Introduction

After more than a decade of the discovery of Fe-based superconductors1, the origin of superconductivity is still under debate2. Spin fluctuations between the hole pockets at the Brillouin zone center and the electron pockets at the zone edges have been considered as a candidate medium of the electron pairing, which typically leads to sign-reversed s ± pairing3,4. This mechanism has been supported by the spin resonance mode observed by neutron scattering measurements below the superconducting gap5. Apart from the important role of magnetism6, other factors, and mechanisms have also been proposed in Fe-based superconductors7,8,9,10,11,12. For example, orbital fluctuations have been proposed to enhance spin fluctuation-mediated pairing8 or directly account for the high Tc in iron pnictides7,9. Angle-resolved photoemission spectroscopy (ARPES) measurements on iron chalcogenides revealed the disappearance of dxy-orbital spectral weight with increasing temperature, implying universal orbital-selective correlation effects12.

Among the Fe-based superconductors, FeSe has the simplest crystal structure and provides an ideal platform for the study of the pairing mechanism13. While the Tc of bulk FeSe is only ~8 K, it can be enhanced by inter-layer interactions: both the intercalation between FeSe layers and SrTiO3 (STO) substrates for the monolayer FeSe can enhance the Tc by a factor of over four14,15,16,17. Two possible pairing mechanisms were proposed for the enhanced superconductivity in monolayer FeSe—interfacial charge-transfer and electron-phonon coupling (EPC)14,18,19,20,21,22. However, these scenarios call for verification in an effective monolayer material without the substrate and related phonons.

LiOH-intercalated FeSe (FeSe11111) is a pure bulk single-crystalline superconductor with a Tc of over 40 K16,17, which satisfies these conditions. Due to the intercalation, the distance between FeSe layers becomes as large as 9 Å in FeSe11111, leading to a highly two-dimensional structure. ARPES experiments also reflect the similarity between FeSe11111 and monolayer FeSe/STO in terms of Fermi surface topology, band structure, and the gap symmetry23. As such, FeSe11111 resembles the monolayer FeSe, avoiding the interface to the STO substrate, and therefore provides a unique opportunity to elucidate the origin of the increased Tc evolving from bulk FeSe to monolayer FeSe/STO. Studies of FeSe11111 recently lead to an even more exciting observation of a Majorana zero mode implying its nontrivial topology24. Therefore, it is appealing to investigate the excitations in FeSe11111 as a crucial step toward understanding the pairing mechanism in iron chalcogenides.

To identify the nature of excitations in FeSe11111 and their connection to superconductivity, we employ the resonant inelastic x-ray scattering (RIXS) technique, which is sensitive to multiple excitations25 and has been widely used to study Fe-based superconductors26,27,28,29,30,31. We uncover the excitations in FeSe11111 films grown on a LaAlO3 substrate with Tc ≃ 42 K (SC42K) and Tc ≃ 28 K (SC28K) (“METHODS”). The higher Tc accompanies a larger c lattice parameter and a lower Fe vacancy concentration in the FeSe-layer32,33,34. We identified six excitations in both samples, including four Raman-like excitations and two fluorescence-like excitations. The four Raman features are located at ~0.1, ~0.3, ~0.7, and ~2.5 eV. Through their momentum and temperature dependence, we attribute the ~0.3 and ~0.7 eV features to orbital excitations and the ~2.5 eV feature to multiple spin excitations. These orbital excitations are sensitive to the distorted tetrahedral crystal field and, accordingly, the lattice geometry. Therefore, our study suggests the lattice-induced orbital fluctuations as an ingredient to enhance the Tc of iron chalcogenides.

Results

Excitations in FeSe11111

Figure 1a shows the crystal structure of the FeSe-based superconductor FeSe11111 in the one-Fe unit notation. It consists of anti-PbO-type FeSe layers sandwiched by anti-PbO-type (Li/Fe)OH layers along c-direction and belongs to the space group P4/nmm (No. 129)16. The schematics of the RIXS experimental geometry is shown in Fig. 1b. The accessible reciprocal space at Fe L3-edge is displayed in Fig. 1c. We state coordinates in units of 2π/a, i.e., 1 corresponds to one reciprocal lattice unit (r.l.u.).

