Site-specific electronic and magnetic excitations of the skyrmion material Cu2OSeO3

The manifestation of skyrmions in the Mott-insulator Cu2OSeO3 originates from a delicate balance between magnetic and electronic energy scales. As a result of these intertwined couplings, the two symmetry-inequivalent magnetic ions, Cu-I and Cu-II, bond into a spin S = 1 entangled tetrahedron. However, conceptualizing the unconventional properties of this material and the energy of the competing interactions is a challenging task due to the complexity of this system. Here we combine X-ray Absorption Spectroscopy and Resonant Inelastic X-ray Scattering to uncover the electronic and magnetic excitations of Cu2OSeO3 with site-specificity. We quantify the energies of the 3d crystal-field splitting for both Cu-I and Cu-II, fundamental for optimizing model Hamiltonians. Additionally, we unveil a site-specific magnetic mode, indicating that individual spin character is preserved within the entangled-tetrahedron picture. Our results thus provide experimental constraints for validating theories that describe the interactions of Cu2OSeO3, highlighting the site-selective capabilities of resonant spectroscopies. The complex interactions in spin spiral systems play an important role for the emergence of multiferroicity and topological magnetic order. Here, the authors investigate the magnetism associated with the two inequivalent Cu positions in the model system Cu2OSeO3, observing site-specific electronic structure and associated magnetic excitations.

S ince their first observation in magnetic solids 1,2 , skyrmionsnano-sized, topological spin objects-immediately attracted enormous interest, thanks to their unique mobility properties in response to low current and electric fields [3][4][5] . Skyrmions are consequently appealing for energy-efficient applications and their generation in insulators is furthermore attractive due to reduced heat dissipation and fast switching response. Cu 2 OSeO 3 is one of the few known Mott-insulators hosting skyrmions [6][7][8][9] . Normally, chiral, noncentrosymmetric, cubic magnetic materials are potential hosts for skyrmions 10 , but in the multiferroic Cu 2 OSeO 3 , the formation of such topological states furthermore arises from the delicate balance between the super-exchange couplings and the Dzyaloshinskii-Moriya (DM) interactions [11][12][13][14] .
While Cu 2 OSeO 3 shares the same P2 1 3 space group as the skyrmion-prototype MnSi 1 , the fundamental magnetic unit behind the skyrmion nucleation within the ferrimagnetic phase is believed to be a composite Cu 4 tetrahedron with an effective spin S = 1 11 and involving two differently coordinated Cu ions, Cu-I and Cu-II. Several studies [11][12][13]15 identified the microscopic interactions between the individual ions within the magnetic building block as crucial to unravel the quantum nature of the skyrmions in Cu 2 OSeO 3 and to explain the emergence of other unconventional phases 9,16,17 .
As most experimental works focused so far on site-averaged magnetic properties of Cu 2 OSeO 3 13,15,18,19 , the site-specific magnetic response and the local electronic structure of the Cu 3d valence states remain marginally understood 20,21 . Nonetheless, the latter is fundamentally related to the DM interactions-responsible for the helical order and the skyrmion formation 22 -as they are explained through multi-orbital and multisite hopping paths, involving both Cu-I and Cu-II. On the other hand, the site-specific magnetic response can elucidate the validity of the Cu 4 tetrahedron picture at the microscopic level, exploiting the single spin point of view.
Hence, the complex nature of Cu 2 OSeO 3 calls for site-specific magnetic and electronic investigations. These results will build prerequisite information for extracting all the interactions underlying the skyrmion generation in this system-by experimentally validating microscopic theoretical models-ultimately unveiling the real energy balance that stabilizes the skyrmion phase. Such information is crucial for future skyrmion applications, as it promises the possibility for designing optimized materials-i.e., with an extended skyrmion pocket in the magnetic phase diagram-where the key interactions can be tuned either by film thickness, electric field, pressure, or strain 6,8,23,24 .
Here, we combine X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) to unveil the electronic and magnetic excitations associated with the two inequivalent Cu sites in the ferrimagnetic phase of Cu 2 OSeO 3 (T C ≃ 57 K). Using density functional theory (DFT) and RIXS cross section calculations, we disentangled the resonant energies associated with Cu-I and Cu-II ions. Capitalizing on this finding, we determine the site-specific spectral fingerprints and extract the orbital symmetries and energies of the crystal-field split 3d levels for both Cu-I and Cu-II. Furthermore, we reveal a site-specificity of the medium-energy magnon mode around 35 meV. Our results thus provide an experimental constraint for theories aimed at quantifying the competing energy terms underlying the skyrmion formation. More broadly, this approach can be extended to the design of devices and heterostructures with improved skyrmion properties, highlighting the importance of resonant techniques when dealing with multisite complex systems.

