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Frustrated electron hopping from the orbital configuration in a two-dimensional lattice

Nature Physics volume 21, pages 1260–1266 (2025)Cite this article

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Abstract

Electron hopping on spatially periodic lattices gives rise to intriguing electronic behaviour. For example, hopping on the geometrically frustrated two-dimensional kagome, dice and Lieb lattices yields electronic band structures with both massless Dirac-like and perfectly dispersion-less, flat bands. As materials featuring the dice and Lieb lattice structures are scarce, an alternative approach proposes to leverage atomic orbitals to realize the characteristic electron hopping of geometrically frustrated lattices. This strategy promises to expand the list of candidate materials with frustrated electron hopping, but is yet to be shown in experiments. Here we demonstrate frustrated hopping in the van der Waals intermetallic Pd5AlI2, emerging from the arrangement of atomic orbitals in a primitive square lattice. Using angle-resolved photoemission spectroscopy and quantum oscillation measurements, we reveal that the band structure of Pd5AlI2 includes linear Dirac-like bands intersected at their crossing point by a locally flat band—an essential characteristic of frustrated hopping in Lieb and dice lattices. Moreover, this compound shows exceptional chemical stability, with its unusual bulk band structure and metallicity persisting in ambient conditions down to the monolayer limit. Hence, our results showcase a way to realize electronic structures characteristic of geometrically frustrated lattices in non-frustrated systems.

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Fig. 1: Frustrated hopping from orbital decoration and the vdW metal Pd5AlI2.
Fig. 2: Bulk electronic structure and transport behaviour of Pd5AlI2.
Fig. 3: Electronic transport and fermiology towards the monolayer limit.
Fig. 4: Thickness-dependent Hall response and survey of vdW metals.

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Data availability

The data that support the findings of this study are presented in the published figures and tables and are available via the Harvard Dataverse repository at https://doi.org/10.7910/DVN/NAD0B9. Further data are available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank J. L. Pack, M. A. Kapfer, E. J. Telford, S. E. Turkel and J. G. Checkelsky for fruitful discussions. Research on novel hopping models was conducted as part of the Programmable Quantum Materials, an Energy Frontier Research Center, funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, under award no. DE-SC0019443 (X.R., C.R.D., A.N.P., R.Q., X.Z. and M.D.). X.R. acknowledges support from a Brown Investigator Award, a program of the Brown Institute for Basic Sciences at the California Institute of Technology. The synthesis and structural characterization of the 2D metal were supported by the NSF MRSEC program through the Center for Precision-Assembled Quantum Materials at Columbia University, award no. DMR-2011738 (X.R., C.R.D., A.N.P. and X.Z.). X.Z. acknowledges support for STM/STS experiments by the Air Force Office of Scientific Research under award no. FA9550-22-1-0389. The Physical Property Measurement System used to perform vibrating-sample magnetometry and electrical transport measurements was purchased with financial support from the National Science Foundation through a supplement to award no. DMR-1751949 (X.R.). A.D. acknowledges support from the Simons Foundation Society of Fellows (grant no. 855186). High-magnetic-field measurements were performed at the National High Magnetic Field Laboratory, supported by the National Science Foundation Cooperative Agreement no. DMR-1644779 and the State of Florida. ARPES measurements used resources at the 21-ID (Electron Spectro-Microscopy) beamline of the National Synchrotron Light Source II, a US DOE, Office of Science User Facility, operated for the DOE, Office of Science, Brookhaven National Laboratory, under contract no. DE-SC0012704.

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Author notes

  1. Aravind Devarakonda

    Present address: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA

  2. Zhi Lin

    Present address: Department of Physics, Boston University, Boston, MA, USA

  3. These authors contributed equally: Aravind Devarakonda, Christie S. Koay, Daniel G. Chica.

Authors and Affiliations

  1. Department of Physics, Columbia University, New York, NY, USA

    Aravind Devarakonda, Sebastian Rossi, Madisen A. Holbrook, Abhay N. Pasupathy, Raquel Queiroz & Cory R. Dean

  2. Department of Chemistry, Columbia University, New York, NY, USA

    Christie S. Koay, Daniel G. Chica, Morgan Thinel, Sae Young Han, Michael E. Ziebel, Milan Delor, Xiaoyang Zhu & Xavier Roy