Fig. 1: RIXS spectra of FeSe11111.
figure 1

a Crystal structure of FeSe11111 in the one-Fe unit cell notation73. b RIXS experimental geometry. The sample a-b plane lies perpendicular to the scattering plane. c Reciprocal space in the one-Fe unit cell representation. The gray shaded area indicates the accessible momenta at Fe L3-edge. d A representative RIXS spectrum of SC28K taken at a photon energy of 709 eV at 50 K. The fitting components are overlaid and the gray solid line is the fit to the data. (Inset) An enlarged view of the spectra below 1 eV. e RIXS intensity map (in log scale) versus energy loss and detuning energy across the Fe L3-edge at 25 K for SC42K. The white line is Fe L3-edge X-ray absorption spectrum, measured via total fluorescence yield (TFY). The black dashed lines are guides to the eyes. f Selected RIXS spectra of incident-energy detuning measurements. Each spectrum and fitting component were shifted vertically for clarity. Vertical ticks indicate two fluorescence-like features.

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Figure 1d shows a representative RIXS spectrum of FeSe11111 collected at the Fe L3-edge resonant energy (709 eV) using π-polarized incident x-rays. Intriguingly, the RIXS spectrum of FeSe11111 highly resembles that of monolayer FeSe, including the weak fluorescence background and broad features at around 0.5 and 3 eV31. This indicates that these features arise from the FeSe-layer instead of the Li-Fe layer. Moreover, we obtained much better statistics of RIXS spectra in FeSe11111 than those in monolayer FeSe due to the larger sample volume. It is noteworthy that, due to the metallic nature of Fe-based superconductors, the fluorescence background is substantially strong in the RIXS spectra of bulk FeSe28,29,31 and iron pnictides26,27. Here, the weak fluorescence background is probably due to the absence of hole Fermi surface pockets at the Γ point, which closes the particle-hole scattering channel and allows us to clearly identify the excitations in FeSe11111. To quantitatively analyze the data, we fitted the RIXS spectra with Gaussian profiles and identified six excitations, as shown in Fig. 1d, which are located at ~0.1, ~0.3, ~0.7, ~1.1, ~2.5, and ~4.8 eV, respectively [see Supplementary Note 4 for fitting details].

Since the incident-energy detuning RIXS measurements can distinguish between Raman-like excitations and fluorescence-like excitations35,36, we measured a series of RIXS spectra across Fe L3-edge in steps of 0.25 eV for SC42K (Fig. 1e). For Raman-like features, the emitted photon energy shifts by the same energy as the incident photon energy. For fluorescence-like features, the energy loss changes by the same energy as the incident photon energy25. The features at around 1.1 and 4.8 eV at the Fe L3-edge resonant energy track the incident energy [see Fig. 1e, f], which are typical fluorescence behaviors. They correspond to the Fe 3d to 2p fluorescence decay reported also in iron pnictides and Fe 3d to Se 4p fluorescence decay, respectively26,37,38. The Raman-like features include an evident shoulder at ~0.1 eV and three excitations at around 0.3, 0.7, and 2.5 eV, which are captured by our atomic multiplet calculations as discussed below. We obtained similar detuning results in SC28K (see Supplementary Fig. 4).

Momentum and T c dependence

To understand the Raman-like excitations, we measured the detailed momentum-resolved RIXS maps along two high-symmetry directions, [0, 0] − [−H, 0], and [0, 0] − [−H, −H], as shown in Fig. 2. The elastic peaks were subtracted to manifest the low-energy excitations. The fittings of RIXS spectra are shown in the bottom spectra in Fig. 2a–f and also in Supplementary Fig. 6. The Raman-like excitations at ~0.1, 0.3, 0.7, and 2.5 eV barely change with momenta along the two high-symmetry directions. The two excitations at ~0.3 and 0.7 eV have similar half width at half maximum (HWHM) between 0.2 and 0.3 eV (see Supplementary Fig. 7), which is much broader than our energy resolution (~0.08 eV). They resemble those around 0.5 eV in monolayer FeSe, which were attributed to spin excitations31. It should be noticed that this comparison is not limited by energy resolution since the energies of these excitations (200–700 meV) are more than 2.5 times the resolution of our RIXS experiment (~80 meV). These excitations have similar energies in both SC28K and SC42K, irrespective of varying Tc (Fig. 2g). We also measured the momentum-dependent RIXS spectra at 95 K and 200 K (Fig. 2c–f), which showed the persistence of these Raman-like excitations to high temperatures. While their energies do not change with temperature (Fig. 2g–i), the intensities descend with increasing temperature (Fig. 2a–f). We will discuss the temperature dependence in more detail below.

