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Pseudogap in a crystalline insulator doped by disordered metals


Key to our understanding of how electrons behave in crystalline solids is the band structure that connects the energy of electron waves to their wavenumber. Even in phases of matter with only short-range order (liquid or amorphous solid), the coherent part of electron waves still has a band structure. Theoretical models for the band structure of liquid metals were formulated more than five decades ago1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, but, so far, band-structure renormalization and the pseudogap induced by resonance scattering have remained unobserved. Here we report the observation of the unusual band structure at the interface of a crystalline insulator (black phosphorus) and disordered dopants (alkali metals). We find that a conventional parabolic band structure of free electrons bends back towards zero wavenumber with a pseudogap of 30–240 millielectronvolts from the Fermi level. This is wavenumber renormalization caused by resonance scattering, leading to the formation of quasi-bound states in the scattering potential of alkali-metal ions. The depth of this potential tuned by different kinds of disordered alkali metal (sodium, potassium, rubidium and caesium) allows the classification of the pseudogap of p-wave and d-wave resonance. Our results may provide a clue to the puzzling spectrum of various crystalline insulators doped by disordered dopants16,17,18,19,20, such as the waterfall dispersion observed in copper oxides.

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Fig. 1: Band structure of liquid metals and experimental systems.
Fig. 2: Electronic structure of black phosphorus doped by disordered alkali metals.
Fig. 3: Resonance scattering in liquid metals and band renormalizations.
Fig. 4: Density of states in liquid metals and spectral simulations.
Fig. 5: Magnitude of the pseudogap and phase diagram.

Data availability

The data that support the findings of this study are available within the paper and from the corresponding author upon reasonable request. Source data are provided with this paper.


  1. Edwards, S. F. The electronic structure of disordered systems. Phil. Mag. 6, 617–638 (1961).

    Article  ADS  Google Scholar 

  2. Edwards, S. F. The electronic structure of liquid metals. Proc. R. Soc. Lond. A 267, 518–540 (1962).

    Article  ADS  CAS  Google Scholar 

  3. Ziman, J. M. A theory of the electrical properties of liquid metals. I: The monovalent metals. Phil. Mag. 6, 1013–1034 (1961).

    Article  ADS  CAS  Google Scholar 

  4. Ziman, J. M. The T matrix, the K matrix, d bands and l-dependent pseudo-potentials in the theory of metals. Proc. Phys. Soc. 86, 337–353 (1965).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  5. Anderson, P. W. & McMillan, W. L. Multiple-scattering theory and resonances in transition metals. In Proc. International School of Physics “Enrico Fermi” Course 37 (ed. Marshall, W.) 50–86 (Academic, 1967).

  6. Morgan, G. J. Electron transport in liquid metals II. A model for the wave functions in liquid transition metals. J. Phys. C 2, 1454–1464 (1969).

    Article  ADS  Google Scholar 

  7. Gyorffy, B. L. Electronic states in liquid metals: a generalization of the coherent-potential approximation for a system with short-range order. Phys. Rev. B 1, 3290–3299 (1970).

    Article  ADS  Google Scholar 

  8. Schwartz, L. & Ehrenreich, H. Single-site approximations in the electronic theory of liquid metals. Ann. Phys. 64, 100–148 (1971).

    Article  ADS  CAS  Google Scholar 

  9. Faber, T. E. An Introduction to the Theory of Liquid Metals (Cambridge Univ. Press, 1972).

  10. Olson, J. J. Anderson–McMillan prescription for the density of states of liquid iron. Phys. Rev. B 12, 2908–2916 (1975).

    Article  ADS  CAS  Google Scholar 

  11. Chang, K. S., Sher, A., Petzinger, K. G. & Weisz, G. Density of states of liquid Cu. Phys. Rev. B 12, 5506–5513 (1975).

    Article  ADS  CAS  Google Scholar 

  12. Oglesby, J. & Lloyd, P. Some single-site structure-independent approximations in condensed materials. J. Phys. C 9, 2879–2886 (1976).

    Article  ADS  CAS  Google Scholar 

  13. Mott, N. F. Metal–insulator transition. Rev. Mod. Phys. 40, 677–683 (1968).

    Article  ADS  CAS  Google Scholar 

  14. Mott, N. F. Conduction in non-crystalline materials: III. Localized states in a pseudogap and near extremities of conduction and valence bands. Phil. Mag. 19, 835–852 (1968).

    Article  ADS  Google Scholar 

  15. Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

    Article  ADS  CAS  Google Scholar 

  16. Lu, D. et al. Angle-resolved photoemission studies of quantum materials. Annu. Rev. Condens. Matter Phys. 3, 129–167 (2012).

