–Fock value of 3d SOC and a 30% reduction in Slater integrals, the calculations explain measured RIXS features well. The RIXS dd excitations of tetrahedral Cu2+ are mainly composed of the hole transitions within the t2 states and those from the t2 to the e states. The corresponding RIXS peaks are labeled A and B, respectively. The value of 10Dq determines the energy position of peak B, while the value of the Jahn-Teller splitting Δe controls the spectral line shape of B, as shown in Supplementary Fig. 3(a),(b). Similarly, Supplementary Fig. 3(c) shows that the Jahn-Teller splitting Δt2 dictates the energy position of peak A. Through the comparison of the RIXS data with calculations, we found that the crystal field parameters are: 10Dq = −0.72 ± 0.05 eV, Δe = 50 ± 20 meV, and Δt2 = 270 ± 50 meV. The CuO4 tetrahedron was found to be slightly compressed along the z axis, lifting the orbital degeneracy of dxy, dyz, dzx states. The Cu L3-edge RIXS spectra provide explicit spectroscopy evidence for the local distortion in CuAl2O4. Furthermore, the calculated RIXS intensity at the L2 edge nearly vanishes when no local distortion was included (dashed gray curves), in contrast to the calculations with CuO4 structural distortions (black curves). On the other hand, the RIXS spectra measured at the resonance of Oh Cu L-edge show only a single structure, indicating a small distortion in the Cu2+ octahedron. The calculated RIXS spectra of Oh Cu sites with 10Dq = 1.6 eV reproduce the experimental data measured at 930.8 eV and 950.2 eV (Fig. 3(b)).

Fig. 3: Cu L-edge RIXS spectra recorded at selected incident photon energies in comparison with multiplet calculations.
figure 3

a RIXS spectra measured with the incident photon energies tuned to the resonance of Td Cu2+ compared with calculated RIXS. The solid black lines plot the calculated spectra of Td Cu2+ with the Jahn-Teller distortion; the dashed gray lines are for those in a spin-orbital entangled state. The calculated RIXS intensity of Td Cu2+ in the Jeff = 1/2 state vanishes. See the main text for the discussion on the origin of peaks A and B. b Measured RIXS of Oh Cu2+ in comparision with calculations. The solid black lines plot the calculated spectra of Oh Cu2+ without a distortion. The incident photon energies Ein indicated in (a) and (b) correspond to the red dashed lines shown in Fig. 2. Both experimental and calculated RIXS spectra of Cu L2 edge are magnified by a factor of four.

Our results indicate that in CuAl2O4, an expected spin-orbit Mott insulator, the Td Cu site is locally compressed. Through comparing the measured RIXS intensities with calculations of compressed CuO4 tetrahedron and CuO6 octahedron, we obtained an amount of site disorder 37.5%, in line with the value obtained from diffraction results. These results indicate that the spin-orbital entangled ground state is destabilized against the Jahn-Teller distorted ground state. From a general perspective, when we start from S = 1/2 and increase the SOC strength, the Jeff = 1/2 state admixes only perturbatively. And then at some critical SOC, we have a drastic transition to Jeff = 1/2, see Fig. 6 in Ref. 28. Thus, at small SOC as in Cu, one would expect a small admixture of Jeff = 1/2, to the ground state. In fact, the distorted ground state from our simulations based on RIXS data has 92% overlap with the dxy orbital expected for the JT ground state. In addition, one can measure spectra of X-ray magnetic circular dichroism (XMCD) in Cu L-edge absorption to further examine the Jahn-Teller distorted ground state of CuAl2O4 by applying a high magnetic field. Figure 4 plots XMCD spectra simulated by using the electronic parameters from RIXS. Clearly, the XMCD spectral line shape at the L2-edge of the Jeff = 1/2 is significantly different from that of the Jahn-Teller state, as shown in Fig. 4(d). Also, the orbital moment the Jahn-Teller state is expected to be quenched; a future XMCD experimental study will be helpful for further clarification.

Fig. 4: Simulation of Cu L-edge XMCD.
figure 4

ac The calculated XAS and XMCD spectra of Cu2+ tetrahedron in the spin-orbit coupling (SO) scheme, Cu2+ tetrahedron in the Jahn-Teller (JT) coupling scheme and Cu2+ octahedron, respectively. The calculations are down with an external magnetic field 1 T alone the z direction, and the polarization of incident X-ray lies on the xy plane. d The thick green line is the calculated XMCD spectra from 37.5% Oh Cu2+ and 62.5% Td Cu2+ with Jahn-Teller distortion; the dashed black line is the result of 37.5% Oh Cu2+ and 62.5% spin-orbital entangled Td Cu2+.

