Electron correlation and incipient flat bands in the Kagome superconductor CsCr₃Sb₅

Abstract

Correlated kagome materials exhibit a compelling interplay between lattice geometry, electron correlation, and topology. In particular, the flat bands near the Fermi level provide a fertile playground for novel many-body states. Here we investigate the electronic structure of CsCr₃Sb₅ using high-resolution angle-resolved photoemission spectroscopy and ab-initio calculations. Our results suggest that Cr 3d electrons are intrinsically incoherent, showing strong electron correlation amplified by Hund’s coupling. Notably, we identify incipient flat bands close to the Fermi level, which are expected to significantly influence the electronic properties of the system. Across the density-wave-like transition at 55 K, we observe a drastic enhancement of the electron scattering rate, which aligns with the semiconducting-like property at high temperatures. These findings establish CsCr₃Sb₅ as a strongly correlated Hund’s metal with incipient flat bands near the Fermi level, which provides an electronic basis for understanding its novel properties compared to the weakly correlated AV₃Sb₅.

The authors investigate the electronic structure of kagome CsCr₃Sb₅ using high-resolution angle-resolved photoemission spectroscopy and ab-initio calculations. The results identify CsCr₃Sb₅ as a strongly correlated Hund’s metal with incipient flat bands near the Fermi level.

Introduction

Kagome materials with corner-sharing triangular networks in their lattice structures exhibit rich and fascinating properties, such as spin liquid states^1–3, topological quantum phases^4–8, fractional quantum Hall effect^9–11, and many-body emergent phenomena^12–15. The characteristic electronic structure of a typical kagome lattice includes the flat band, the Dirac fermions at the K point, and the van Hove singularity (vHS) at the M point (Fig. 1a). If the Fermi level (E_F) is tuned to the flat band, novel magnetic states such as fractional quantum Hall effect and quantum anomalous Hall effect may emerge, while the vHS and the Dirac fermions at the E_F can induce unconventional superconductivity, charge-density wave (CDW), and/or other novel properties^16–18. Up to date, many different kagome materials with characteristic kagome electronic structures have been discovered and extensively studied^5,8,18–22. However, the investigation of intrinsic kagome-related many-body ground states remains inadequate since the flat bands are usually situated far from E_F, and their impact on the electronic properties of the system requires further exploration.

Fig. 1. Basic properties of CsCr₃Sb₅.

a Schematic illustration of the characteristic band structure of a kagome lattice including the flat band, the Dirac fermion at the K point, and the van Hove singularity (vHS) at the M point. b The crystal structure of CsCr₃Sb₅ in which Cr atoms form a planar kagome lattice. c Kagome structure of Cr d orbitals. For simplicity, only the d_xz orbital is shown. d In-plane magnetic susceptibility (χ) as a function of temperature measured at H = 5 T. e In-plane resistivity (ρ) as a function of temperature shows a peak near 55 K. f, g Comparison between the calculated band structure of CsCr₃Sb₅ (f) and CsV₃Sb₅ (g) by the density functional theory (DFT). The colored lines indicate similar dispersions of the two materials.

Among the abundant kagome materials, AV₃Sb₅ (A = K, Rb, and Cs) have attracted great attention due to their intriguing emergent properties^15,23, such as the interplay/competition between unconventional superconductivity and CDW^12–15, time- and/or rotational-symmetry broken phases^24–31, giant anomalous Hall effect related to the chirality of the charge order^27,32, the observation of pair-density waves^33,34, and putative loop current order^35–37. In the exploration of AV₃Sb₅ materials, it has been a common consensus that the vHS near E_F plays a key role in the CDW and possibly also in the superconductivity^38,39, while the attractive flat band was observed far away from E_F⁴⁰.