    Article  CAS  Google Scholar 

  17. Graf, J. et al. Universal high energy anomaly in the angle-resolved photoemission spectra of high temperature superconductors: possible evidence of spinon and holon branches. Phys. Rev. Lett. 98, 067004 (2007).

    Article  ADS  CAS  Google Scholar 

  18. Inosov, D. S. et al. Momentum and energy dependence of the anomalous high-energy dispersion in the electronic structure of high temperature superconductors. Phys. Rev. Lett. 99, 237002 (2007).

    Article  ADS  CAS  Google Scholar 

  19. Kim, Y. K., Sung, N. H., Denlinger, J. D. & Kim, B. J. Observation of a d-wave gap in electron-doped Sr2IrO4. Nat. Phys. 12, 37–41 (2016).

    Article  CAS  Google Scholar 

  20. Uchida, M. et al. Pseudogap of metallic layered nickelate R2−xSrxNiO4 (R = Nd, Eu) crystals measured using angle-resolved photoemission spectroscopy. Phys. Rev. Lett. 106, 027001 (2011).

    Article  ADS  CAS  Google Scholar 

  21. Baumberger, F., Auwärter, W., Greber, T. & Osterwalder, J. Electron coherence in a melting lead monolayer. Science 306, 2221–2224 (2004).

    Article  ADS  CAS  Google Scholar 

  22. Kim, K. S. & Yeom, H. W. Radial band structure of electrons in liquid metals. Phys. Rev. Lett. 107, 136402 (2011).

    Article  ADS  Google Scholar 

  23. Kim, J. et al. Observation of tunable band gap and anisotropic Dirac semimetal state in black phosphorus. Science 349, 723–726 (2015).

    Article  CAS  Google Scholar 

  24. Baik, S. S., Kim, K. S., Yi, Y. & Choi, H. J. Emergence of two-dimensional massless Dirac fermions, chiral pseudospins, and Berry’s phase in potassium doped few-layer black phosphorus. Nano Lett. 15, 7788–7793 (2015).

    Article  ADS  CAS  Google Scholar 

  25. Kiraly, B. et al. Anisotropic two-dimensional screening at the surface of black phosphorus. Phys. Rev. Lett. 123, 216403 (2019).

    Article  ADS  CAS  Google Scholar 

  26. Tian, Z. et al. Isotropic charge screening of anisotropic black phosphorus revealed by potassium adatoms. Phys. Rev. B 100, 085440 (2019).

    Article  ADS  CAS  Google Scholar 

  27. Negulyaev, N. N. et al. Melting of two-dimensional adatom superlattices stabilized by long-range electronic interactions. Phys. Rev. Lett. 102, 246102 (2009).

    Article  ADS  CAS  Google Scholar 

  28. Hirata, A. et al. Direct observation of local atomic order in a metallic glass. Nat. Mater. 10, 28–33 (2011).

    Article  ADS  CAS  Google Scholar 

  29. Sakurai, J. J. & Napolitano, J. Modern Quantum Mechanics (Cambridge Univ. Press, 2017).

  30. Meyer, A., Nestor, C. W. Jr & Young, W. H. Pseudo-atom phase shifts of liquid metals and alloys. Adv. Phys. 16, 581–590 (1967).

    Article  ADS  CAS  Google Scholar 

  31. Kang, M. et al. Evolution of charge order topology across a magnetic phase transition in cuprate superconductors. Nat. Phys. 15, 335–340 (2019); correction 15, 721 (2019).

    Article  ADS  CAS  Google Scholar 

  32. Voit, J. et al. Electronic structure of solids with competing periodic potentials. Science 290, 501–503 (2000).

    Article  ADS  CAS  Google Scholar 

  33. Park, S. R. et al. Electronic structure of electron-doped Sm1.86Ce0.14CuO4: strong pseudogap effects, nodeless gap, and signatures of short-range order. Phys. Rev. B 75, 060501 (2007).

    Article  ADS  Google Scholar 

  34. Lee, W. S. et al. Abrupt onset of a second energy gap at the superconducting transition of underdoped Bi2212. Nature 450, 81–84 (2007).

    Article  ADS  CAS  Google Scholar 

  35. Presland, M. R., Tallon, J. L., Buckley, R. G., Liu, R. S. & Flower, N. E. General trends in oxygen stoichiometry effects on Tc in Bi and Tl superconductors. Physica C 176, 95–105 (1991).

    Article  ADS  CAS  Google Scholar 

  36. Fischer, Ø., Kugler, M., Maggio-Aprile, I., Berthod, C. & Renner, C. Scanning tunneling spectroscopy of high-temperature superconductors. Rev. Mod. Phys. 79, 353–419 (2007).