Fig. 2: Momentum-dependent RIXS spectra of FeSe11111 at 25, 95, and 200 K.
figure 2

a, c, e Selected RIXS spectra of SC28K along two high-symmetry directions as indicated in Fig. 1c at 20, 95, and 200 K, respectively. Each spectrum is shifted vertically for clarity. The spectra were collected at the Fe L3 resonant peak energy of 709 eV. Spectra were normalized to the high energy-loss range of [4, 12] eV. The elastic peak was subtracted from the data to visualize the low-energy features. b, d, f Similar measurements on SC42K. Examples of the six peak decompositions described in the text are shown in the bottom spectra. g, h, i Summary of fitted energies of four Raman features along two high-symmetry directions for both SC28K and SC42K at 25, 95, and 200 K, respectively. To avoid specular reflection, we did not measure RIXS spectra at zero momentum transfer. The size of the markers corresponds to the spectral weight of the features. Error bars were the standard deviation of the fits.

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The dispersionless behavior of the Raman-like features suggests that the spectral peaks should be of local nature. Different from d9 copper oxide superconductors where the low-energy states lie in a single 3d orbital, the hybridization and interactions among five partially filled 3d orbitals in Fe allow rich atomic excitations, which can be reflected in the RIXS spectra. Using atomic multiplet theory with a distorted tetrahedral crystal field environment (CFE), we were able to reproduce theoretical spectral peaks at the experimentally observed excitation energies [see comparisons in Fig. 3a, b]. By projecting the excited-state wavefunctions to the maximal-overlapped Fock states, the nature of each excitation can be addressed approximately in a single-particle picture [see Fig. 3c]. The ~0.3 and 0.7 eV features, which are more evident in experiments, correspond to d − d excitations between the eg − eg and eg − t2g orbitals, respectively. These excitations contain both spin-flip and non-spin-flip contributions, which can be further distinguished through polarization control (See Fig. 3d and SI). Due to the locality of these low-energy d − d excitations, their energy scales are primarily determined by crystal field and lattice geometry instead of in-plane hoppings or Coulomb interactions. Therefore, the energy scales of these excitations are robust against varying Tc. We notice that RIXS measurements on FeS complexes with a distorted tetrahedral CFE for Fe2+ also observed two peaks at 0.32 and 0.58 eV attributed to d − d excitations39. RIXS experiments on PrFeAsO0.7 (Tc = 42 K) revealed a ~0.5 eV feature assigned to d − d orbital excitation as well40.

Fig. 3: Experimental and theoretical RIXS spectra.
figure 3

a The same RIXS spectrum as that in Fig. 1d while the elastic peak was subtracted from data, leaving only four Raman features. The colored solid lines are fitting results of each Raman feature. The gray solid line is the sum of the fitting results of the four Raman peaks. b RIXS spectra simulated by atomic multiplet theory. The broad peak at ~0.5 eV consists of 0.3 and 0.7 eV excitations, which are respectively d − d transitions among the eg − eg and eg − t2g orbitals. The broad 2.5 eV high-energy peak consists of multiple excited states of total spin quantum number S = 1. c Schematic cartoons illustrating the spin and orbital characters of the ground state and the low-energy excited states. d Dynamical spin structure factor S(q = 0, ω) and (non-spin-flip) orbital structure factor O(q = 0, ω) calculated using the multiplet model defined in the text.

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In addition to these two low-energy peaks, our simulation exhibits substantial intensity in a 2 eV window around the energy loss ω ~2.5 eV, which consists of multiple spectral peaks when the spectral broadening is reduced. These excited states mainly exhibit a change of total spin quantum number ΔS = 1. The energy scale for these high-energy excitations is mainly determined by the Racah parameters B and C between d − d orbitals. In our calculation, we used B = 0.12 eV and C = 0.4 eV, which lead to a typical Hund’s coupling ~0.8−0.9 eV for Fe2+. When only a single Fe site is present, the Racah parameter A provides only an overall energy shift of the atomic multiplet levels, and it is simply set to zero here. Since B and C arise from multipole interactions, they are more difficult to be screened (unlike A due to monopole interaction). Therefore, it is expected that the high-energy 2.5 eV feature will not be strongly affected by charge screening and the change of Tc, which is also consistent with the experiment.

