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Three-dimensional collective charge excitations in electron-doped copper oxide superconductors


High-temperature copper oxide superconductors consist of stacked CuO2 planes, with electronic band structures and magnetic excitations that are primarily two-dimensional1,2, but with superconducting coherence that is three-dimensional. This dichotomy highlights the importance of out-of-plane charge dynamics, which has been found to be incoherent in the normal state3,4 within the limited range of momenta accessible by optics. Here we use resonant inelastic X-ray scattering to explore the charge dynamics across all three dimensions of the Brillouin zone. Polarization analysis of recently discovered collective excitations (modes) in electron-doped copper oxides5,6,7 reveals their charge origin, that is, without mixing with magnetic components5,6,7. The excitations disperse along both the in-plane and out-of-plane directions, revealing its three-dimensional nature. The periodicity of the out-of-plane dispersion corresponds to the distance between neighbouring CuO2 planes rather than to the crystallographic c-axis lattice constant, suggesting that the interplane Coulomb interaction is responsible for the coherent out-of-plane charge dynamics. The observed properties are hallmarks of the long-sought ‘acoustic plasmon’, which is a branch of distinct charge collective modes predicted for layered systems8,9,10,11,12 and argued to play a substantial part in mediating high-temperature superconductivity10,11,12.

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Fig. 1: Plasmons in a layered electron gas and dispersive charge excitations in electron-doped copper oxides.
Fig. 2: Three-dimensionality of the zone centre excitations.
Fig. 3: Out-of-plane plasmon dispersion.
Fig. 4: Doping dependence of the plasmon.

Data availability

Raw data are included for Figs. 1b–e, 2, 3a, 4, and Extended Data Figs. 1–4. The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.


  1. Damascelli, A., Hussain, Z. & Shen, Z. X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473 (2003).

    ADS  CAS  Article  Google Scholar 

  2. Fujita, M. et al. Progress in neutron scattering studies of spin excitations in high-T c cuprates. J. Phys. Soc. Jpn. 81, 011007 (2012).

    ADS  Article  Google Scholar 

  3. Tajima, S. Optical studies of high-temperature superconducting cuprates. Rep. Prog. Phys. 79, 094001 (2016).

    ADS  Article  Google Scholar 

  4. Singley, E. J., Basov, D. N., Kurahashi, K., Uefuji, T. & Yamada, K. Electron dynamics in Nd1.85Ce0.15CuO4+δ: evidence for the pseudogap state and unconventional c-axis response. Phys. Rev. B 64, 224503 (2001).

    ADS  Article  Google Scholar 

  5. Ishii, K. et al. High-energy spin and charge excitations in electron-doped copper oxide superconductors. Nat. Commun. 5, 3714 (2014).

    CAS  Article  Google Scholar 

  6. Lee, W. S. et al. Asymmetry of collective excitations in electron- and hole-doped cuprate superconductors. Nat. Phys. 10, 883 (2014).

    CAS  Article  Google Scholar 

  7. Dellea, G. et al. Spin and charge excitations in artifcial artificial hole- and electron-doped infinite layer cuprate superconductors. Phys. Rev. B 96, 115117 (2017).

    ADS  Article  Google Scholar 

  8. Grecu, D. Plasmon frequency of the electron gas in layered structures. Phys. Rev. B 8, 1958 (1973).

    ADS  CAS  Article  Google Scholar 

  9. Fetter, A. L. Electrodynamics of a layered electron gas II. Periodic array. Ann. Phys. 88, 1 (1974).

    ADS  Article  Google Scholar 

  10. Kresin, V. Z. & Morawitz, H. Layer plasmons and high-T c superconductivity. Phys. Rev. B 37, 7854 (1988).

    ADS  CAS  Article  Google Scholar 

  11. Ishii, Y. & Ruvalds, J. Acoustic plasmons and cuprate superconductivity. Phys. Rev. B 48, 3455 (1993).

    ADS  CAS  Article  Google Scholar 

  12. Bill, A., Morawitz, H. & Kresin, V. Z. Electronic collective modes and superconductivity in layered conductors. Phys. Rev. B 68, 144519 (2003).