Moreover, the non-magnetic and weakly correlated kagome physics of AV₃Sb₅ can be further enriched by introducing magnetism and strong electron correlations. By replacing vanadium with chromium, the sibling compound CsCr₃Sb₅ provides an ideal platform to investigate the impact of magnetism and electronic correlation effect in the AV₃Sb₅-type kagome systems⁴¹. Indeed, recent investigations reveal many novel properties of this new kagome material, including superconductivity up to 6.4 K under pressure, frustrated altermagnetism, density-wave-like order, and non-Fermi liquid behavior in the high-temperature state^41–43. Compared to the non-magnetic and weakly correlated AV₃Sb₅, incipient flat bands (IFB, flat bands that extend in a small portion of Brillouin zone and locate away from E_F)⁴⁴ originated from correlated Cr 3d orbitals appear near E_F, which may dominate the correlated electronic properties of the system. These novel electronic characteristics, together with the prototypical kagome electronic structure of CsCr₃Sb₅, require systematic experimental studies of its electronic structure.

In this work, by performing high-resolution angle-resolved photoemission spectroscopy (ARPES) measurements, we systematically investigate the electronic structure of CsCr₃Sb₅ single crystals. Our experiments reveal weakly correlated Sb 5p states and intrinsically incoherent Cr 3d states, in overall agreement with our density functional theory (DFT) and dynamical mean-field theory (DMFT) calculations. Compared to CsV₃Sb₅, the vHSs and Dirac fermions characterizing the kagome lattice locate far away from E_F. Interestingly, we observe a flat Cr 3d band at about 80 meV below E_F, confirming the existence of IFB close to E_F. Moreover, the electron scattering rate, as manifested by the spectral broadening, is drastically enhanced above the density-wave transition temperature (~ 55 K), which is related to the semiconducting-like behavior of the system at high temperatures. These findings, in good consistency with the theoretical calculations^45,46, identify CsCr₃Sb₅ as a strongly correlated Hund’s metal with IFB. Our work provides a foundation for further exploration of the correlated kagome material with unconventional superconductivity.

Results and discussion

Basic electronic structure of CrCr₃Sb₅

CsCr₃Sb₅ crystallizes in a similar layered structure of AV₃Sb₅ with the space group of P6/mmm (Fig. 1b)⁴¹. In each unit cell, the Cr atoms constitute a two-dimensional kagome lattice with Sb atoms occupying the center of the hexagons. It is mainly the Cr 3d orbitals that form the kagome electronic states near E_F, as schematically shown in Fig. 1c. Previous X-ray scattering and nuclear magnetic resonance experiments revealed an intertwined CDW and spin-density wave (SDW) below 55 K⁴¹. Consistently, our susceptibility measurement suggests an antiferromagnetic transition at 55 K (Fig. 1d), accompanied by a semiconducting-like to metal transition in the resistivity measurement (Fig. 1e), which is a manifestation of the intertwined density-wave-like order in the system. Fig 1f, g compare the DFT-calculated electronic structure of CsCr₃Sb₅ and CsV₃Sb₅. In general, the two compounds show similar electronic structures characterizing the kagome lattice. Compared to CsV₃Sb₅, where the vHSs are close to E_F, the vHSs and Dirac fermions in CrCr₃Sb₅ locate further away from E_F, suggesting their relatively irrelevance in the novel transport properties. Prominently, compared to CsV₃Sb₅, the bandwidth of the dispersive bands near E_F is much smaller, and the flat bands located close to E_F in CsCr₃Sb₅ (Fig. 1f), which are believed to play a key role in the unconventional superconductivity^44,47,48.

Fig 2a shows the calculated three-dimensional Fermi surface (FS) of CsCr₃Sb₅. There is a nearly cylindrical electron pocket and two cylindrical hole pockets with similar volumes at k_||= 0 (the $\overline{\mathrm{\Gamma }}$ point), together with a warped cylindrical electron pocket around k_||= 0.66 Å⁻¹ (the $\overline{M}$ point). At certain k_z, there is no Fermi crossing along $\overline{\mathrm{\Gamma }}\overline{K}\overline{M}$, leaving holes in the cylindrical hole pockets around $\overline{\mathrm{\Gamma }}$ (see the yellow Fermi pockets in Fig. 2a). The experimental FS in the k_x-k_z plane shows a weak but resolvable k_z dispersion of the bands around k_||= 0 (the $\overline{\mathrm{\Gamma }}$ point), confirming the quasi-two-dimensionality of the electronic structure (Fig. 2b, also see the Supplementary Information). The FS in the k_x-k_y plane shows a hexagonal structure (Fig. 2c). We observe a circular electron pocket around the $\overline{\mathrm{\Gamma }}$ point, together with a spectral-weight patch around the $\overline{M}$ point. With increasing binding energy, the circular electron pocket around the $\overline{\mathrm{\Gamma }}$ point shrinks, and a star-of-David-shape pattern can be resolved at − 0.4 eV, as indicated by the black dashed lines in Fig. 2d.

Fig. 2. Overall band structures of CsCr₃Sb₅.

a Three-dimensional plot of the calculated Fermi surface of CsCr₃Sb₅. b k_z dispersion measured in the photon energy range between 60 eV and 110 eV. c, d The k_x-k_y map of ARPES intensity at the Fermi level (E_F) and − 0.4 eV. e–g Band dispersion along $\overline{\mathrm{\Gamma }}\overline{K}$ measured using different photon polarizations in a large energy range. The black arrow in (g) indicates the band that contributing to the star-of-David structure in (d). h Comparison between the calculated density-of-states (DOS) and integrated energy distribution curve (EDC). In panels (c, d, and g), the data measured with linear-horizontally (LH) and linear-vertically (LV) polarized photons were merged for completeness of the electronic structure. Data in (c–g) were measured using 93 eV photons. All data were collected at 6 K.

Fig 2e–g shows the experimental band dispersions along high symmetry directions in a large energy range. Since the electronic states near E_F are mainly contributed by Cr 3d orbitals, the data strongly depend on the light polarization. In general, the spectra can be roughly divided into three segments in energy: the dispersive bands near E_F, an M-shape feature around − 1.5 eV, and weakly dispersive bands around − 3 eV. These features contribute peaks in the integrated energy distribution curve (EDC) that is in overall consistency with the calculated density-of-states (DOS), as compared in Fig. 2h.

Incipient Flat bands near E_F

Figure 3 shows the fine electronic structure along high-symmetry directions near E_F. The experiment shows an overall agreement with the calculation (Fig. 3a, b), as shown by the data overlaid with the renormalized calculation in the Supplementary Information. The energy bands originated from Sb p orbitals (Supplementary Information) are well captured by the calculation. Specifically, they form the vHS at the Γ point, which shows electron-like dispersion in the k_x-k_y plane and hole-like dispersion along k_z, as schematically shown in Fig. 3c. We notice that the electron band around $\overline{\mathrm{\Gamma }}$ has mixed p and d orbital components, suggesting a p-d interaction (Supplementary Information). For the Cr 3d orbital bands, the spectra are generally much broader. It is noteworthy that while non-trivial surface states related to the Z₂ topological electronic structure were observed in AV₃Sb₅⁴⁰, here we do not find clear evidence for the surface states.

Fig. 3. Comparison between experimental and calculated fine electronic structure of CsCr₃Sb₅ near E_F.

a Density functional theory (DFT) calculation of the orbital-projected electronic structure of CsCr₃Sb₅ along high-symmetry directions. b ARPES spectra along high-symmetry directions. c Schematic illustration of the vHS at Γ (blue circle in a) with hole-like and electron-like dispersion along k_z and k_x/y, respectively. d Zoom-in plot of band dispersions along ΑΗ (left) and the corresponding EDC at Α (right). e DFT + dynamical mean-field theory (DMFT) calculation of the electronic structure, with on-site Coulomb interaction U = 5 eV and Hund’s coupling J_H = 0.88 eV. The red arrows indicate the flat band close to E_F (Supplementary Information). The calculated result with U = 5 eV and J_H = 0.5 eV is shown in the Supplementary Information. Data were collected at 6 K. For completeness, data measured with LH- and LV-polarized photons were merged.