Temperature dependence

To further explore the nature of the Raman excitations, we performed temperature-dependent measurements from 25 to 275 K on both SC28K and SC42K (Fig. 4). It is clear that the intensity at the energy-loss region of 0.2–0.8 eV decreases with temperature (Fig. 4a, c). Moreover, the ratio of the intensity between two excitations at ~0.3 and ~0.7 eV varies with temperature (Fig. 4b, d), which supports our fitting with two peaks (details of the fitting shown in Supplementary Fig. 8). The temperature dependence of the integrated intensity of the four Raman features is shown in Fig. 4e–h. Intriguingly, the excitations at ~0.3 and ~0.7 eV decrease by more than 50% from 25 to 275 K, while the features at ~0.1 and ~2.5 eV change only slightly with temperature in both samples. It is also known that the change of Tc is connected to different Fe vacancy concentrations32,33,34. This can affect the oxidation state so that Fe valency may be reduced from 2+, which can slightly affect the ratio between the 0.3 and 0.7 eV peaks.

Fig. 4: Temperature dependence of SC28K and SC42K at q = [−0.2, 0].
figure 4

Spectra were normalized to the integrated fluorescence intensity from energy loss 4 to 12 eV. a, c Energy vs temperature RIXS maps measured on SC28K and SC42K, respectively. The elastic peak was subtracted to manifest the low-energy excitations. b, d The waterfall plots correspond to (a) and (c), respectively. Examples of the fitting components are shown in the bottom spectra. e–h Temperature dependence of the fitted integrated intensity of these four Raman features, respectively. The integrated intensity was normalized to that of 25 K for comparison. Temperature evolution of the spectral weight of dxy orbital measured by ARPES in monolayer FeSe/STO (green markers) is shown in (g) for comparison12.

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The temperature dependence of the 0.3 and 0.7 eV spectral features suggests the primary orbital nature of these excitations. Previous magnetic susceptibility measurements on SC28K and SC42K displayed a drop or deviation from the Curie-Weiss behavior around ~120 K, implying a magnetic transition or fluctuation17,33. In Fe-based superconductors, magnetic transitions or fluctuations are usually accompanied by an orbital weight redistribution. Previous studies have demonstrated that the spectral intensity can exhibit a more apparent temperature dependence when orbital fluctuations are present41. To demonstrate the relevance of orbital fluctuations, we simulate the dynamical spin and (non-spin-flip) orbital structure factors S(q = 0, ω) and O(q = 0, ω) separately (see Fig. 3d). Orbital excitations with spin-flip character is included in the S(q, ω). As expected from the single-particle picture in Fig. 3c, the two low-energy Raman peaks at ~0.3 and ~0.7 eV are present in both orbital and spin channels. A full-polarization study by measuring the polarizations of both incident and outgoing x-rays would help us to quantitatively determine the orbital and spin contributions to the peaks at ~0.3 and 0.7 eV. Such full-polarization investigation is suggested for future more advanced studies. Nevertheless, it is possible to determine qualitatively the orbital and spin contributions by performing temperature-dependent measurements. Considering the experimental measurements integrated over all scattering polarizations, the contributions from these two channels are approximated at 2:1. Since the onset of orbital excitations can cause a rapid transfer of the spectral weight away from coherent peaks41, a rough estimation is that the intensity of these d − d excitations drops by 60% at high temperature due to the fluctuation of orbitals, consistent with the experimental observations for SC28K and SC42K. Moreover, we notice the suppression of the ~0.7 eV peak involving the dxy orbital is consistent with ARPES measurements on monolayer FeSe that show the disappearance of dxy spectral weight at high temperature12 [see Fig. 4g]. This agreement implies substantial orbital fluctuations and orbital redistribution in high-Tc iron chalcogenides. We would like to stress that the two low-energy peaks at ~0.3 and 0.7 eV are both orbital excitations no matter whether the spin-flip is involved. Therefore, the mixture of spin channels does not weaken our conclusion about orbital fluctuations.

