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Bogoliubov quasiparticle on the gossamer Fermi surface in electron-doped cuprates

Nature Physics volume 19, pages 1834–1840 (2023)Cite this article

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Abstract

Electron-doped cuprates consistently exhibit strong antiferromagnetic correlations, leading to the prevalent belief that antiferromagnetic spin fluctuations mediate Cooper pairing in these unconventional superconductors. However, early investigations showed that although antiferromagnetic spin fluctuations create the largest pseudogap at hot spots in momentum space, the superconducting gap is also maximized at these locations. This presented a paradox for spin-fluctuation-mediated pairing: Cooper pairing is strongest at momenta where the normal-state low-energy spectral weight is most suppressed. Here we investigate this paradox and find evidence that a gossamer—meaning very faint—Fermi surface can provide an explanation for these observations. We study Nd2–xCexCuO4 using angle-resolved photoemission spectroscopy and directly observe the Bogoliubov quasiparticles. First, we resolve the previously observed reconstructed main band and the states gapped by the antiferromagnetic pseudogap around the hot spots. Within the antiferromagnetic pseudogap, we also observe gossamer states with distinct dispersion, from which coherence peaks of Bogoliubov quasiparticles emerge below the superconducting critical temperature. Moreover, the direct observation of a Bogoliubov quasiparticle permits an accurate determination of the superconducting gap, yielding a maximum value an order of magnitude smaller than the pseudogap, establishing the distinct nature of these two gaps. We propose that orientation fluctuations in the antiferromagnetic order parameter are responsible for the gossamer states.

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Fig. 1: Momentum-dependent normal-state electronic structure of NCCO.
Fig. 2: Temperature-dependent spectra reveal the origin of the low-energy peak.
Fig. 3: Momentum dependence of Bogoliubov quasiparticles.
Fig. 4: Spin orientational fluctuations restore gossamer states.

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Data availability

The data presented in this work are available via the Stanford Digital Depository at https://purl.stanford.edu/gt676zx4703. Source data are provided with this paper.

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Acknowledgements

We thank S. A. Kivelson, L. Taillefer, Y. Wang, B. Moritz and Y. Zhong for fruitful discussions. Technical assistance and valuable discussions with C. Polly, B. Thiagarajan and H. Fedderwitz are gratefully acknowledged. The work at Stanford University and Stanford Institute for Materials and Energy Sciences is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515 (K.-J.X., M.H., S.-D.C., J.H., Y.H., C.L., C.R.R., Y.S.L., T.P.D., D.-H. Lu and Z.-X.S.). Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract no. DE-AC02-76SF00515 (K.-J.X., D.-H. Lu, M.H., S.-D.C., J.H., Y.H. and Z.-X.S.). Z.-X.S. acknowledges the hospitality of KTH and the Swedish Research Council during his sabbatical, when this collaboration started. We acknowledge MAX IV Laboratory for time on the Bloch beamline under Proposal 20210256 (Q.G., C.L., M.H.B. and O.T.). Research conducted at MAX IV, a Swedish national user facility, is supported by the Swedish Research Council under contract 2018-07152, the Swedish Governmental Agency for Innovation Systems under contract 2018-04969 and Formas under contract 2019-02496. The work at KTH and MAX IV was supported by Swedish Research Council grants 2019-00701 and 2019-03486, as well as The Knut and Alice Wallenberg Foundation grant 2018.0104 (Q.G., C.L., M.H.B. and O.T.). D.-H. Lee was funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract no. DEAC02-05-CH11231 (Quantum Material Program KC2202). Y.H. acknowledges support from the NSF CAREER award no. DMR-2239171.

Author information

Author notes

  1. Su-Di Chen

    Present address: Department of Physics, University of California, Berkeley, CA, USA

  2. These authors contributed equally: Ke-Jun Xu, Qinda Guo, Makoto Hashimoto, Zi-Xiang Li.

Authors and Affiliations

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

    Ke-Jun Xu, Su-Di Chen, Junfeng He, Yu He, Costel R. Rotundu, Young S. Lee, Thomas P. Devereaux & Zhi-Xun Shen

  2. Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA, USA

    Ke-Jun Xu, Su-Di Chen, Junfeng He, Yu He, Costel R. Rotundu, Young S. Lee, Thomas P. Devereaux & Zhi-Xun Shen