Results and discussion
Soft X-ray resonant spectroscopies of Cu 2 OSeO 3 . Resonant spectroscopies with their chemical sensitivity provide unique advantages in the study of multisite compounds. RIXS is furthermore helpful when there is a need to study the valence electronic structure as it probes the charge-neutral, dipole-forbidden, ddexcitations of a system, enabling it to reconstruct its ground state energy levels [25][26][27][28][29][30] . In particular, for 3d elements, such excitations can be accessed using the L 3 edge resonance in the soft x-ray range, promoting electrons from the 2p 3/2 core states to the 3d valence states. In the present study on Cu 2 OSeO 3 , we use the Cu L 3 edge at 930 eV to probe the magnetic Cu ions. Figure 1a shows the crystallographic unit cell of Cu 2 OSeO 3 , containing 16 Cu atoms arranged in four tetrahedrons. Each tetrahedron (dashed orange line in Fig. 1a) consists of one Cu-I and three Cu-II ions. Below the ferrimagnetic ordering temperature T C , the Cu-I spin aligns anti-parallel to the Cu-II spins. As each Cu brings a spin momentum of 1/2, this configuration yields a total spin momentum of S = 1 for the tetrahedron unit. For the XAS and RIXS measurements, we prepared a single crystal with [100] surface normal. The sample orientation used throughout the experiment is displayed in the inset of Fig. 1b. Figure 1b presents the Cu L 3 XAS spectrum of Cu 2 OSeO 3 (blue line) acquired in terms of Total Fluorescence Yield (TFY), at T = 45 K. The line-shape and the peak at 930.9 eV are consistent with previous measurements 31 , although our interpretation differs from ref. 20 as explained later on in the text. Since a single Cu 3d 9 site hosts one hole, its XAS spectrum is expected to be a Lorentzian curve with a 2p core-hole lifetime-dominated Fig. 1 Overview of the crystal structure, experimental configuration, and Cu L 3 X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) data. a Cu 2 OSeO 3 unit cell (a = 8.925 Å). The orange dashed line highlights the magnetic tetrahedron unit. b Cu L 3 XAS spectrum in terms of total fluorescence yield (TFY), at T = 45 K. TFY * is the spectrum after correction for self-absorption and saturation effects. c RIXS energy map with linear color scale for the intensity, also at T = 45 K. The thin solid line displays the partial fluorescence yield (PFY) signal, obtained integrating the RIXS spectra up to 10 eV. The inset in panel b displays the sample orientation used for these measurements, with [100] and [01-1] axes lying within the scattering plane. Incoming π polarized x-rays were used for all measurements.
width of 0.3-0.5 eV 32,33 . While the enhanced width of~1.2 eV and the asymmetric line-shape suggest distinct contributions from the Cu-I and Cu-II sites, we underline that their energy splitting ΔE CuII-CuI should be intrinsically small. This is supported by TFY * (green dashed line), corrected TFY for self-absorption and saturation effects 34,35 , that still displays an asymmetric single-peaked line-shape with~1 eV large width.
No clear consensus has been reached so far on the value of ΔE CuII-CuI . Previous DFT+U calculations estimated ΔE CuII-CuI to be~0.2 eV 21,22 , while resonant x-ray scattering proposed ΔE CuII-CuI to be~2 eV 20 . Here, we quantify ΔE CuII-CuI and resolve the 3d electronic structure for each Cu site using RIXS at the Cu L 3 edge. The spectra acquired by varying the incident photon energy within the range~928-933 eV are plotted versus energy loss and gathered in a color map (see Fig. 1c). This displays two main features: the quasi-elastic line around zero-energy loss and a broad multipeaked structure between 1 and 2 eV. The latter excitations are interpreted as intra-site dd-excitations stemming from the local crystal field in agreement with ellipsometry data 21 and other RIXS measurements on Cu 2+ systems 26,27 .
