Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Catalogue of flat-band stoichiometric materials

An Author Correction to this article was published on 08 July 2022

This article has been updated

Abstract

Topological electronic flattened bands near or at the Fermi level are a promising route towards unconventional superconductivity and correlated insulating states. However, the related experiments are mostly limited to engineered materials, such as moiré systems1,2,3. Here we present a catalogue of the naturally occuring three-dimensional stoichiometric materials with flat bands around the Fermi level. We consider 55,206 materials from the Inorganic Crystal Structure Database catalogued using the Topological Quantum Chemistry website4,5, which provides their structural parameters, space group, band structure, density of states and topological characterization. We combine several direct signatures and properties of band flatness with a high-throughput analysis of all crystal structures. In particular, we identify materials hosting line-graph or bipartite sublattices—in either two or three dimensions—that probably lead to flat bands. From this trove of information, we create the Materials Flatband Database website, a powerful search engine for future theoretical and experimental studies. We use the database to extract a curated list of 2,379 high-quality flat-band materials, from which we identify 345 promising candidates that potentially host flat bands with charge centres that are not strongly localized on the atomic sites. We showcase five representative materials and provide a theoretical explanation for the origin of their flat bands close to the Fermi energy using the S-matrix method introduced in a parallel work6.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Fig. 1: An illustration of the three possible types of flat band.
Fig. 2: Crystal and band structures of the representative flat-band materials.

Data availability

All data are available in the Supplementary Information and through our public website, the Materials Flatband Database (https://www.topologicalquantumchemistry.fr/flatbands).

Change history

References

  1. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

    Article  ADS  CAS  Google Scholar 

  2. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Article  ADS  CAS  Google Scholar 

  3. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Article  ADS  CAS  Google Scholar 

  4. Vergniory, M. G. et al. A complete catalogue of high-quality topological materials. Nature 566, 480–485 (2019).

    Article  ADS  CAS  Google Scholar 

  5. Vergniory, M. G. et al. All topological bands of all stoichiometric materials. Preprint at https://arxiv.org/abs/2105.09954 (2021).

  6. Călugăru, D. et al. General construction and topological classification of crystalline flat bands. Nat. Phys. 18, 185–189 (2022).

    Article  Google Scholar 

  7. Kumar, P., Peotta, S., Takasu, Y., Takahashi, Y. & Törmä, P. Flat-band-induced non-Fermi-liquid behavior of multicomponent fermions. Phys. Rev. A 103, L031301 (2021).

    Article  ADS  CAS  Google Scholar 

  8. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    Article  ADS  CAS  Google Scholar 

  9. Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983).

    Article  ADS  Google Scholar 

  10. Moore, G. & Read, N. Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  11. Drozdov, A., Eremets, M., Troyan, I., Ksenofontov, V. & Shylin, S. I. Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature 525, 73–76 (2015).

    Article  ADS  CAS  Google Scholar 

  12. Drozdov, A. et al. Superconductivity at 250 K in lanthanum hydride under high pressures. Nature 569, 528–531 (2019).

    Article  ADS  CAS  Google Scholar 

  13. Tang, E., Mei, J.-W. & Wen, X.-G. High-temperature fractional quantum Hall states. Phys. Rev. Lett. 106, 236802 (2011).

    Article  ADS  Google Scholar 

  14. Neupert, T., Santos, L., Chamon, C. & Mudry, C. Fractional quantum Hall states at zero magnetic field. Phys. Rev. Lett. 106, 236804 (2011).

    Article  ADS  Google Scholar 

  15. Sheng, D., Gu, Z.-C., Sun, K. & Sheng, L. Fractional quantum Hall effect in the absence of Landau levels. Nat. Commun. 2, 389 (2011).

    Article  ADS  CAS  Google Scholar 

  16. Regnault, N. & Bernevig, B. A. Fractional Chern insulator. Phys. Rev. X 1, 021014 (2011).

    Google Scholar 

  17. Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).

    Article  CAS  Google Scholar 

  18. Peri, V., Song, Z.-D., Bernevig, B. A. & Huber, S. D. Fragile topology and flat-band superconductivity in the strong-coupling regime. Phys. Rev. Lett. 126, 027002 (2021).

    Article  ADS  CAS  Google Scholar 

  19. Rhim, J.-W., Kim, K. & Yang, B.-J. Quantum distance and anomalous Landau levels of flat bands. Nature 584, 59–63 (2020).

    Article  ADS  CAS  Google Scholar 

  20. Xie, F., Song, Z., Lian, B. & Bernevig, B. A. Topology-bounded superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 124, 167002 (2020).

    Article  ADS  CAS  Google Scholar 

  21. Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).

    Article  ADS  CAS  Google Scholar 

  22. Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

    Article  ADS  CAS  Google Scholar 

  23. Y. Xu et al. Filling-enforced obstructed atomic insulators. Preprint at https://arxiv.org/abs/2106.10276 (2021).

  24. Mielke, A. Exact ground states for the Hubbard model on the kagome lattice. J. Phys. A 25, 4335–4345 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  25. Tasaki, H. From Nagaoka’s ferromagnetism to flat-band ferromagnetism and beyond: an introduction to ferromagnetism in the Hubbard model. Prog. Theor. Phys. 99, 489–548 (1998).

