Abstract
Electronic flat-band materials host quantum states characterized by a quenched kinetic energy. These flat bands are often conducive to enhanced electron correlation effects and emergent quantum phases of matter1. Long studied in theoretical models2,3,4, these systems have received renewed interest after their experimental realization in van der Waals heterostructures5,6 and quasi-two-dimensional (2D) crystalline materials7,8. An outstanding experimental question is if such flat bands can be realized in three-dimensional (3D) networks, potentially enabling new materials platforms9,10 and phenomena11,12,13. Here we investigate the C15 Laves phase metal CaNi2, which contains a nickel pyrochlore lattice predicted at a model network level to host a doubly-degenerate, topological flat band arising from 3D destructive interference of electronic hopping14,15. Using angle-resolved photoemission spectroscopy, we observe a band with vanishing dispersion across the full 3D Brillouin zone that we identify with the pyrochlore flat band as well as two additional flat bands that we show arise from multi-orbital interference of Ni d-electrons. Furthermore, we demonstrate chemical tuning of the flat-band manifold to the Fermi level that coincides with enhanced electronic correlations and the appearance of superconductivity. Extending the notion of intrinsic band flatness from 2D to 3D, this provides a potential pathway to correlated behaviour predicted for higher-dimensional flat-band systems ranging from tunable topological15 to fractionalized phases16.
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Data availability
The data presented in the main text of this article are available from the Harvard Dataverse53. All other data are available from the corresponding authors upon reasonable request.
Code availability
The codes used for the density functional theory and analytical calculations in this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We appreciate the discussions with J. Cano and J. Singleton. This work was funded, in part, by the Gordon and Betty Moore Foundation EPiQS Initiative (grant no. GBMF9070 to J.G.C.) (instrumentation development, DFT calculations), the Air Force Office of Scientific Research (AFOSR) (award FA9550-22-1-0432) (material synthesis, ARPES) and the NSF (DMR-2104964) (material analysis) and the Center for Advancement of Topological Semimetals, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), through the Ames Laboratory (contract no. DE-AC02-07CH11358) (pulsed-field experiments). P.M.N. acknowledges support from the STC Center for Integrated Quantum Materials (NSF grant DMR-1231319). M.K., S.L., S.P. and J.-H.P. acknowledge support from the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (nos 2022M3H4A1A04074153 and 2020M3H4A2084417). M.L. acknowledges support from the DOE, BES (DE-SC0020148). T.S. acknowledges support from JSPS KAKENHI (grant no. 21K03454). This research used resources of the Advanced Light Source, which is a DOE, Office of Science, user facility (contract no. DE-AC02-05CH11231). This research used resources from the Advanced Photon Source, a DOE, Office of Science, user facility operated for the DOE, Office of Science, by the Argonne National Laboratory (contract no. DE-AC02-06CH11357). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement no. DMR-1644779, the State of Florida and the DOE. We thank the MIT SuperCloud54 and the Lincoln Laboratory Supercomputing Center for providing high-performance computing resources that have contributed to the results reported in this study.
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J.P.W., P.M.N. and R.M. synthesized the materials with support from T.N.L., T.S. and M.L.; J.P.W. and P.M.N. characterized the materials; M.K., D.O., J.P.W. and P.M.N. performed and analysed VUV ARPES experiments with support from A.C., S.L., S.P., J.-H.P., C.J., A.B., E.R., A.R. and E.V.; J.P.W., P.M.N. and R.M. performed and analysed soft X-ray ARPES with support from J.L.M.; J.P.W. and P.M.N. performed and analysed quantum oscillation experiments with support from S.Y.F.Z., A.C., D.G. and J.C.P; S.F. performed DFT calculations; and J.P.W., P.M.N., S.F. and T.S. performed electronic and structural modelling and analysis. All authors contributed to writing the paper. R.C. and J.G.C. coordinated the project.
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Extended data figures and tables
Extended Data Fig. 1 Experimental setup of VUV-ARPES.