The fine details of the dd-excitations are shown in Fig. 2a, where the RIXS spectra are presented as a vertical stack for increasing incident photon energy. While the overall intensity evolution of the spectral-weight is due to changing the photon energy across the absorption resonance, the shape evolution between 930.6 and 932 eV highlights the existence of different sets of crystal-field excitations. The excitations pattern originates from the presence of the two inequivalent Cu species, Cu-I and Cu-II, having different oxygen coordination. However, because of the small splitting between Cu-I and Cu-II energy levels as suggested by the XAS, it is necessary to resort to more involved data analysis for disentangling the site-specific crystal-field excitations and their resonant energies.
Resolving Cu-I and Cu-II electronic structure using DFT and single-ion calculations. Preliminary considerations of the orbital excitations of the two sites can be made by examining the distinct point group symmetries of the crystal field associated with Cu-I, approximately a trigonal bipyramid D 3h , and Cu-II, approximately a square pyramid C 4v 21,36 . These two symmetries naturally lead to different orbital arrangements 21 . The ground state (GS) orbitals can be identified by referring to the local Cartesian axis (see Fig. 2b) that minimizes the Cu-O distance (z 1 for Cu-I yielding a d z 2 GS, and x 2 /y 2 for Cu-II yielding a d x 2 Ày 2 GS), in agreement with refs. 21,22 . To obtain an estimate of the site-specific orbital character and energy of the crystal-field split 3d levels in Cu 2 OSeO 3 , we perform DFT combined with Wannier90 calculations. Details on the DFT part can be found in Supplementary Note 1. Tight binding (TB) HamiltoniansĤ CF CuÀI andĤ CF CuÀII consisting of 3d orbitals from four Cu-I sites and from twelve Cu-II sites are then formulated. From the calculated 3d energies, we can directly extract the crystalfield excitation energies for each Cu site and use them as a guide to fit the experimental data. Together with the schematics of Fig. 2b, we thus expect two dd-excitations for Cu-I, labeled as E1 and E3, and four dd-excitations for Cu-II, labeled as E2, E4, E5, and E6. Note that E0 corresponds to the zero-energy transition, obtained when the initial and final 3d levels coincide.
To address the Cu-I and Cu-II resonant energies as well as their respective orbital energies, we simulate XAS and RIXS spectra based on a single-atom model using the EDRIXS code 37 . Details are collected in Supplementary Note 2. The intensity of the calculated RIXS spectra accounts for all atoms within the unit cell, neglecting interference effects between them. Furthermore, the experimental geometry and the incoming polarization projections are included as well. By using the theoretical energies for the dd-excitations (see Supplementary Table 3) and the RIXS intensity obtained from the atomic model calculations, we can simulate the RIXS spectra associated with each Cu species, see Fig. 2c for Cu-I and Fig. 2d for Cu-II. We performed then a constrained fit where the integrated intensity of each dd transition is fixed to the theoretical model while their energy and width are allowed to vary. As detailed in the Supplementary Note 3, we use the following model to fit the data: where A CuI/II are the site-specific amplitudes at each incident energy andĨ CuI=II RIXS are the site-specific area-normalized theoretical RIXS spectra.
The fitted spectra are presented in Figs. 3a-p, where each panel refers to specific incident energy and displays the raw data (open dots), the fitted Gaussian components for each dd-excitations (solid filling in aquamarine color for Cu-I and purple color for Cu-II), and their sum (pink solid line). Figure 3q-r summarize the fitted dd-excitation center positions, corresponding to the E1-E6 values. Notably, the extracted peak positions are reasonably constant across the scanned incident energy range, validating the reliability of the fitting model, while their averages and errors are summarized in Table 1.
From these results, we conclude that the experimental ddexcitation energies are well reproduced by the eigenvalues of thê H CF CuÀI andĤ CF CuÀII Hamiltonians (within 10-20%), and confirm that the effective D 3h symmetry for Cu-I ion and the distorted C 4v symmetry for Cu-II ion are good approximations for the real material despite small distortions from these idealized symmetries. Furthermore, by plotting the A CuI and A CuII amplitudes as a function of the incident photon energy, we obtain the experimental resonant profiles for Cu-I and Cu-II sites (see Fig. 3s). By summing the A CuI and A CuII amplitudes across the Cu L 3 edge (pink dots in Fig. 3s) we can well reproduce the XAS of Cu 2 OSeO 3 (blue solid line): this good agreement corroborates the consistency of the analysis. After extracting the respective maxima position from the A CuI (~930.6 eV) and A CuII (~931.15 eV) profiles, we can additionally quantify ΔE CuII-CuĨ 0.55 ± 0.05 eV, as the energy difference between the Cu-I and Cu-II resonances. This value is a bit larger than our theoretical estimate (~0.33 eV, refer to Supplementary Note 1) and previous DFT+U work 21,22 , while it strongly differs from the value reported in ref. 20 ,~2 eV. Our result finally legitimates the approximation of neglecting interference effects between the two inequivalent Cu species, since ΔE CuII-CuI ≳ Γ CuL .