  3. Department of Applied Physics, Stanford University, Stanford, CA, USA

    Ke-Jun Xu, Su-Di Chen, Yu He, Cong Li, Young S. Lee & Zhi-Xun Shen

  4. Department of Applied Physics, KTH Royal Institute of Technology, Stockholm, Sweden

    Qinda Guo, Cong Li, Magnus H. Berntsen & Oscar Tjernberg

  5. Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

    Makoto Hashimoto & Dong-Hui Lu

  6. Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Zi-Xiang Li

  7. University of Chinese Academy of Sciences, Beijing, China

    Zi-Xiang Li

  8. Department of Physics and CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, University of Science and Technology of China, Hefei, China

    Junfeng He

  9. Department of Applied Physics, Yale University, New Haven, CT, USA

    Yu He

  10. Department of Materials Science and Engineering, Stanford University, Stanford, CA, USA

    Thomas P. Devereaux

  11. Department of Physics, Stockholm University, Stockholm, Sweden

    Andreas Rydh

  12. Department of Physics, University of California, Berkeley, CA, USA

    Dung-Hai Lee

  13. Material Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Dung-Hai Lee

  14. Department of Physics, Stanford University, Stanford, CA, USA

    Zhi-Xun Shen

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  1. Ke-Jun Xu
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Contributions

K.-J.X., M.H., O.T. and Z.-X.S. conceived the experiment. K.-J.X., J.H., C.R.R. and Y.S.L. synthesized the samples. K.-J.X., M.H. and D.-H. Lu performed the ARPES measurements at SSRL. Q.G., C.L. and M.H.B. performed the ARPES measurements at MAX IV. A.R. performed the specific heat measurements. K.-J.X., Q.G., M.H., S.-D.C., Y.H., T.P.D., D.-H. Lee, O.T. and Z.-X.S. analysed and interpreted the ARPES data. Z.-X.L. and D.-H. Lee conceived the theoretical model and performed the numerical calculations. K.-J.X., M.H., D.-H. Lee and Z.-X.S. wrote the manuscript with input from all authors.

Corresponding authors

Correspondence to Dung-Hai Lee, Oscar Tjernberg or Zhi-Xun Shen.

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Extended data

Extended Data Fig. 1 Energy distribution curves of the normal-state spectra in Nd1.85Ce0.15CuO4.

a-f, energy-momentum spectra from the corresponding cuts 1–6 in main Fig. 1g. Red dots track the position of the peak associated with AF and green triangles track a higher energy hump feature (see main text). Grey dots in (e) track the only observable broad feature in this cut. Vertical black dashed line indicates the AF zone boundary. Grey line indicates the Fermi energy EF. Error bars correspond to the width of peaks in Energy distribution curve (EDC) fittings. g–l, EDCs of the momentum region indicated by the black double arrows in the respective cuts in a–f. Red, green, and grey dots indicate the same respective features as a-f. Black diamonds in l indicates the zone boundary dispersion and reconstructed electron pocket. The thick black line indicates the Fermi momentum kF, and the grey vertical line indicates EF.

Source data

Extended Data Fig. 2 Spectral fittings in the normal state spectra.

a–e, Normal spectra for cuts 1–5 that are the same as that in main Fig. 1. Small red arrow points to the Fermi momentum. f–j, Fits of the EDC curves. Grey dots are the raw kF EDC data, with the fittings shown by the overlaying black curve. The EDCs are fitted with 3 Lorentzian in g–i and 2 Lorentzian for f and j. Each fit is also convolved with the Fermi-Dirac function.

Source data

Extended Data Fig. 3 Momentum distribution curves of gossamer states.

MDCs at the hot spot, same as cut 4 in the main text Fig. 1, showing peaks within the AF pseudogap (which has a broad gap edge at around 100 meV). The dispersion of the gossamer states is extracted by fitting the MDC peaks with Lorentzian peaks.

Source data

Extended Data Fig. 4 Momentum-dependent matrix element normalization.

a, Fermi surface mapping of an NCCO sample where the surface was heavily dosed with K atoms, such that there are no visible AF spectral signatures. b, angular dependence of the kF spectral weight of the surface-K-dosed sample (red dots) as in a. The black curve is a high order polynomial fit to extract the angle-dependent normalization factor. Horizontal error bars correspond to the uncertainty in the absolute momentum determination.