(a) Experimental geometry of VUV-ARPES. (b) The simulated trajectory of photoelectrons for the photon energy-dependent ARPES experiment. Photoelectron trajectories from 80 eV to 200 eV were plotted as red and green curves with 5 eV steps. Black solid lines correspond to the bulk Brillouin zones. 3D Brillouin zones truncated by (c) kx-kz plane at ky = 0, (d) kx-ky plane at kz = 0, and (e) kx-ky plane at kz = π/c.
Extended Data Fig. 2 Additional analysis of the flat bands in CaNi2.
(a) Schematics of the 3D Brillouin zones truncated by kz = 0 plane of the first Brillouin zone. The truncation exposes the kz = 0 plane of the first Brillouin zone and the kz = 2/3 × π/c plane of the second Brillouin zones as marked with blue areas in (b). (c) Schematics of the 3D Brillouin zones truncated by kz = π/c plane of the first Brillouin zone. The truncation exposes the kz = π/c plane of the first Brillouin zone and the kz = −1/3 × π/c plane of the second Brillouin zones as marked with blue areas in (d). In (c),(d), we defined a new momentum point Γ1 as the projection of Γ to the kz = −1/3 × π/c plane. (e),(h),(k),(n) ARPES spectra of CaNi2 measured along the momentum directions marked with dashed-lines in (a-d). The corresponding second derivative plots and DFT band structures are shown in (f),(i),(l),(o) and (g),(j),(m),(p), respectively. The red, blue, and green arrows in (f),(i),(l),(o) mark the positions of FB1, FB2, and FB3, respectively.
Extended Data Fig. 3 Pyrochlore band features in CaNi2.
(a-c) Dispersion of CaNi2 around the Brillouin zone center Γ. The spectrum in (a) (obtained from the photon energy-dependent ARPES measurements) was normalized using the intensity of the flat band at −0.58 eV. We summed two spectra collected by photons with linear vertical and linear horizontal polarization to minimize matrix element effects. The black arrow marks the quadratic band touching points (QBT) expected from the model pyrochlore tight-binding calculation. (d) ARPES spectrum obtained with 175 eV photons with linear vertical polarization. The black arrows indicate the Dirac nodes at W. (e) Second derivative plot of (d). (f) The DFT band dispersion of CaNi2 to be compared with (d),(e). (g) EDC plots of CaNi2 showing the FB1, FB2, and FB3 dispersions (red, blue, and green circles, respectively). (h) The energy-momentum dispersion along the high symmetry directions. The dispersions along different high symmetry directions were obtained from the following measurements: the Γ-K-X dispersion from the kz = 0 Fermi surface obtained using 217 eV photons, the Γ-L dispersion from the photon-energy dependence measurements from 135 eV to 175 eV photons, and the L-U-W-L dispersion from the kz = π/c Fermi surface obtained using 175 eV photons.
Extended Data Fig. 4 Topological band features of CaRh2.
(a) The ARPES spectrum of Ca(Rh0.98Ru0.02)2 along the Γ-K-X direction measured with 127 eV photons. The flat band corresponds to the FB2 in CaNi2 and the dispersive band with band top at Γ are marked with black arrows. Detailed dispersions of the flat band and the dispersive band are obtained from the corresponding EDC analyses in (c) and overlaid on top of the ARPES spectrum in (a). The double-headed arrows in (a) and (c) indicate the SOC gap between the flat band and the dispersive band. (b) DFT band structure of CaRh2 for comparison with the experimental dispersions in (a),(c). (d,e) Wide energy-range ARPES spectrum of Ca(Rh0.98Ru0.02)2 and its second derivative plot measured along the Γ-K-X direction. (f) DFT band structure of CaRh2 for comparison with (d,e). The red arrows in (d-f) mark the symmetry-protected Dirac crossing at X.
Extended Data Fig. 5 Saddle point dispersion at L.