The conclusion drawn from the RIXS data does not disagree with density functional theory (DFT) results of Ref. 15. It is shown in Ref. 15 that Jeff = 1/2 state can be realized in a narrow range of parameters used in the calculations. For example, the ground state changes if we change Hubbard U. Moreover, our calculations show that the choice of double counting also changes the ground state wavefunction. Figure 5 demonstrates that even the choice of DFT+U+SOC calculation scheme may affect the result. Only a delicate balance between Coulomb interaction and SOC stabilizes the Jeff = 1/2 state, and our experimental result shows that most probably this idealized situation is not realized in CuAl2O4.

Fig. 5: Total energies as obtained in DFT + U + SOC with two different implementations of the + U term (black curve - spin-polarization is allowed only in “U” part, so-called LDA + U; red - spin polarization is possible in both DFT and “U” parts, LSDA + U).
figure 5

Minimum at c/a = 1 corresponds to SOC Jeff = 1/2 state, while at c/a < 1 to Jahn-Teller S = 1/2. These results illustrate that the type of ground state strongly depends on the details of the DFT calculations.


The spin-orbit materials have become an important class of systems demonstrating exceptional physical properties defined by competition of various interactions such as strong electronic correlations, vibronic and SOCs, etc. Experimental diagnosis of their ground state is a challenging task, which, however, unravels mechanisms lying behind the physical effects observed in these materials. Using RIXS and crystal-field multiplet calculations, we demonstrate that this method is an effective probe of the spin-orbital entangled Jeff = 1/2 state in 3d transition-metal oxides. Being applied to spin-orbit candidate material CuAl2O4 it shows that, contrary to previous expectations, a competing Jahn-Teller configuration is stabilized in this material, and the tetragonal splittings of the e and t2 orbitals are Δe = 50 meV and Δt2 = 270 meV, respectively. These results suggest that the Td Cu site is locally compressed. Neither neutron powder diffraction23 nor single-crystal X-ray diffraction29 studies observed anomalous atomic displacement in CuAl2O4, but total scattering measurements like pair-distribution-function analysis will be an excellent probe to study the local structure.


Sample synthesis

A stoichiometric mixture of Al2O3 (99.9%) and CuO (99.9%) was used for the synthesis of CuAl2O4. The mixture was pressed into a pellet and then annealed at 1193 K for 84 h and 1293 K for 38 h (with several intermediate grindings) in the air on a Pt foil. X-ray powder diffraction data were measured at room temperature on a RIGAKU MiniFlex600 diffractometer using Cu Kα radiation (2θ range of 8–140°, a step width of 0.02°, and a scan speed of 1 deg/min). The X-ray data were analyzed by the Rietveld method using RIETAN-200030. The sample was single-phase with sharp reflections. The distribution of Cu2+ cations between the tetrahedral 8a site and octahedral 16d site was refined with a constraint on the total chemical composition. The experimental, calculated, and difference X-ray diffraction profiles and the main refinement results are shown in Supplementary Fig. 1.

XAS and RIXS measurements

All XAS and RIXS measurements were performed at the AGM-AGS spectrometer of beamline 41A at Taiwan Photon Source31. This AGM-AGS beamline is based on the energy compensation principle of grating dispersion. The energy bandwidth of incident X-ray was 314.5 meV while keeping the total energy resolution of RIXS at 90 meV. The sample was at room temperature during the measurements. Both XAS and RIXS measurements were carried out using a linear horizontally (π) polarized X-ray. The XAS spectrum was measured with a normal-incident X-ray in the total electron yield mode. For the RIXS measurement, the incidence angle was 45°, and the scattering angle was fixed at 90°.

Multiplet calculations

Simulations of RIXS, XAS, and XMCD spectra were performed with the full multiplet code through QUANTY, a script language to calculate many-body eigenenergy, XAS, and RIXS spectra32,33. The on-site Coulomb interaction, the crystal field, and the 2p and 3d SOC (ζ2p&ζ3d) were included in the calculations. The intra-atomic Coulomb interaction of 3d electrons is described by the radial part of the direct Coulomb interactions F2(3d, 3d) and F4(3d, 3d). The interaction between core-hole and 3d electrons is described by F2(2p, 3d) and exchange interactions G1(2p, 3d), G3(2p, 3d). The Hartree–Fock values of ζ2p (13.498 eV) and ζ3d (0.102 eV), and a scaling factor 70% for Slater integrals were used in the calculations. The calculations of tetrahedral and octahedral Cu2+ were conducted separately with crystal field of −0.72 eV and 1.55 eV, respectively. The calculation of RIXS and XAS spectra are an average of three possible geometries as described in Supplementary Fig. 2.

DFT calculations

DFT calculations were performed using VASP package34 and PBE type of the exchange-correlation functional35. We used 125 k-points for the Brillouin zone integration and chose U correction according to ref. 36 with Hubbard U = 7 eV37,38 and Hund’s exchange JH = 1 eV.