Source data

Extended Data Fig. 5 Temperature dependence of Brillouin zone diagonal and Brillouin zone boundary spectra for Nd1.85Ce0.15CuO4.

a, b, Temperature dependence of the kF EDCs at the Brillouin zone diagonal (a) and Brillouin zone boundary (b), with Brillouin zone position shown in the respective insets. Measurement temperatures are indicated by the colored legends. The grey vertical line in (a) is a guide to the eye highlighting the persistence of the AF feature through the superconducting transition temperature of 25 K.

Source data

Extended Data Fig. 6 Specific heat reveals small superconducting transition anomaly.

a, Difference specific heat (left) and magnetic susceptibility (right) as a function of temperature. The difference specific heat is extracted by subtracting the zero field curve by the 2T curve. The low temperature drop off is from the field dependence of the Nd Schottky anomaly. The bare band expectation based on Bardeen-Cooper-Schrieffer (BCS) theory is derived from the calculated electronic specific heat in c. b, comparison of the zero field and finite field C/T curves for extraction of the electronic specific heat. 2T is enough to suppress the transition to a lower temperature such that the full height of the jump is preserved in the difference curve. c, Calculated electronic specific heat from the bare band tight binding parameters, for NCCO (black curve) and LSCO (red curve). See Methods sections VII and VIII for additional information of the electronic specific heat calculation and derivation of the expected BCS specific heat jump size.

Source data

Extended Data Fig. 7 Excluding significant doping inhomogeneity.

a, ARPES spectra of cut 5 in Fig. 1. Diagonal black line is the dispersion extracted from fitting the momentum distribution curves (MDCs). b, Dispersion extracted from MDC fitting. The red lines are bounds on the momentum deviation of the dispersion. c, Fermi surface mapping constructed using intensity from ±10 meV of EF. Green line indicates the approximate radius of the large Fermi pocket. We can place an upper bound on the doping uncertainty by the momentum bounds in b. Here, the effect of the AF reconstruction is broad enough to facilitate the extraction of an approximately linear dispersion in this energy range. The intensity between EF and EB ~ 100 meV is from the in-gap residual states, and the intensity at EB > ~100 meV is dominated by the AF states. If the AF and in-gap states originate from distinct doping regions, then we expect the dispersions of the in-gap states and AF states to be offset in momentum. The deviation from a linear dispersion is bounded to be about \(\Delta k \sim\)0.007 Å−1. We note that while the individual MDC widths are larger than this momentum uncertainty, the collective noise in the dispersion fittings show an error smaller than \(\Delta k \sim\)0.007 Å−1. This momentum uncertainty can be translated to a Fermi surface volume uncertainty by the following approximation \(\Delta {A}_{{FS}} \sim 2\pi r\left({k}_{F}\right)\bullet\)∆k = 0.025 ± 0.003 Å−2, where r(kF) is the approximate radius of the large Fermi surface that is nearly circular in this doping regime. We note that the uncertainty in r(kF), which we use a generous estimate of ±0.05 Å−1, is about 10% and linearly affects ∆AFS. The doping uncertainty is the ∆n = ∆AFS/AFS = 0.01 ± 0.001 Å−2. The doping inhomogeneity upper bound is about 1% (or ±0.5% from the nominal 15% doping), and thus excludes significant doping phase separation in our measured samples.

Source data

Extended Data Fig. 8 Symmetrized spectra in overdoped sample reached by surface K dosing.

a–c, symmetrized EDC at the zone diagonal (a), hot spot (b), and zone boundary (c) kF. The momentum location of the EDC is shown in the respective insets. Here, little to no signature of the incipient antiferromagnetic gap is observed anywhere in momentum. No superconducting gap is observed within the experimental temperature and resolution. We note that while the experimental resolution is ~4 meV, this number is defined by the broadening of the Fermi Dirac cutoff, and typically one can observe gap features much smaller than the experimental resolution. Here, the extremely overdoped side is reached via surface K dosing, which introduces additional electrons into the system. From the Fermi surface volume, we estimate the doping to be about 19%.

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Xu, KJ., Guo, Q., Hashimoto, M. et al. Bogoliubov quasiparticle on the gossamer Fermi surface in electron-doped cuprates. Nat. Phys. 19, 1834–1840 (2023). https://doi.org/10.1038/s41567-023-02209-x

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  • DOI: https://doi.org/10.1038/s41567-023-02209-x

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