    ADS  Article  Google Scholar 

  13. Bozovic, I. Plasmons in cuprate superconductors. Phys. Rev. B 42, 1969 (1990).

    ADS  CAS  Article  Google Scholar 

  14. Levallois, J. et al. Temperature-dependent ellipsometry measurements of partial coulomb energy in superconducting cuprates. Phys. Rev. X 6, 031027 (2016).

    Google Scholar 

  15. Fink, J., Knupfer, M., Atzkern, S. & Golden, M. Electronic correlation in solids, studies using electron energy-loss spectroscopy. J. Elec. Spectrosc. Rel. Phenom. 117/118, 287–309 (2001).

    Article  Google Scholar 

  16. Leggett, A. J. Where is the energy saved in cuprate superconductivity? J. Phys. Chem. Solids 59, 1729 (1998).

    ADS  CAS  Article  Google Scholar 

  17. Leggett, A. J. Cuprate superconductivity: dependence of T c on the c-axis layering structure. Phys. Rev. Lett. 83, 392 (1999).

    ADS  CAS  Article  Google Scholar 

  18. Greco, A., Yamase, H. & Bejas, M. Plasmon excitations in layered high-T c cuprates. Phys. Rev. B 94, 075139 (2016).

    ADS  Article  Google Scholar 

  19. Jia, C., Wohlfeld, K., Wang, Y., Moritz, B. & Devereaux, T. P. Using RIXS to uncover elementary charge and spin excitations. Phys. Rev. X 6, 021020 (2016).

    Google Scholar 

  20. Ament, L. J. P., van Veenendaal, M., Devereaux, T. P., Hill, J. P. & van den Brink, J. Resonant inelastic X-ray scattering studies of elementary excitations. Rev. Mod. Phys. 83, 705 (2011).

    ADS  CAS  Article  Google Scholar 

  21. Huang, H. Y. et al. Raman and fluorescence characteristics of resonant inelastic X-ray scattering from doped superconducting cuprates. Sci. Rep. 6, 19657 (2016).

    ADS  CAS  Article  Google Scholar 

  22. Markiewicz, R. S., Hasan, M. Z. & Bansil, A. Acoustic plasmons and doping evolution of Mott physics in resonant inelastic X-ray scattering from cuprate superconductors. Phys. Rev. B 77, 094518 (2008).

    ADS  Article  Google Scholar 

  23. Kim, J. et al. Comparison of resonant inelastic X-ray scattering spectra and dielectric loss functions in copper oxides. Phys. Rev. B 79, 094525 (2009).

    ADS  Article  Google Scholar 

  24. Meevasana, W., Devereaux, T. P., Nagaosa, N., Shen, Z. X. & Zaanen, J. Calculation of overdamped c-axis charge dynamics and the coupling to polar phonons in cuprate superconductors. Phys. Rev. B 74, 174524 (2006).

    ADS  Article  Google Scholar 

  25. Lee, W. C. Superconductivity-induced changes in density-density correlation function enabled by Umklapp processes. Phys. Rev. B 91, 224503 (2015).

    ADS  Article  Google Scholar 

  26. Sarkar, T. et al. Fermi surface reconstruction and anomalous low-temperature resistivity in electron-doped La1−xCexCuO4. Phys. Rev. B 96, 155449 (2017).

    ADS  Article  Google Scholar 

  27. Mitrano, M. et al. Anomalous density fluctuation in a strange metal. Proc. Natl Acad. Sci. USA 115, 5392 (2018).

    ADS  CAS  Article  Google Scholar 

  28. Ishii, K. et al. Observation of momentum-dependent charge excitations in hole-doped cuprates using resonant inelastic X-ray scattering at the oxygen K edge. Phys. Rev. B 96, 115148 (2017).

    ADS  Article  Google Scholar 

  29. Scalapino, D. J. et al. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383 (2012).

    ADS  CAS  Article  Google Scholar 

  30. Clarke, D. G., Strong, S. P. & Anderson, P. W. Incoherence of single particle hopping between Luttinger liquids. Phys. Rev. Lett. 72, 3218 (1994).