In contrast, the orbital O(q = 0, ω) has almost no contribution to the high-energy regime. Thus, the 2.5 eV Raman feature originates primarily from local spin-flip d − d excitations, accounting for its weak temperature dependence. This mechanism can be further tested in the future by the temperature dependence of RIXS spectra employing a polarimeter for full-polarization measurements. In addition to the above three features at higher energies, another Raman feature at ~0.1 eV is also dispersionless within our energy resolution (~80 meV), which is absent in the multiplet model calculation. ARPES and electron energy-loss spectroscopy (EELS) measurements on monolayer FeSe reported a phonon mode at ~90–100 meV arising from the STO substrate18,20,42. However, our FeSe11111 thin-film has a different substrate LaAlO3 and the thickness of the film is several hundreds of nanometers. A previous Raman study on FeSe11111 measured phonons only up to 400 cm−1 43. Nevertheless, the presence of oxygen and hydrogen within (Li/Fe)OH layers may contribute to a 0.1 eV phonon excitation. Another possibility relates to an optical spin-wave mode, which is typically much less dispersive compared to the acoustic spin-wave mode. In this case, the system exhibits short- or long-range magnetism involving block spins in an enlarged magnetic unit cell44. However, INS studies of FeSe11111 have not reported an optical magnon around 0.1 eV45, which renders this origin less likely.

Discussion

The dispersionless excitations in FeSe11111 and monolayer FeSe contrast the dispersive spin excitations observed in bulk FeSe, reflecting their intrinsic differences. Nevertheless, the absence of orbital excitations in bulk FeSe might be due to these orbital excitations being broader and therefore harder to isolate from the large fluorescence background in bulk FeSe28,29,31, which remains to be explored in future studies. Our theory-experiment comparison suggests that a minimal multiplet model involving the Fe 3d orbitals can quantitatively address these dispersionless features. Informed by experimental results, our simulation ignores nonlocal interactions, which usually cause dispersive magnon excitations. That said, an orbital-dominated mechanism does not contradict and may coexist with the recently proposed scenario of hybridized magnons in a two-band Hubbard model31. We notice that previous INS studies reveal the spin excitations dispersing up to ~140 meV near [π, π] in FeSe11111, but their spectral weight is much lower than that in bulk FeSe due to doping effects45,46, which would make it difficult to detect by RIXS within the accessible momenta near [0, 0]. Our observation of orbital excitations and strong temperature-dependent orbital fluctuations is still compatible with Hund’s metal picture, which can lead to an orbital-selective Mott phase at zero temperature. The degrees of orbital fluctuations at elevated temperatures will be dependent on the competition between different energy scales, such as Hund’s coupling and crystal-field splitting.

Orbital-selective Cooper pairing, manifesting as highly anisotropic superconducting energy gaps, has been discovered in both bulk FeSe and monolayer FeSe using Bogoliubov quasiparticle interference imaging47,48. By raising the temperature, ARPES studies have observed strong orbital-dependent correlation effects in monolayer FeSe where the dxy orbital completely loses its spectral weight12. However, this effect is not found in bulk FeSe49. Here we observe that FeSe11111 displays similar orbital excitations as those in monolayer FeSe31. Moreover, we find the spectral weights of orbital excitations at ~0.3 and 0.7 eV decrease with increasing temperature, and the latter involving dxy orbitals shows stronger temperature-dependent behavior, consistent with ARPES results12. Since the evident change of orbital spectral weights occurs not only in FeSe11111 but also in monolayer FeSe, both exhibiting a relatively high transition temperature Tc, it suggests that the orbital fluctuations may play important roles in superconductivity for these materials.

In our FeSe11111 material, the prominent orbital excitations at 0.3 and 0.7 eV are attributed to d − d transitions among the eg − eg and eg − t2g orbitals. In iron pnictides, it was shown that the t2g orbitals are necessary to lead to the ferro-orbital fluctuations9. The lattice-induced orbital fluctuation effect, in turn, can favor superconductivity. Moreover, when the electron-phonon coupling matrix elements for all five 3d orbitals are included, the critical value of phonon-mediated interactions that separates the phases of orbital fluctuation and orbital order is reduced to half compared to that in the three-orbital model. In these regards, all five 3d orbitals and their fluctuations can be important to superconductivity. This mechanism is consistent with the previous scanning tunneling microscopy (STM) studies in both monolayer FeSe and FeSe1111150,51, in favor of the phonon-mediated pairing enhancement scenario18 or orbital fluctuation-mediated pairing mechanism9. The absence of the dxz/dyz hole Fermi surface may also suppress the nematic order that competes with superconductivity52, underscoring the crucial role of the orbitals in the pairing.