    Article  ADS  CAS  Google Scholar 

  37. Jung, S. W. et al. Black phosphorus as a bipolar pseudospin semiconductor. Nat. Mater. 19, 277–281 (2020).

    Article  ADS  CAS  Google Scholar 

  38. March, N. H. Liquid Metals (Cambridge Univ. Press, 1990).

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We thank S.-S. Baik, Y.-W. Son and E.-G. Moon for helpful discussions; and D. Song and C. Kim for providing samples. This work was supported by National Research Foundation (NRF) of Korea (grant numbers NRF-2020R1A2C2102469, NRF-2017R1A5A1014862, NRF-2020K1A3A7A09080364 and NRF-2021R1A3B1077156), and the Yonsei Signature Research Cluster Program of 2021 (2021-22-0004). This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract number DE-AC02-05CH11231. The part of this work on copper oxides was carried out with the support of the Diamond Light Source (beamline I05).

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Authors and Affiliations



S.H.R., M.H. and D.Y.P. performed ARPES experiments with help from C.J., E.R. and A.B. S.H.R. and M.H. performed model calculations and simulations with help from D.Y.P. K.S.K. conceived and directed the project. S.H.R., M.H., D.Y.P. and K.S.K wrote the manuscript with contributions from all other co-authors.

Corresponding author

Correspondence to Keun Su Kim.

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The authors declare no competing interests.

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Peer review information Nature thanks Andrés Santander and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Key assumptions in structural simulations.

ah, Let us consider the long-range order of dopants whose density is 1 × 1014 cm−2 (a). If the position of dopants is fully randomized (b), no structure is expected in its FFT (f). Considering that each dopant is ionized by donating its valence electron, there should be Coulomb repulsion between the ionized dopants, which prevents them from being located close to each other in a certain range corresponding to the radius of hard spheres. If the position of dopants is randomized under this hard-sphere assumption with no further interactions considered (c), one can always find the radial structure factor in its FFT (g). The radial shape itself in the structure factor reflects the presence of a mean interatomic distance (or short-range order) generic to ionic dopants randomly distributed under the repulsive interaction. This is the structure factor required for the resonance effect in the theory of liquid metals1,2,3,4,5,6,7,8,9,10,11,12, which is clearly distinguished from both crystalline (a, e) and fully random (b, f) cases. In the case of monovalent dopants on a crystalline insulator, the area of a circle whose radius is half the radial peak is the same as that enclosed by the Fermi surface in any shape due to charge conservation. To account for the isotropic magnitude of a pseudogap over the anisotropic Fermi surface of C1, the anisotropy of the structure factor is additionally considered based on a recent STM study25, where the radial and anisotropic structure factor was found at the density of 1.8 × 1013 cm−2. When the anisotropy factor of 0.42 taken from the anisotropic kF values in the Fermi surface is applied to the hard-sphere model (d), the radial structure factor becomes anisotropic, as seen in h.

Extended Data Fig. 2 Momentum distribution curves.

ARPES data of Na-doped black phosphorus taken along ks (left) and kx (right) directions (those in Fig. 2e, f). ARPES intensity is plotted as a function of k at the constant energy shown on the vertical axis. Red dots mark the peak position of k distribution curves to show the backward-bending band dispersion.

Extended Data Fig. 3 Constant-energy map of ARPES spectra.

ARPES intensity maps of bulk black phosphorus whose surface is doped by Na at nd = 1.7 × 1014 cm−2 (top row) and K at nd = 1.0 × 1014 cm−2 (bottom row). Data are plotted at the constant energy marked on top of each panel as a function of kx and ky in the surface Brillouin zone of black phosphorus. Unlike the pair of C2 bands that cross EF, the C1 band shows a clear signature of a pseudogap with a magnitude of 235 meV for Na and 65 meV for K. This magnitude of the pseudogap is found to be nearly isotropic with in-plane k directions. The size of the area enclosed by the C1 contour of Na decreases towards EF, which reflects the backward-bending band dispersion.

Extended Data Fig. 4 Dependence of the pseudogap on photon energy and samples.

a, Photon-energy dependence of ARPES data taken from black phosphorus whose surface is doped by K at nd = 1.8 × 1014 cm−2. A pseudogap of 30 meV at this nd is robustly observed for a photon energy of 106–125 eV (red arrow) in contrast to the C2 band that crosses EF (blue arrow). b, c, Sample dependence of ARPES data taken from black phosphorus doped by Na at nd = 1.7 × 1014 cm−2 (b) and K at nd = 1.0 × 1014 cm−2 (c). The pseudogap of 235 meV for Na and 65 meV for K is observed without exception for all of the samples.