    ADS  CAS  Article  Google Scholar 

  31. Sachdev, S. & Chowdhury, D. The novel metallic states of the cuprates: Fermi liquids with topological order, and strange metals. Prog. Theor. Exp. Phys. 2016, 12C102 (2016).

    Article  MATH  Google Scholar 

  32. Pile, D. Perspective on plasmonics. Nat. Photon. 6, 714–715 (2012).

    Google Scholar 

  33. Braicovich, L. et al. The simultaneous measurement of energy and linear polarization of the scattered radiation in resonant inelastic soft X-ray scattering. Rev. Sci. Instrum. 85, 115104 (2014).

    ADS  CAS  Article  Google Scholar 

  34. Kung, Y. F. et al. Characterizing the three-orbital Hubbard model with determinant quantum Monte Carlo. Phys. Rev. B 93, 155166 (2016).

    ADS  Article  Google Scholar 

  35. Blankenbecler, R., Scalapino, D. J. & Sugar, R. L. Monte Carlo calculations of coupled boson-fermion systems. I. Phys. Rev. D 24, 2278 (1981).

    ADS  CAS  Article  Google Scholar 

  36. Jarrell, M. & Gubernatis, J. E. Bayesian inference and the analytic continuation of imaginary-time Monte Carlo data. Phys. Rep. 269, 133 (1996).

    ADS  MathSciNet  CAS  Article  MATH  Google Scholar 

  37. Fetter, A. L. Electrodynamics of a layered electron gas. I. Single layer. Ann. Phys. 81, 367 (1973).

    ADS  Article  Google Scholar 

  38. Turlakov, M. & Leggett, A. J. Sum rule analysis of umklapp processes and Coulomb energy: application to cuprate superconductivity. Phys. Rev. B 67, 094517 (2003).

    ADS  Article  Google Scholar 

  39. Moretti Sala, M. et al. Energy and symmetry of dd excitations in undoped layered cuprates measured by Cu L 3 resonant inelastic X-ray scattering. New J. Phys. 13, 043026 (2011).

    ADS  Article  Google Scholar 

  40. Le Tacon, M. et al. Intense paramagnon excitations in a large family of high-temperature superconductors. Nat. Phys. 7, 725 (2011).

    Article  Google Scholar 

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This work is supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-76SF00515. L.C. acknowledges support from the Department of Energy, SLAC Laboratory Directed Research and Development funder contract under DE-AC02-76SF00515. RIXS data were taken at beamline ID32 of the European Synchrotron Radiation Facility (ESRF, Grenoble, France) using the ERIXS spectrometer designed jointly by the ESRF and the Politecnico di Milano. G.G. and Y.Y.P. were supported by the by ERC-P-ReXS project (2016-0790) of the Fondazione CARIPLO and Regione Lombardia, in Italy. R.L.G. and T.S. acknowledge support from NSF award DMR-1708334. Computational work was performed on the Sherlock cluster at Stanford University and on resources of the National Energy Research Scientific Computing Center, supported by the US DOE under contract number DE-AC02-05CH11231.

Reviewer information

Nature thanks D. M. Casa, D. van der Marel and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations



W.S.L., G.G., L.B., T.P.D. and Z.X.S. conceived the experiment. M. Hepting, W.S.L., L.C., R.F., Y.Y.P., G.G., M. Hashimoto, K.K. and N.B.B. conducted the experiment at ESRF. M. Hepting, L.C. and W.S.L. analysed the data. E.W.H., W.C.L., B.M. and T.P.D. performed the theoretical calculations. T.S., J.-F.H., C.R.R., Y.S.L. and R.L.G. synthesized and prepared samples for the experiments. M. Hepting, B.M. and W.S.L. wrote the manuscript with input from all authors.

Corresponding authors

Correspondence to Z. X. Shen, T. P. Devereaux or W. S. Lee.