Prominently, we reveal a flat band close to E_F, as indicated by the red arrows in Fig. 3b. The flat band with a band bottom at about 80 meV below E_F can be better resolved from the data along $\overline{\mathrm{\Gamma }}\overline{K}$ and the EDC at $\overline{\mathrm{\Gamma }}$ shown in Fig. 3d (also see the Supplementary Information). This flat band close to E_F may play an important role in the transport properties of CsCr₃Sb₅ and contribute to the electronic specific-heat coefficient 105 mJ/(K·mol), much larger than CsV₃Sb₅⁴¹. The observation of the flat band in the ARPES experiment suggests that the DFT-calculated flat bands are pushed closer to E_F. To understand this difference, we perform DFT + DMFT calculation, which can better capture the strong electron correlations of Cr 3d electrons. The calculated spectral function at Coulomb interaction U = 5 eV and Hund’s coupling J_H = 0.88 eV⁴⁵ is shown in Fig. 3e. Interestingly, the flat bands above E_F in Fig. 3a are indeed pushed closer to E_F and strongly renormalized, leaving the incipient flat band with prominent spectral weight mainly around the $\overline{\mathrm{\Gamma }}$ point. It is likely that the E_F of CsCr₃Sb₅ crystals are slightly raised up by self-doping or surface effects and the flat bands, similar to the situation in CsV₃Sb₅⁴⁹. Therefore, the flat band can be experimentally observed by the ARPES experiment.

Moreover, the DFT + DMFT calculation suggests that the electron correlation effect shows a minor impact on the Sb p orbital bands but strongly broadens Cr d orbital bands, well capturing the intrinsic characteristics of our ARPES spectra. We emphasize that the strong Hund’s coupling is essential in understanding the strong renormalization of the effective mass and the enhanced electron self-energy⁴⁵. As shown in the Supplementary Information, the calculation with different Hund’s coupling suggests that the strong broadening of the Cr 3d orbitals are intrinsically due to the Hund’s coupling. The agreement between our experiment and DMFT calculation thus confirms that CsCr₃Sb₅ is a strongly correlated Hund’s metal with incipient flat bands.

Enhanced scattering rate across the density-wave-like transition

Although CsCr₃Sb₅ shows a density-wave-like transition near 55 K⁴¹, as confirmed by our susceptibility and resistivity measurements, we did not observe clear signatures of band folding, splitting, or shift related to the transition with synchrotron-based ARPES (Fig. 4a and b). To better resolve the dispersion around the $\overline{\mathrm{\Gamma }}$ point and track the temperature evolution of the band structure, we conducted laser-ARPES measurements with improved energy and momentum resolutions. The Sb p orbital band around $\overline{\mathrm{\Gamma }}$ exhibits strong intensity with a slight change of the dispersion slope as approaching E_F, possibly due to the interaction with Cr 3d orbitals (Fig. 4c and the data after dividing the Fermi-Dirac distribution function in the Supplementary Information). Fig 4d shows the temperature evolution of our laser-ARPES spectra. Again, no clear change of the spectra, such as gap opening is observed across the density-wave transition at 55 K, as can be seen from the corresponding EDCs near E_F in Fig. 4e.

Fig. 4. Temperature evolution of the electronic structure.

a, b ARPES spectra measured at 6 K (a) and 75 K (b) with 52 eV photons. c Laser-ARPES spectra measured at 40 K showing the electron band around the $\overline{\mathrm{\Gamma }}$ point. d Laser-ARPES spectra measured at different temperatures. The black line is the extracted band dispersion by fitting the momentum-distribution curves (MDCs) to Lorentzians. e EDCs at the Fermi momentum at different temperatures. f the MDCs at E_F at different temperatures with the fitting results overlaid. The corresponding temperatures are indicated in (e) (dots). g Temperature dependence of the full-width-at-half-maximum (FWHM) of the MDCs at E_F. Error bars represent the confidence intervals of the fitting under 95% confidence level. Data in (c–g) were collected using a 7 eV laser.

The metallic band structure, however, deviates from the observed semiconducting-like resistivity above 55 K. We notice that the spectra are suddenly broadened across the density-wave transition as shown by the MDCs near E_F (Fig. 4f). By fitting the MDCs of the Sb p orbital band at E_F to Lorentzians (black lines in Fig. 4f), we extract the full-width-at-half-maximum (FWHM) of the MDCs and plot the results as a function of temperature in Fig. 4g. Apparently, with increasing temperature, the FWHM quickly rises above about 50 K, suggesting an enhanced electron scattering rate at high temperatures, which is likely due to the enhanced fluctuation effect and may explain the semiconducting-like behavior of CsCr₃Sb₅ above 55 K. Further experimental and theoretical exploration are required to unravel the connection between the electronic structure and density-wave transition in this material.

Discussion

The replacement of vanadium by chromium greatly changes the electronic structure of AV₃Sb₅, making CsCr₃Sb₅ an ideal kagome material to explore correlated physics such as unconventional superconductivity and magnetism. On the one hand, the extra valance electrons in chromium raise the E_F of CsCr₃Sb₅ close to the flat bands compared to CsV₃Sb₅, leaving IFBs near E_F. On the other hand, the Coulomb interaction between Cr 3d electrons induces strong electron correlation, which is further enhanced by the Hund’s coupling. Therefore, the scattering rate or the imaginary part of electron self-energy is greatly enhanced, making the electronic states intrinsically incoherent, as manifested by the broad ARPES spectra of Cr 3d electrons. We emphasize that only the Hubbard U cannot induce the experimentally observed large effective mass⁴¹ and scattering rate (see Figs. 3b, e, and the Supplementary Information), while the correlation effects are sensitive to the Hund’s coupling⁴⁵, verifying CsCr₃Sb₅ as a strongly correlated Hund’s metal.

The IFB and the strong electron correlation of CsCr₃Sb₅ provide an electronic basis for understanding its novel properties, such as large effective mass, magnetism, and unconventional superconductivity. Firstly, the cooperation of Hund’s coupling and the IFBs significantly enhances the orbital-dependent electron correlation effect^45,46, similar to the physics in iron-based superconductors^50–52. Moreover, the IFB can boost the Kondo-like effect, such as heavy fermion in Hund’s metal, according to the previous theoretical calculation. It is, therefore, possible to tune the correlated physics of CsCr₃Sb₅ by controlling the portion and width of the IFBs⁴⁵. Secondly, Hund’s coupling induces the localization of the magnetic moments and intrinsically incoherent Cr 3d states. In the paramagnetic state, the fluctuation of local spin moments further enhances the scattering rate of electrons, as observed in our temperature-dependent measurements, which is related to the weak semiconducting-like transport property at high temperatures. With decreasing temperature, the system is expected to undergo an incoherence-to-coherence crossover by screening the local moments⁴⁵, and the relatively coherent flat bands can be observed near E_F, which is also consistent with our observation of the reduced scattering rate with decreasing temperature. Finally, the incipient flat band is believed to be crucial for unconventional superconductivity^44,47,48. It has been shown that the application of pressure can effectively tune the portions and width of the incipient flat bands⁴⁵, which may be important for the pressurized superconductivity in CsCr₃Sb₅⁴¹. It is noteworthy that similar physics, including the flat bands near E_F, the correlation enhanced by Hund’s coupling, and the competition between density-wave and pressurized superconductivity, has been revealed in the nickelates La_n+1Ni_nO_3n+1^53–56.

In summary, we systematically investigate the electronic structure of the correlated kagome material CsCr₃Sb₅. Our work reveals the characteristic electronic structure including the vHS and flat bands. The electronic scattering is drastically enhanced across the density-wave transition, providing an understanding of the semiconducting-like transport at high temperatures. Consistent with the DFT + DMFT calculations, our experimental results identify that CsCr₃Sb₅ is a strongly correlated Hund’s metal with IFBs. Our work provides crucial information for understanding the novel properties of CsCr₃Sb₅ in comparison to the weakly-correlated and non-magnetic AV₃Sb₅ compounds.