Site-dependent magnetic excitations. The magnetic properties of Cu 2 OSeO 3 , as well as the magnon modes, have been explained so far as emanating from the effective S = 1 Cu 4 tetrahedra [11][12][13]15,18 , rather than from individual Cu spins, yielding the definition of "entangled-tetrahedron ground state". However, some discrepancies between the Cu 4 tetrahedra model and the spin excitations have been reported in ref. 38 . Benefiting from the unique site sensitivity offered by RIXS as demonstrated above, we investigate the site-dependence of the magnon modes in Cu 2 OSeO 3 , to assess possible contributions of the two inequivalent Cu sites into the spin excitations improving our current understanding of this complex system.
The magnon spectrum of Cu 2 OSeO 3 has been studied so far by several techniques, e.g., inelastic neutron scattering, Raman, Infrared, ESR, and THz spectroscopy 13,15,18,39,40 . These results consistently reported (i) inter-tetrahedron ferromagnons below 15 meV; (ii) intra-tetrahedron, medium-energy magnon branches between 30-40 meV; (iii) high-energy (>50 meV) phonon modes and multi-magnons. In Fig. 4 we display high-resolution RIXS  Energies of the Cu-I and Cu-II dd-excitations, extracted from the fit of the resonant inelastic X-ray scattering (RIXS) measurements. The errors are defined as the standard deviation associated with the least square fit results.
spectra of Cu 2 OSeO 3 at q = [1.3,0,0] r.l.u., measured at the Cu L 3 edge. From Fig. 4a, a long tail up to 100 meV can be observed. With reference to the magnon dispersion measured by inelastic neutron scattering 15,18 , at q = [1.3,0,0] r.l.u., we expect the first component around 12 meV from the ferromagnon mode, a second one around 35 meV from the medium-energy magnon branches. A phonon mode around 54 meV is furthermore expected, consistently with Raman and Infrared data 39,40 . Moreover, we attribute the highenergy spectral-weight around 70-80 meV to multi-magnons. This assignment is further supported by the temperature dependence of Fig. 4a 41 , where the spectral-weight of the magnetic components is enhanced below T C , up to 100 meV. Using resolution limited Gaussian peaks (FWHM = 30 meV) for fitting the elastic peak (P1, 0 meV), the ferromagnon (P2, 12 meV), the medium-energy magnon (P3, 35 meV), the phonon (P4, 54 meV) and a wider Gaussian for high-energy multi-magnon component (P5, 80 meV, FWHM = 40 meV), we can accurately reproduce the RIXS spectrum at T = 45 K, see Fig. 4b.
To investigate the site-dependent character of the magnetic excitation, we collected RIXS spectra as a function of incident photon energy across the Cu L 3 edge, while leaving q unchanged. The resulting data are presented in Fig. 4c. As expected, an overall amplitude renormalization of the whole spectrum takes place due to the absorption effect. However, by tracking the individual P1-P5 components through the fitting analysis, we can extract the intensity behavior for each individual excitation as a function of the photon energy, see Fig. 4d. The intensity profile of the elastic peak P1, the ferromagnon mode P2, the phonon mode P4, and the multimagnon mode P5 track well with the TFY absorption profile (peaked at~903.9 eV), within the error bars. This result suggests these modes do not have any specific or noticeable Cu-site-dependence. Interestingly, instead, the intensity profile of the medium-energy magnon mode P3 displays a pronounced resonance at the Cu-I sites,~930.6 eV, clearly standing out beyond the error-bar scale. We interpret this peculiar behavior considering the 35 meV component of the RIXS spectra dominated by magnon modes with B and C character 12,13 : these intra-tetrahedron modes correspond to rotating the Cu-I minority spins through the J AF S interaction, while leaving the Cu-II spins unaltered. Hence, our result highlights that individual Cu spin character persists in the mediumenergy magnon modes, simultaneously with the entangledtetrahedron nature reported so far for the magnetic excitations of Cu 2 OSeO 3 11,12 . This finding recalls the magnetic dual nature proposed for MnSi 42 , thus highlighting the complexity of these systems.