  3. Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, NY, USA

    Asish K. Kundu & Abhay N. Pasupathy

  4. National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York, NY, USA

    Asish K. Kundu, Anil Rajapitamahuni & Elio Vescovo

  5. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA

    Zhi Lin

  6. Department of Chemistry, Indiana University, Bloomington, IN, USA

    Alexandru B. Georgescu

  7. Research Center for Electronic and Optical Materials, National Institute of Materials Science, Tsukuba, Japan

    Kenji Watanabe

  8. Research Center for Materials Nanoarchitectonics, National Institute of Materials Science, Tsukuba, Japan

    Takashi Taniguchi

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Contributions

C.S.K. and D.G.C. synthesized and characterized the bulk crystals. S.Y.H. supported with the energy-dispersive X-ray spectroscopy and scanning electron microscopy experiments, and M.E.Z. supported with the magnetization measurements. A.D. fabricated the nanodevices and performed the transport characterization. A.D., C.S.K. and D.G.C. conducted the high-field measurements. M.T. performed the STM characterization. A.K.K. performed the ARPES measurements with support from A.R. and E.V. K.W. and T.T. provided the hexagonal boron nitride crystals. Z.L., A.B.G., A.D. and R.Q. performed the theoretical calculations. A.N.P., R.Q., C.R.D. and X.R. supervised the project. A.D., C.R.D. and X.R. wrote the manuscript with input from all authors.

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Correspondence to Aravind Devarakonda, Abhay N. Pasupathy, Raquel Queiroz, Cory R. Dean or Xavier Roy.

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Extended data

Extended Data Fig. 1 SEM imaging and EDS of Pd5AlI2.

a SEM image of Pd5AlI2 single crystal outlined in white. b EDS from a representative area scan on the crystal shown. Using statistics collected from ten area scans using the Al content as the reference, the Pd stoichiometry is 4.783 ± 0.174, Al stoichiometry is 1.0, and I stoichiometry is 1.843 ± 0.049.

Extended Data Fig. 2 ARPES electronic structure and photon energy dependence.

ARPES E(k) intensity map (LH polarization) along the \(\overline{\rm{M}}-\overline{{\Gamma}}-\overline{\rm{M}}\) line of the surface BZ at various incoming photon energies hν, a 127 eV b 136 eV and c 142 eV, probing distinct slices of kz, marked in e. The weak variation of these maps evidences the quasi-2D nature of the electronic structure. d ARPES Fermi surface cross-sections in the kx – ky plane (LV polarization) at various kz. The cross-sections are similarly invariant with kz. e An ARPES kz – k|| map along the \(\overline{\rm{M}}-\overline{{\Gamma}}-\overline{\rm{M}}\) line (LH polarization) exhibits cylindrical FSs for all three pockets. The dashed lines show the approximate kz sampled by the E(k) maps a – c. ARPES FS maps (hν = 125 eV) at f 100 meV g 150 meV h and 750 meV below EF. The experimental maps show good correspondence to DFT calculated FSs at i 100 meV j 150 meV and k 645 meV below EF.

Extended Data Fig. 3 Pd5AlI2 AFM topography and optical contrast.

a Topography of terraced Pd5AlI2 flake exfoliated onto SiO2/Si wafer measured by AFM with b linecut along magenta segment showing single Pd5AlI2 layer steps. c Optical microscope image of the same flake showing d contrast in the red, green, and blue color channels between regions of varying thickness. e Spinning a layer of PMMA over the flake f enhances optical contrast, making it easier to distinguish regions with different layer number.

Extended Data Table 1 Single crystal X-ray refinement
Full size table
Extended Data Table 2 Survey of exfoliated van der Waals metals
Full size table

Supplementary information

Supplementary Information

Supplementary Sections I–IV, Figs. 1–5 and Discussion.

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Devarakonda, A., Koay, C.S., Chica, D.G. et al. Frustrated electron hopping from the orbital configuration in a two-dimensional lattice. Nat. Phys. 21, 1260–1266 (2025). https://doi.org/10.1038/s41567-025-02953-2

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