We stress that the orbital excitations referred to in this work involve both spin-flip and non-spin-flip excitations. Therefore, our conclusion does not exclude the role of spin fluctuations in the pairing mechanism, which has also been proposed in FeSe1111145,53,54. For example, the phase-sensitive quasiparticle interference technique has revealed a sign-changing order parameter in Zn-doped FeSe1111153 and INS studies have observed the spin resonance below Tc in FeSe1111145,54, indicating the important role of spin fluctuations in electron pairing. Thus, as a layered material without STO substrate, FeSe11111 and its excitations provide a unique opportunity to elucidate the evolution from bulk to monolayer FeSe materials, which suggests the importance of both orbital and spin degrees of freedom for high-Tc iron selenides.

Methods

Sample preparation

The high-quality single-crystalline FeSe11111 films were synthesized by the matrix-assisted hydrothermal epitaxy (MHE) technique55,56. Their crystalline quality was characterized by x-ray diffraction (XRD) on a 9 kW Rigaku SmartLab x-ray diffractometer equipped with two Ge(220) monochromators at room temperature (Supplementary Fig. 1). Both the XRD patterns exhibit a single preferred (00l) orientation and the full width at half maximum (FWHM) is around 0.3∘ of the x-ray rocking curves for the (006) reflection, showing the highly crystalline quality of these films (Supplementary Fig. 1). Their Tc values are defined at the onset temperatures of the diamagnetism by magnetic measurements (Supplementary Fig. 1), performed with a 1 Oe magnetic field applied along the c-axis using a SQUID magnetometer (MPMS XL1, Quantum Design).

RIXS measurements

We performed high-resolution RIXS experiments by using the Super Advanced x-ray Spectrometer (SAXES) at the Advanced Resonant Spectroscopy (ADRESS) beamline of the Swiss Light Source, Paul Scherrer Institut, Switzerland57,58. The energy resolution was 80 meV (FWHM). We cleaved both samples in an ultra-high vacuum. We used the grazing-incident geometry, which provided stronger signals than the grazing-out geometry (Supplementary Fig. 3). All samples were aligned with the surface normal (001) in the scattering plane. π polarized x-rays were used for the RIXS measurements since the signals were similar between π- and σ-polarizations (Supplementary Fig. 3). The incident energy was tuned to the peak maximum of Fe L3-edge (709 eV), except for detuning measurements. The angle between incoming and outgoing x-rays was fixed to 130 degrees, resulting in a constant value of the total momentum transfer. We refer to reciprocal space coordinates in the conventional one-Fe unit cell with a = b = 2.680 Å, c = 9.318 Å for SC42K and c = 9.261 Å for SC28K33. The momentum transfer Q is defined as Q = Ha* + Kb* + Lc* where a* = 2π/a, b* = 2π/b, and c* = 2π/c. q is the projection of Q in the ab-plane. We measured ab-plane momentum transfer scans by varying the grazing-incident angle to the sample surface. Each RIXS spectrum was measured for 30 min except for the spectrum of incident energy-dependent RIXS maps, which was measured for 5 min.

Atomic multiplet theory

To model the experimental spectra, we make two assumptions about the Hamiltonian and the RIXS cross-section. First, we consider only multiplet interactions in the atomic limit59. In this case, our calculation can capture local spin or orbital flip excitations but not collective spin waves (magnon). Nevertheless, this assumption is appropriate in our system, as (i) the experiment exhibits little momentum dependence and (ii) we are not aware of any lattice model with short or long-range magnetism capable of exhibiting a complete flat spin-wave dispersion. The resulting multiplet Hamiltonian containing all Fe 3d orbitals reads