Note: during the review process of the manuscript, we notice that several independent papers have investigated the electronic structures of CsCr₃Sb₅ and Cs(V_1-xCr_x)₃Sb₅ with consistent results^57–59.

METHODS

Sample growth and characterization

Single crystals of CsCr₃Sb₅ were grown via a self-flux method. The details of the crystal growth can be seen in ref. ⁴¹. Hexagonal-shaped crystalline flakes with a typical size of 0.5 × 0.5 × 0.02 mm³ were harvested for the ARPES measurement. Before the ARPES experiment, crystals from the same batch were characterized by X-ray diffraction, energy-dispersive X-ray spectroscopy, and measurements of electrical resistivity and magnetic susceptibility. The resistivity measurement was conducted using a standard four-terminal method. The magnetic measurements were performed on a Magnetic Property Measurement System (MPMS-3, Quantum Design).

ARPES measurements

Synchrotron-based ARPES measurements were conducted at beamline 5-2 in Stanford synchrotron lightsource (SSRL, proposal No. S-XV-ST-6370A). The samples were cleaved in-situ under ultra-high vacuum below 1 × 10^-10mbar. Data were collected with a Scienta DA30L electron analyzer. The total energy and angular resolutions were set to 15 meV and 0.2°, respectively. ARPES experiments were also repeated at beamline 03U and Dreamline in Shanghai Synchrotron Radiation Facility (SSRF).

Laser-ARPES measurements were conducted at Tsinghua University. The 7-eV laser was generated by frequency doubling in a KBBF crystal and focused on the sample by an optical lens with a beam spot of about 20 μm. CsCr₃Sb₅ single crystals were cleaved in-situ under ultra-high vacuum below 5 × 10^-11mbar. Data were collected by a Scienta DA30L electron analyzer. The total energy and angular resolutions were set to 3 meV and 0.2°, respectively.

DFT calculations

First-principles band structure calculations were performed using Vienna ab initio simulation package (VASP)⁶⁰ with a plane wave basis. The exchange-correlation energy was considered under Perdew-Burke-Ernzerhof (PBE) type generalized gradient approximation (GGA)⁶¹ with spin-orbit coupling included. The cutoff energy for the plane-wave basis was set to 500 eV. A Γ-centered k-point mesh of 12 × 12 × 8 was adopted in the self-consistent calculations.

DFT + DMFT calculations

The single-site DFT + DMFT calculations are performed with full charge self-consistency using the DFT+eDMFT code^62,63 based on the WIEN2K package⁶⁴. We choose a large hybridization energy window from -10 to 10 eV, including all the five correlated Cr 3d orbitals and other non-correlated orbitals such as Sb p orbitals, to capture the strong hybridization effect between the Sb p and Cr 3d orbitals. Rotationally-invariant local Coulomb interaction Hamiltonian is used, which is parameterized by on-site Hubbard U and Hund’s coupling J_H. We choose the “exact” double-counting scheme⁶⁵. The continuous-time quantum Monte Carlo (CTQMC)⁶⁶ is used as an impurity solver. The temperature for CTQMC is 100 K. The self-energy on real frequency is obtained by the analytical continuation method of maximum entropy.

Author contributions

L.X.Y. and G.H.C. conceived the experiments. Y.D.L. carried out ARPES measurements with the assistance of X.D., W.X.Z., K.Y.Z., Y.Q.H., S.Y.Z., H.K.C., J.Y.L., C.P., Y.H.Y., D.H.L., M.H., Z.K.L., and Y.L.C. Ab-initio calculations were performed by X.D. and S.Q.W. DFT + DMFT calculations were performed by Y.L.W. Single crystals were synthesized and characterized by Y.L. and G.H.C. The paper was written by Y.D.L. and L.X.Y. All authors contributed to the scientific planning and discussion.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data sets that support the findings of this study are available from the corresponding author upon request.

Competing interests

The authors declare that they have no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-58487-x.