Abstract

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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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.

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Acknowledgements

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.

Author information

Author notes

    • L. Chaix

    Present address: Université Grenoble Alpes, CNRS, Institut Néel, Grenoble, France

    • Y. Y. Peng

    Present address: Department of Physics and Seitz Materials Research Lab, University of Illinois, Urbana, IL, USA

    • J.-F. He

    Present address: Department of Physics, University of Science and Technology of China, Hefei, China

Affiliations

  1. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA

    • M. Hepting
    • , L. Chaix
    • , E. W. Huang
    • , B. Moritz
    • , J.-F. He
    • , C. R. Rotundu
    • , Y. S. Lee
    • , Z. X. Shen
    • , T. P. Devereaux
    •  & W. S. Lee
  2. Department of Physics, Stanford University, Stanford, CA, USA

    • E. W. Huang
  3. Dipartimento di Fisica, Politecnico di Milano, Milan, Italy

    • R. Fumagalli
    • , Y. Y. Peng
    • , L. Braicovich
    •  & G. Ghiringhelli
  4. European Synchrotron Radiation Facility (ESRF), Grenoble, France

    • K. Kummer
    • , N. B. Brookes
    •  & L. Braicovich
  5. Department of Physics, Binghamton University, Binghamton, NY, USA

    • W. C. Lee
  6. Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

    • M. Hashimoto
  7. Department of Physics, Center for Nanophysics and Advanced Materials, University of Maryland, College Park, MD, USA

    • T. Sarkar
    •  & R. L. Greene
  8. CNR-SPIN, Politecnico di Milano, Milan, Italy

    • G. Ghiringhelli

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Contributions

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.

Competing interests

The authors declare no competing interests.

Corresponding authors

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

Extended data figures and tables

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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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