    Article  ADS  CAS  Google Scholar 

  26. Bergman, D. L., Wu, C. & Balents, L. Band touching from real-space topology in frustrated hopping models. Phys. Rev. B 78, 125104 (2008).

    Article  ADS  Google Scholar 

  27. Liu, Z., Liu, F. & Wu, Y.-S. Exotic electronic states in the world of flat bands: from theory to material. Chin. Phys. B 23, 077308 (2014).

    Article  ADS  Google Scholar 

  28. Ma, D.-S. et al. Spin–orbit-induced topological flat bands in line and split graphs of bipartite lattices. Phys. Rev. Lett. 125, 266403 (2020).

    Article  ADS  CAS  Google Scholar 

  29. Chiu, C. S., Ma, D.-S., Song, Z.-D., Bernevig, B. A. & Houck, A. A. Fragile topology in line-graph lattices with two, three, or four gapped flat bands. Phys. Rev. Res. 2, 043414 (2020).

    Article  CAS  Google Scholar 

  30. Inorganic Crystal Structure Database (ICSD) (Fachinformationszentrum Karlsruhe, 2015); https://icsd.products.fiz-karlsruhe.de/.

  31. Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964).

    Article  ADS  MathSciNet  Google Scholar 

  32. Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).

    Article  ADS  MathSciNet  Google Scholar 

  33. Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. Phys. Rev. B 48, 13115–13118 (1993).

    Article  ADS  CAS  Google Scholar 

  34. Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).

    Article  CAS  Google Scholar 

  35. Ivantchev, S., Kroumova, E., Madariaga, G., Pérez-Mato, J. M. & Aroyo, M. I. SUBGROUPGRAPH: a computer program for analysis of group–subgroup relations between space groups. J. Appl. Crystallogr. 33, 1190–1191 (2000).

    Article  CAS  Google Scholar 

  36. Ivantchev, S. et al. SUPERGROUPS—a computer program for the determination of the supergroups of the space groups. J. Appl. Crystallogr. 35, 511–512 (2002).

    Article  CAS  Google Scholar 

  37. Souza, I., Marzari, N. & Vanderbilt, D. Maximally localized Wannier functions for entangled energy bands. Phys. Rev. B 65, 035109 (2001).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank X. Dai, D. Calugaru, A. Chew, M. Vergniory and C. Chiu for discussions. We acknowledge the computational resources Cobra/Draco in the Max Planck Computing and Data Facility (MPCDF) and Atlas in the Donostia International Physics Center (DIPC). This research also used the resources of the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility operated under contract number DE-AC02-05CH11231. This work is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 101020833). B.A.B. and N.R. were also supported by the US Department of Energy (grant number DE-SC0016239), and were partially supported by the National Science Foundation (EAGER grant number DMR 1643312), a Simons Investigator grant (number 404513), the Office of Naval Research (ONR grant number N00014-20-1-2303), the Packard Foundation, the Schmidt Fund for Innovative Research, the BSF Israel US foundation (grant number 2018226), the Gordon and Betty Moore Foundation through grant number GBMF8685 towards the Princeton theory programme, and a Guggenheim Fellowship from the John Simon Guggenheim Memorial Foundation. A.Y., N.P.O., R.J.C., L.M.S., B.A.B. and N.R. were supported by the NSF-MRSEC (grant number DMR-2011750). A.Y. was supported by NSF-DMR-1904442. B.A.B., L.M.S. and N.R. acknowledge financial support from the Schmidt DataX Fund at Princeton University made possible through a major gift from the Schmidt Futures Foundation. L.M.S. acknowledges financial support from the Packard and Sloan Foundation. L.E. was supported by the Government of the Basque Country (Project IT1301-19) and the Spanish Ministry of Science and Innovation (PID2019-106644GB-I00). C.F. was supported by the European Research Council (ERC) advanced grant number 742068 ‘TOP-MAT’, Deutsche Forschungsgemeinschaft (DFG) through SFB 1143, and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat (EXC 2147, project number 390858490). S.S.P.P. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—project number 314790414.

Author information

Authors and Affiliations

Authors

Contributions

B.A.B. and N.R. conceived this work; N.R. and M.-R.L. performed the high-throughput calculations with the help from L.E. and Y.X.; Y.X., D.-S.M., Z.-D.S., M.-R.L., L.E. and N.R. worked out the theoretical explanations for the flat-band materials detailed in Supplementary Section F; the material lists in Supplementary Section H were manually selected by Y.X., M.-R.L., Z.-D.S., M.J. and N.R.; N.R. built the flat-band material database; D.-S.M. performed the ab initio ferromagnetic calculations advised by Y.X.; M.J., L.S. and C.F. helped curate the list of materials to find the most experimentally relevant. All authors discussed the results and wrote the main text and Methods; Y.X., Z.-D.S., M-.R.L., D.-S.M., M.J., L.E. and N.R. wrote the Supplementary Information.

Corresponding authors

Correspondence to Nicolas Regnault, Yuanfeng Xu or B. Andrei Bernevig.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks David Carpentier and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

This file contains supplementary text, equations, tables, figures and references.

Peer Review File

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Regnault, N., Xu, Y., Li, MR. et al. Catalogue of flat-band stoichiometric materials. Nature 603, 824–828 (2022). https://doi.org/10.1038/s41586-022-04519-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-022-04519-1

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing