Resonant inelastic X-ray scattering as a probe of Jeff = 1/2 state in 3d transition-metal oxide

The state with effective total moment Jeff = 1/2 stabilized by the spin-orbit coupling is known to suppress Jahn-Teller distortions and may induce a strong exchange anisotropy. This in turn may lead to the formation of an elusive spin-liquid state in real materials. While recent studies have demonstrated that such a situation can be realized in 3d transition-metal compounds such as those based on Co2+ and Cu2+, diagnosis of Jeff = 1/2 state remains challenging. We show that resonant inelastic X-ray scattering is an effective tool to probe this state and apply it to CuAl2O4, material where Cu2+ ions were previously proposed to be in the Jeff = 1/2 state. Our results unambiguously demonstrate that, contrary to previous expectations, a competitive (to Jeff = 1/2) Jahn-Teller state realizes in this compound.


INTRODUCTION
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 physics 1,2 . 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 liquid 3,4 . Since the SOC constant is large for heavy elements 5 , the investigations were initially concentrated on the 4d and 5d transition-metal compounds such as α − RuCl 3 and (Li,Na) 2 IrO 3 4,6,7 . 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 J eff = 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 Co 2+ ions having octahedral coordination or Cu 2+ 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 state [10][11][12][13] . Strong exchange anisotropy was also predicted for Cu 2+ ions occupying A sites in spinels such as CuAl 2 O 4 , if the ground state is characterized by J eff = 1/2 14,15 . 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 J eff = 1/2 states, which may lead to anisotropic exchange interactions. Applying this technique to CuAl 2 O 4 , we show that the vibronic coupling suppresses the formation of the J eff = 1/2 state in this material.
CuAl 2 O 4 belongs to the A-site spinel system, in which the Cu 2+ ions site at the center of the tetrahedral A-sites and the nonmagnetic Al 3+ ions are located at the center of octahedral B-sites. The 3d states of tetrahedral Cu 2+ are split into t 2 and e states because of the T d crystal field of the four oxygen ions, as illustrated in Fig. 1. An atomic t 5 2 configuration is therefore realized with a single hole in the upper t 2 manifold, making CuAl 2 O 4 a possible candidate of the spin-orbit Mott insulator because the on-site Coulomb interaction amplifies the effect of relativistic SOC 14,15,21 . The t 2 degeneracy can be lifted through the following two channels. In the presence of SOC, the triply degenerate t 2 states are split into states of effective total angular momenta J eff = 1/2 and 3/2 through the approximation of effective orbital angular momentum L eff = 1 for t 2 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 T d 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 CuAl 2 O 4 is a spin-orbital-entangled J eff = 1/2 Mott insulator 14,15 . X-ray and neutron diffraction results also show that the crystal structure of CuAl 2 O 4 at ambient pressure is in cubic phase without evidence of tetragonal distortion 22 . 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 CuAl 2 O 4 , where about 30% of Cu 2+ ions occupy the octahedral sites 23,24 . To unravel the ground state of CuAl 2 O 4 , 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 1 of local Jahn-Teller distortion in the tetrahedral sites, in contrast to the scenario of spin-orbital entanglement.

RESULTS
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 Cu 2+ is in the pure J eff = 1/ 2 ground state, the L 2 edge is forbidden due to the dipole selection rule, i.e., the transition from 2p 1/2 to J eff = 1/2 is not allowed 25,26 Consistent with recent XAS results of single-crystal CuAl 2 O 4 22 , our data show that the L 3 -edge XAS contains two distinct features; they arise from the transition to the unoccupied 3d states of tetrahedral and octahedral Cu 2+ . In addition, we observed nonvanishing Cu L 2 -edge XAS intensity, implying the existence of octahedral Cu 2+ or tetrahedral Cu 2+ which has a ground state with a Jahn-Teller distortion. Although previous XAS study concluded that the L 2 XAS intensity solely originates from the octahedral Cu 2+ site 22 , the measured XAS can also be explained by the scenario of the coexistence of octahedral Cu 2+ and tetrahedral Cu 2+ of a spin-half ground state. In other words, whether CuAl 2 O 4 is a J eff = 1/2 Mott insulator remains an open question. To resolve this issue, we resort to RIXS measurements to  separate the contribution of octahedral Cu 2+ to the L 2 absorption from that of tetrahedral sites by examining the incident-energy dependence of Cu L 2 -edge RIXS.
Resonant inelastic X-ray scattering RIXS has been proved to be a powerful probe of crystal-field excitation from different site symmetry 27 . 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   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 Cu 2+ . 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 Cu 2+ . Interestingly, similar features occur at Cu L 2 -edge RIXS map as well. The Cu L 2 -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 L 2 -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 L 3 -edge XAS spectrum.
To understand the electronic structure of CuAl 2 O 4 , we analyzed RIXS data through crystal-field multiplet calculations. The RIXS spectra measured at the L 3 resonance energy of T d Cu 2+ , i.e., 930 eV, were compared with the atomic multiplet calculations of a single Cu 2+ ion in the crystal field produced by four ligands O 2− , 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 Cu 2+ are mainly composed of the hole transitions within the t 2 states and those from the t 2 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 Δt 2 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 Δt 2 = 270 ± 50 meV. The CuO 4 tetrahedron was found to be slightly compressed along the z axis, lifting the orbital degeneracy of d xy , d yz , d zx states. The Cu L 3edge RIXS spectra provide explicit spectroscopy evidence for the local distortion in CuAl 2 O 4 . Furthermore, the calculated RIXS intensity at the L 2 edge nearly vanishes when no local distortion was included (dashed gray curves), in contrast to the calculations with CuO 4 structural distortions (black curves). On the other hand, the RIXS spectra measured at the resonance of O h Cu L-edge show only a single structure, indicating a small distortion in the Cu 2+ octahedron. The calculated RIXS spectra of O h Cu sites with 10Dq = 1.6 eV reproduce the experimental data measured at 930.8 eV and 950.2 eV (Fig. 3(b)).
Our results indicate that in CuAl 2 O 4 , an expected spin-orbit Mott insulator, the T d Cu site is locally compressed. Through comparing the measured RIXS intensities with calculations of compressed CuO 4 tetrahedron and CuO 6 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 J eff = 1/2 state admixes only perturbatively. And then at some critical SOC, we have a drastic transition to J eff = 1/2, see Fig. 6 in Ref. 28 . Thus, at small SOC as in Cu, one would expect a small admixture of J eff = 1/ 2, to the ground state. In fact, the distorted ground state from our simulations based on RIXS data has 92% overlap with the d xy 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 CuAl 2 O 4 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 L 2 -edge of the J eff = 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.
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 J eff = 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 J eff = 1/2 state, and our experimental result shows that most probably this idealized situation is not realized in CuAl 2 O 4 .

DISCUSSION
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 J eff = 1/2 state in 3d transition-metal oxides. Being applied to spin-orbit candidate material CuAl 2 O 4 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 t 2 orbitals are Δe = 50 meV and Δt 2 = 270 meV, respectively. These results suggest that the T d Cu site is locally compressed. Neither neutron powder diffraction 23 nor single-crystal X-ray diffraction 29 studies observed anomalous atomic displacement in CuAl 2 O 4 , but total scattering measurements like pairdistribution-function analysis will be an excellent probe to study the local structure.

Sample synthesis
A stoichiometric mixture of Al 2 O 3 (99.9%) and CuO (99.9%) was used for the synthesis of CuAl 2 O 4 . 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-2000 30 . The sample was single-phase with sharp reflections. The distribution of Cu 2+ cations between the tetrahedral 8a site and Fig. 5 Total energies as obtained in DFT + U + SOC with two different implementations of the + U term (black curve -spinpolarization is allowed only in "U" part, so-called LDA + U; redspin polarization is possible in both DFT and "U" parts, LSDA + U). Minimum at c/a = 1 corresponds to SOC J eff = 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.
H.Y. Huang et al. 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 Source 31 . 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°.