Conclusions
By combining resonant spectroscopies, DFT, and single-ion calculations, we elucidated the electronic and magnetic excitations of the multisite skyrmion material Cu 2 OSeO 3 . We identified the L 3 resonant energies for Cu-I and Cu-II ionic species present in this system. With this unique information at hand, we revealed the site-resolved 3d electronic structure in terms of crystal-field splittings, and moreover, the difference between the Cu-I and Cu-II ground state energies of about 0.55 eV. Due to the difficulty in accurately calculating these quantities for strongly correlated electron systems, our work provides theorists with an experimental benchmark for fine-tuning microscopic models of Cu 2 OSeO 3 , hence for extrapolating the competing energy terms, i.e., hopping, exchange integrals and DM interactions. Furthermore, we revealed an unexpected sitedependent character (Cu-I) for the medium-energy magnon branch: this result demonstrates that individual spin character is preserved in specific magnon modes, suggesting that local spin behavior coexists with the entangled nature of the magnetic ground state, explained by means of S = 1 tetrahedra rather than single spins.
More broadly, as the magnetism in Cu 2 OSeO 3 is determined by the competition between several interactions (super-exchange interactions, DM interactions, and crystal-anisotropy stemming from the spin-orbital interactions), our results on site-specific magnetic and electronic ground state excitations should be regarded as a prerequisite for validating future and past theoretical models of Cu 2 OSeO 3 dedicated to the microscopic understanding of e.g., the skyrmion phase, the magnetic chirality, and the multiferroicity.
Finally, our finding overall highlights the complexity of this skyrmion material and, at the same time, the relevance of using advanced spectroscopies to reveal site-specific information. The method presented here can be extended to thin films and heterostructures of Cu 2 OSeO 3 as well as devices (e.g., under the application of electric field) to elucidate the evolution of the site-specific excitations, and thus of the energy balance between the interactions contributing to the skyrmion formation. Fig. 4 Magnetic excitations in Cu 2 OSeO 3 . a High-resolution resonant inelastic X-ray scattering (RIXS) spectra measured at an incident energy of 930.9 eV, q = [1.3,0,0] r.l.u. for T = 27, 45, and 100 K, with π polarized light. The zero-energy was determined with reference to a carbon tape placed on the sample. b Fitting analysis of the 45 K RIXS spectrum, using five Gaussian peaks with a fixed width, fixed center position (allowing ±5% variation w.r.t. the values presented in the main text), and free amplitude. Raw data were displayed as open dots. The fitting sum is the solid line. The error bars are defined assuming a Poisson distribution of the single-photon counted events. c Repeating the fitting analysis for all RIXS spectra measured across the Cu L 3 resonance. d Summary of the integrated intensity for each fitting component, as a function of the incident photon energy (dot symbols). The error bars are extracted from the fitting, through the error propagation method. The smoothed lines underneath are a guide for the eyes. The Cu L 3 absorption spectrum is reproduced on top to ease the comparison.

Methods
Sample details. Single crystalline Cu 2 OSeO 3 was prepared by the chemical vapor transport method 9,15 . A Curie temperature of T C ≃ 57 K was extracted from the sample used in this study, in line with ref. 43 .
XAS and RIXS measurements. The XAS and RIXS experiments were performed at the SIX 2-ID beamline of NSLS-II 44 . The XAS data of Fig. 1b was measured in total fluorescence yield (TFY), at an incident angle of θ in = 20 ∘ . The energy resolution and experimental geometry used for the RIXS measurements were: ΔE = 50 meV (FWHM) and θ in = 20 ∘ /2Θ = 90 ∘ for the crystal-field study; ΔE = 30 meV (FWHM) and θ in = 75 ∘ /2Θ = 150 ∘ for the spin excitation study. All the measurements used π-polarized x-ray photons.
Calculations. Details about the DFT and single-ion calculations are available respectively in Supplementary Notes 1, 2.

Data availability
Data that support the findings of this study are available upon reasonable request from the corresponding authors.