(a) Experimental Fermi surface of Ca(Rh0.98Ru0.02)2. The red and yellow-dashed lines are the momentum directions crossing L point, along which the ARPES spectra in (c) and (f) are acquired, respectively. (b) DFT Fermi surface for CaRh2. (c-e) ARPES spectrum, second derivative plot, and DFT band structure along the kx direction marked in a. (f-h) Corresponding plots along the ky direction marked in a. In panels c,f, we overlaid the DFT dispersions to highlight the hole-like band along kx and the electron-like band along ky, together forming the saddle point at L. Red arrows in panels d,g also mark the saddle point dispersion. We note that the flat-like features in the second derivative plots are artifacts from the Fermi cutoff. In addition to the saddle point dispersion, we detected an additional highly dispersing band crossing the Fermi level between the L and W in the ARPES spectra of c. This band is not reproduced in the bulk DFT band calculation and is potentially of surface origin.
Extended Data Fig. 6 Quantum oscillations in CaNi2.
(a) Polynomial-subtracted magnetic torque data used to extract dHvA frequencies for CaNi2 sample T1 (main) and (b) sample T2 taken at base temperature of the He-3 cryostat. (c,d) Corresponding fast Fourier transform amplitudes (dashed lines are guides to the eye labelled with the orbits identified from calculations). (e) Orientation of θ relative to the crystallographic high symmetry directions for all data in this figure. (f) Observed and calculated quantum oscillation frequencies for CaNi2 as a function of field-tilt angle. (g) Temperature dependent oscillation data taken at θ = 63° relative to the [100] direction. (h) Corresponding temperature-dependent Fourier transform amplitudes. (i) Temperature dependence of the Fourier transform amplitudes at θ = 63°. Solid lines are fittings to the Lifshitz-Kosevich formula where all 0 K values have been normalized to unity from which corresponding experimental effective mass parameters have been extracted.
Extended Data Fig. 7 Quantum oscillations in Ca(Rh0.98Ru0.02)2.
(a) Observed and calculated quantum oscillation frequencies of CaRh2 as a function of field-angle away from [111], with the geometry as defined in panel (d). (b) Polynomial-subtracted magnetic torque data as a function of field taken at base temperature of the He-3 cryostat, (c) corresponding fast Fourier transform amplitudes (dashed lines are guides to the eye labelled with the orbits identified from calculations). (d) Orientation of θ relative to the crystallographic high symmetry directions, corresponding to the “binary” rotation direction. (e) Temperature dependent oscillation data taken at θ = 176° corresponding to the [111] direction. (f) Corresponding temperature-dependent Fourier transform amplitudes. (g) Temperature dependence of the Fourier transform amplitudes. Solid lines are fittings to the Lifshitz-Kosevich formula where all 0 K values have been normalized to unity from which corresponding experimental effective mass parameters have been extracted.
Extended Data Fig. 8 DFT Fermi surfaces of CaNi2.
(a-e) Calculated Fermi sheets of CaNi2 with their corresponding label plotted within one Brillouin zone. (f-h) Schematic showing the locations of characteristic extremal frequencies of the δ Fermi sheet for field along high symmetry directions (f) shows the two lowest frequency branches δ1 and δ2 for B || [111] (g) δ3 for B || [100], and (h) δ4 for B || [011].
Extended Data Fig. 9 Fermi surfaces of CaNi2.
(a) Fermi surface at kz = π/c and (b) kz = 0. The Brillouin zones are overlaid with black dashed lines. Left and right panels represent the experimental and calculated Fermi surfaces, respectively.
Extended Data Fig. 10 DFT Fermi surfaces of CaRh2.
(a-d) Calculated Fermi sheets of CaRh2 with their corresponding label plotted within one Brillouin zone.
Supplementary information
Supplementary Information
This file contains the following sections: (I) Single-crystal characterization; (II) Supplementary high-field measurements; (III) Comparison of renormalization factors; (IV) Density functional theory calculations; (V) Tight-binding models for the pyrochlore lattice; and additional references.
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Wakefield, J.P., Kang, M., Neves, P.M. et al. Three-dimensional flat bands in pyrochlore metal CaNi2. Nature 623, 301–306 (2023). https://doi.org/10.1038/s41586-023-06640-1
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DOI: https://doi.org/10.1038/s41586-023-06640-1
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