Extended Data Fig. 5 Model calculation for partial waves at d-wave resonance.

A series of Ul, δl, Re(Δk) and Im(Δk) is calculated as described in Methods for the partial waves from l = 0 (top row) to l = 3 (bottom row). The depth of potential V0 is set to 16.4 eV, which corresponds to d-wave resonance at kr = 0.36 Å−1, as shown in Fig. 4a. The d-wave resonance can be identified by δ2 passing through π/2, which is accompanied by Re(Δk) and Im(Δk) taking the form of sin(2δl)/k2 and sin2(δl)/k2, respectively. For l = 0, there is no potential well, in which electron waves can be trapped, and it is impossible to have s-wave resonance.

Extended Data Fig. 6 Comparison of Δk at p-wave, d-wave and f-wave resonances.

A series of Ul, δl, Re(Δk) and Im(Δk) is calculated for p-wave (top row), d-wave (middle row) and f-wave (bottom row) resonances (Methods). V0 was set to 7.4 eV for p-wave resonance, 16.4 eV for d-wave resonance and 27.8 eV for f-wave resonance to have the same kr of 0.36 Å−1 (dotted line in Fig. 4a). The variation of δl, Re(Δk) and Im(Δk) is narrower in k or E width for the higher number of l, which accounts for the smaller magnitude of pseudogap in Fig. 5a.

Extended Data Fig. 7 Spectral simulations compared with experimental ARPES data.

The experimental band structure is taken from black phosphorus whose surface is doped by different kinds of alkali metal (Na, K, Rb and Cs), as marked at the bottom right of each panel. This series of ARPES spectra was taken along ky (left column), ks (middle column) and kx (right column). Each dataset is directly compared with corresponding spectral simulations at p-wave resonance for Na and d-wave resonance for K, Rb and Cs (Methods). The bottom energy of spectral simulations is limited by that of the non-interacting (bare) band, which is the reason that those for Rb and Cs are slightly cut at a binding energy of 0.42–0.45 eV.

Extended Data Fig. 8 Partial-wave analysis with the screened scattering potential.

a, Phase shift δl of the partial waves calculated by the screened scattering potential for the liquid phase of Cs ions, which clearly shows d-wave resonance9. The dotted line is a fit with that calculated by the spherical step potential at V0 = 12.96 eV and rs = 2.1 Å. b, Phase shift δl of partial waves calculated by the screened scattering potential of Na, K, Rb and Cs ions30. K, Rb and Cs ions favour d-wave resonance, whereas the Na ions favour p-wave resonance, which is exactly as observed in our experiments. Our results reproduce other details as well. (1) δl crosses π/2 clearly for K, Rb and Cs ions, whereas that for Na ions is relatively incomplete. (2) The δl values of K and Rb ions are nearly the same. (3) The abruptness of δl, which is related to the magnitude of the pseudogap, increases in the order of Na → K/Rb → Cs.

Extended Data Fig. 9 Evolution of pseudogap with surface doping.

A doping series of APRES data are taken from black phosphorus whose surface is doped by Na, K, Rb and Cs, marked at the bottom right of each panel. Shown together in number is the dopant density nd in units of 1014 cm−2. It can be clearly seen that the magnitude of the pseudogap progressively decreases to zero with increasing nd, as summarized in the phase diagram (Fig. 5b). The hole-like states developing inside of the C1 band (pronounced for Na at nd = 3.3 × 1014 cm−2) are the valence band of black phosphorus23. The additional feature near EF and at the Γ point (pronounced for K at nd = 2.9 × 1014 cm−2) is another conduction band of black phosphorus reproduced in first-principles band calculations24.

Extended Data Fig. 10 Resonance effect for the waterfall dispersion in copper oxides.

a, ARPES data taken from Bi1.5Pb0.5Sr2CaCu2O8+δ (Tc = 88 K) along the nodal direction with a photon energy of 98 eV. nd is estimated from Tc (88 K)35, pseudogap (32 meV)36 and Luttinger count to be 1.2 × 1014−1.4 × 1014 cm−2, corresponding to kr = 0.29 ± 0.01 Å−1. b, Spectral simulations for resonance scattering at kr = 0.29 Å−1 (Methods). The grey curve represents the bare band. The waterfall dispersion in a is reproduced remarkably well by our spectral simulations in b.

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Ryu, S.H., Huh, M., Park, D.Y. et al. Pseudogap in a crystalline insulator doped by disordered metals. Nature 596, 68–73 (2021).

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