$$\begin{array}{l}{{{{\mathcal{H}}}}}_{3d}=\mathop{\sum}\limits_{\alpha}{U}_{\alpha }{d}_{\uparrow}^{(\alpha ){\dagger} }{d}_{\uparrow}^{(\alpha )}{d}_{\downarrow }^{(\alpha ){\dagger} }{d}_{\downarrow}^{(\alpha )}+\frac{1}{2}\mathop{\sum}\limits_{{{\alpha \ne \beta }}\atop{{\sigma {\sigma}^{\prime}}}}{U}_{\alpha \beta}^{\prime}{d}_{\sigma}^{(\alpha ){\dagger} }{d}_{\sigma}^{(\alpha )}{d}_{{\sigma}^{\prime}}^{(\beta ){\dagger} }{d}_{{\sigma }^{\prime}}^{(\beta )}\\ \,\qquad\quad+\,\mathop{\sum}\limits_{\alpha ,\sigma}{\varepsilon}_{\alpha }{d}_{\sigma }^{(\alpha ){\dagger} }{d}_{\sigma }^{(\alpha )}+\frac{1}{2}\mathop{\sum}\limits_{{{\alpha \ne \beta }}\atop{{\sigma {\sigma}^{\prime}}}}{J}_{\alpha \beta}{d}_{{\sigma}^{\prime}}^{(\alpha ){\dagger} }{d}_{\sigma }^{(\beta ){\dagger} }{d}_{\sigma }^{(\alpha )}{d}_{{\sigma}^{\prime}}^{(\beta )}.\end{array}$$
(1)

Here, \({d}_{\sigma }^{(\alpha )}\) (\({d}_{\sigma }^{(\alpha ){\dagger} }\)) annihilates (creates) a 3dα electron with spin σ (α = x2 − y2, z2, xy, yz, and xz). εα denotes the site energy of orbital α. In a distorted tetragonal CFE, εα can be parameterized by three parameters 10Dq, δ, and μ60. We use 10Dq = 0.485 eV, δ = −0.12 eV, and μ = −0.32 eV. Here, 10Dq represents the site energy separation between eg and t2g orbitals, and its typical value is ~0.5 eV for a tetrahedral CFE in iron-based superconductors. δ and μ describe additional splitting between the eg and t2g orbitals due to a lowered crystal symmetry. The intra-orbital U and inter-orbital \({U}^{\prime}\) interactions, as well as Hund’s coupling and pair hopping J, can be parameterized by the Racah parameters A, B, and C. In the case of a single Fe atom, A only provides an overall energy shift, so it is simply set to zero. Without any fine-tuning, we use B = 0.12 eV and C = 0.40 eV, leading to a typical value of ~0.8–0.9 eV for Fe2+ Hund’s coupling, independent of charge screening and the change of Tc. The Hamiltonian is solved by exact diagonalization (ED) to obtain the energy eigenvalues and eigenstates. It is noted that within reasonable CFE and interaction parameters, a high spin S = 2 ground state is always favored, which agrees with our further ab initio CASPT2 simulations (see below). If the ground-state spin quantum number is reduced from S = 2, then the spectral intensity ratio between the 0.3 and 0.7 eV peaks also will change61. This can happen with charge or orbital fluctuation effects, which are always present to some extent in the experiment but not captured in the atomic multiplet calculation.

The RIXS cross-section is defined as

$$I({{{\bf{q}}}},\omega ,{\omega }_{{{{\rm{in}}}}})=\frac{1}{\pi N}{{{\rm{Im}}}}\left\langle {{\Psi }}\right|\frac{1}{{{{{\mathcal{H}}}}}_{3d}-E_g-\omega -i\delta }\left|{{\Psi }}\right\rangle ,$$
(2)

where

$$\left|{{\Psi}}\right\rangle =\frac{1}{N}\mathop{\sum}\limits_{{{{\bf{k}}}}} {{{{\mathcal{D}}}}}_{{{{\bf{k}}}}+{{{\bf{q}}}}}^{{\dagger} }\frac{1}{{{{{\mathcal{H}}}}}_{3d}+{H}_{{{{\rm{core}}}}}-E_g-{\omega}_{{{{\rm{in}}}}}-i{{\Gamma }}}{{{{\mathcal{D}}}}}_{{{{\bf{k}}}}}| G\rangle ,$$
(3)