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Extended data figures and tables

Extended Data Fig. 1 Fits of the RIXS spectra.

a, Fits of LCCO (x = 0.175) RIXS spectra at in-plane momentum transfer positions q = (0.045 0) and (0.095 0), representative of all fits performed in the scope of this work. The model uses a Gaussian for the elastic peak (green) and anti-symmetrized Lorentzians for all other contributions in the spectrum, convoluted with the energy resolution (here ΔE = 68 meV) via Gaussian convolution. The anti-symmetrized Lorentizan is used to ensure zero mode intensity at zero energy loss, as explained in the supplementary information of ref. 40. The peak profiles of the zone centre excitation (plasmon) are shaded in red. b, Full-width at half-maximum (FWHM) of the zone centre excitation (plasmon) as extracted from the fits for momentum transfer along the hh- and h- directions at l* =  0.5, l* =  0.825 and l* =  0.9, corresponding to the fitted peak positions shown in Fig. 2c. Error bars are the standard deviation of the fits. c, FWHM of the zone centre excitation (plasmon) as extracted from the fits for momentum transfer along the out-of-plane direction at h = 0.025. The panel corresponds to the fitted peak positions shown in Fig. 3a.

Extended Data Fig. 2 Raw data and fits of the RIXS spectra.

a, b, Raw RIXS spectra (red) of LCCO (x = 0.175) together with the fits (solid black lines) for momentum transfer along the hh-direction (a) and h-direction (b) at different l*. The spectra are offset in the vertical direction for clarity. c, Raw RIXS spectra together with the fits for momentum transfer along the l*-direction at h = 0.025.

Extended Data Fig. 3 Three-dimensionality of the zone centre excitations in NCCO.

a, b, RIXS intensity maps of NCCO (x = 0.15) for momentum transfer along the h-direction at l* = 0.5 and l* = 0.825. Red and grey symbols indicate least-squares-fit peak positions of the zone centre excitation and the paramagnon, respectively. The inset indicates the probe direction in reciprocal space. c, RIXS intensity map of NCCO (x = 0.15) for momentum transfer along the out-of-plane direction at h = 0.025. White symbols indicate fitted peak positions of the zone centre excitation. Error bars are estimated from the uncertainty in energy-loss reference-point determination (±0.01 eV) together with the standard deviation of the fits.

Extended Data Fig. 4 Fits of the plasmon dispersion in the layered electron gas model.

a, Fits (solid lines) of the mode energies of LCCO (x = 0.175) (red symbols) as a function of in-plane momentum transfer q along the h-direction at l* = 0.5, l* = 0.825 and l* = 0.9. The fit is global, that is, the three l* datasets are fitted simultaneously with the same fit parameter, as described in the Methods. Error bars of the data points are the same as those estimated in Fig. 2c.

Extended Data Fig. 5 Verification of electron doping systematics via dd excitations in the RIXS spectra.

a, dd excitations in RIXS spectra at momentum transfer (0.015, 0, 1) taken from samples with different Ce doping concentrations x. The energy positions of dd excitations shift to higher energy with increasing electron doping, which can be used as an internal reference to verify the doping concentrations. The inset shows a zoom-in of the leading-edge region of the dd excitations. b, The correlation between the Ce concentration x and the energy of the dd-leading edge (inflection point) and the \(3{d}_{{z}^{2}-{r}^{2}}\) peak. All samples show good correlation except for the x = 0.175 sample, indicating a larger uncertainty of its doping concentration. Thus, the x = 0.175 data were not included in Fig. 4. The error bars are estimated from the standard deviation of the fit used to determine the energy of the dd-leading edge and the \(3{d}_{{z}^{2}-{r}^{2}}\) peak.

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Hepting, M., Chaix, L., Huang, E.W. et al. Three-dimensional collective charge excitations in electron-doped copper oxide superconductors. Nature 563, 374–378 (2018).

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  • Acoustic Plasmons
  • Charge Dynamics
  • CuO2 Planes
  • Resonant Inelastic X-ray Scattering (RIXS)
  • RIXS Spectra

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