and the (momentum-resolved) dipole transition operator \({{{{\mathcal{D}}}}}_{{{{\bf{k}}}}}={\sum }_{{{{\bf{i}}}}}{e}^{-{{{\bf{k}}}}\cdot {{{{\bf{r}}}}}_{{{{\bf{i}}}}}}{{{{\mathcal{D}}}}}_{{{{\bf{i}}}}}\) and Eg is the ground state energy. In principle, the intermediate state of a RIXS process contains a core-hole interaction (with core-hole lifetime 1/Γ) \({H}_{{{{\rm{core}}}}}\), describing the interaction between Fe core 2p and valence 3d orbitals. To simplify the calculation and clearly assign the spectral features, we adopt the second commonly used assumption of ultrashort core-hole lifetime approximation62,63, i.e., Γ ≫ Uα. In this case, the RIXS cross-section is reduced to the dynamical spin and orbital structure factors. The dynamical spin structure is defined as

$$S({{{\bf{q}}}},\omega )=\frac{1}{\pi N}{{{\rm{Im}}}}\left\langle G\right|{\rho }_{-{{{\bf{q}}}}}^{(s)}\frac{1}{{H}_{3d}-E_g-\omega -i\delta }{\rho }_{{{{\bf{q}}}}}^{(s)}\left|G\right\rangle ,$$
(4)

where \({\rho }_{{{{\bf{q}}}}}^{(s)}={\sum }_{\alpha \beta }{d}_{{{{\bf{k}}}}+{{{\bf{q}}}}\uparrow }^{(\beta ){\dagger} }{d}_{{{{\bf{k}}}}\downarrow }^{(\alpha )}\) and the orbital excitation is defined as

$$O({{{\bf{q}}}},\omega )=\frac{1}{\pi N}{{{\rm{Im}}}}\left\langle G\right|{\rho }_{-{{{\bf{q}}}}}^{(o)}\frac{1}{{H}_{3d}-E_g-\omega -i\delta }{\rho }_{{{{\bf{q}}}}}^{(o)}\left|G\right\rangle ,$$
(5)

where \({\rho }_{{{{\bf{q}}}}}^{(o)}={\sum }_{\alpha \ne \beta \sigma }{d}_{{{{\bf{k}}}}+{{{\bf{q}}}}\sigma }^{(\beta ){\dagger} }{d}_{{{{\bf{k}}}}\sigma }^{(\alpha )}\). These two structure factors approximate the spin-flip and non-spin-flip RIXS at the ultrashort core-hole lifetime limit. We note that the charge structure factor N(q, ω) also contributes to the non-spin-flip cross-section, but its inelastic response vanishes for q = 0 due to charge conservation. Using the atomic multiplet model, we only present the response function for q = 0 in this paper.

Ground-state configuration from ab initio quantum chemistry

To determine the ground-state spin configuration as a starting point for the ED calculation, we perform two different ab initio simulations for a single unit cell Fe(II)[Se2−]4 complex. Due to the expectation of strong multi-reference effects, we complement the PBE0 DFT simulation with a CASPT2 simulation with (10e, 12o) active space. As shown in Table 1, both simulations conclude with a ground state with S = 2, justifying the selection of CFE in our atomic multiplet simulations. More specifically, the DFT simulations predict relatively large energy differences among all spin sectors compared to the CASPT2 results. This is because the lower-spin state exhibits a stronger multi-reference effect as a result of quantum fluctuations. Without accounting for this effect, DFT over-estimated the energy of these lower-spin states. In contrast, the energy differences obtained by CASPT2 are relatively small, accounting for the experimental observation of reduced magnetic moment with the presence of itinerancy.

Table 1 Ground-state energy for each spin configuration calculated using DFT and CASPT2.
Full size table

For DFT calculation, we used the PBE0 hybrid function (PBE GGA + 20% Hartree-Fock exchange) with def2-TZVP basis set, which has been frequently used in spin splitting energy calculations of transition metal complexes64,65. The simulation is conducted using ORCA 4.066. For CASPT2 calculation, we followed literature recommendations for transition metals to include five 3d orbitals67,68, two σ orbitals describing metal-ligand bonding, and five double-shell d orbitals for mid-row and later transition, resulting in 12 active orbitals. We employ relativistic atomic natural orbital (ANO-rcc) basis sets contracted to [7s6p5d3f2g1h] for Fe and [7s6p4d3f2g] for Se69,70. The 41 core orbitals were frozen in all calculations. The CASPT2 simulation is conducted using Molcas 871.