Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Imaging the energy gap modulations of the cuprate pair-density-wave state

Nature volume 580, pages 65–70 (2020)Cite this article

Subjects

Abstract

The defining characteristic1,2 of Cooper pairs with finite centre-of-mass momentum is a spatially modulating superconducting energy gap Δ(r), where r is a position. Recently, this concept has been generalized to the pair-density-wave (PDW) state predicted to exist in copper oxides (cuprates)3,4. Although the signature of a cuprate PDW has been detected in Cooper-pair tunnelling5, the distinctive signature in single-electron tunnelling of a periodic Δ(r) modulation has not been observed. Here, using a spectroscopic technique based on scanning tunnelling microscopy, we find strong Δ(r) modulations in the canonical cuprate Bi2Sr2CaCu2O8+δ that have eight-unit-cell periodicity or wavevectors Q ≈ (2π/a0)(1/8, 0) and Q ≈ (2π/a0)(0, 1/8) (where a0 is the distance between neighbouring Cu atoms). Simultaneous imaging of the local density of states N(r, E) (where E is the energy) reveals electronic modulations with wavevectors Q and 2Q, as anticipated when the PDW coexists with superconductivity. Finally, by visualizing the topological defects in these N(r, E) density waves at 2Q, we find them to be concentrated in areas where the PDW spatial phase changes by π, as predicted by the theory of half-vortices in a PDW state6,7. Overall, this is a compelling demonstration, from multiple single-electron signatures, of a PDW state coexisting with superconductivity in Bi2Sr2CaCu2O8+δ.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schematic of unidirectional 8a0 PDW order intertwined with a density wave.
Fig. 2: Superconducting gap energy modulations.
Fig. 3: The PDW order parameter amplitude and phase.
Fig. 4: The interplay of N(r) and PDW, and the possible half-vortices.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Fulde, P. & Ferrel, R. A. Superconductivity in a strong spin-exchange field. Phys. Rev. 135, A550 (1964).

    Article  ADS  Google Scholar 

  2. Larkin, A. I. & Ovchinnikov, Yu. N. Inhomogeneous state of superconductors. Sov. Phys. JETP 20, 762–769 (1965).

    MathSciNet  Google Scholar 

  3. Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Colloquium: theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015).

    Article  ADS  CAS  Google Scholar 

  4. Agterberg, D. F. et al. The physics of pair density waves: cuprate superconductors and beyond. Preprint at https://arxiv.org/abs/1904.09687 (2019).

  5. Hamidian, M. H. et al. Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+δ. Nature 532, 343–347 (2016).

    Article  ADS  CAS  Google Scholar 

  6. Agterberg, D. F. & Tsunetsugu, H. Dislocations and vortices in pair-density-wave superconductors. Nat. Phys. 4, 639–642 (2008).

    Article  CAS  Google Scholar 

  7. Berg, E., Fradkin, E. & Kivelson, S. A. Charge-4e superconductivity from pair-density-wave order in certain high-temperature superconductors. Nat. Phys. 5, 830–833 (2009).

    Article  CAS  Google Scholar 

  8. Norman, M. R., Pines, D. & Kallin, C. The pseudogap: friend or foe of high T c? Adv. Phys. 54, 715–733 (2005).

    Article  ADS  CAS  Google Scholar 

  9. Li, Q., Hücker, M., Gu, G. D., Tsvelik, A. M. & Tranquada, J. M. Two-dimensional superconducting fluctuation in stripe-ordered La1.875Ba0.125CuO4. Phys. Rev. Lett. 99, 067001 (2007).

    Article  ADS  CAS  Google Scholar 

  10. Berg, E., Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Striped superconductors: how spin, charge, and superconducting orders intertwine in the cuprates. New J. Phys. 11, 115004 (2009).

    Article  ADS  Google Scholar 

  11. Berg, E. et al. Dynamical layer decoupling in a stripe-ordered high-T c superconductor. Phys. Rev. Lett. 99, 127003 (2007).

    Article  ADS  CAS  Google Scholar 

  12. He, R.-H. et al. From a single-band metal to a high-temperature superconductor via two thermal phase transitions. Science 331, 1579–1583 (2011).

    Article  ADS  CAS  Google Scholar 

  13. Lee, P. A. Amperean pairing and the pseudogap phase of cuprate superconductors. Phys. Rev. X 4, 031017 (2014).

    Google Scholar 

  14. Norman, M. R. & Davis, J. C. Quantum oscillations in biaxial pair density wave state. Proc. Natl Acad. Sci. USA 115, 5389–5391 (2018).

    Article  ADS  CAS  Google Scholar 

  15. Himeda, A., Kato, T. & Ogata, M. Stripe states with spatially oscillating d-wave superconductivity in the two-dimensional t−t′−J model. Phys. Rev. Lett. 88, 117001 (2002).

    Article  ADS  CAS  Google Scholar 

  16. Raczkowski, M., Capello, M., Poilblanc, D., Frésard, R. & Oleś, A. M. Unidirectional d-wave superconducting domains in the two-dimensional t−J model. Phys. Rev. B 76, 140505 (2007).

    Article  ADS  Google Scholar 

  17. Yang, K.-Y., Chen, W. Q., Rice, T. M., Sigrist, M. & Zhang, F.-C. Nature of stripes in the generalized t–J model applied to the cuprate superconductors. New J. Phys. 11, 055053 (2009).

    Article  ADS  Google Scholar 

  18. Loder, F., Graser, S., Kampf, A. P. & Kopp, T. Mean-field pairing theory for the charge-stripe phase of high-temperature cuprate superconductors. Phys. Rev. Lett. 107, 187001 (2011).

    Article  ADS  Google Scholar 

  19. Corboz, P., Rice, T. M. & Troyer, M. Competing states in the t–J model: uniform d-wave state versus stripe state. Phys. Rev. Lett. 113, 046402 (2014).

    Article  ADS  Google Scholar 

  20. Tu, W.-L. & Lee, T.-K. Genesis of charge orders in high temperature superconductors. Sci. Rep. 6, 18675 (2016).

    Article  ADS  CAS  Google Scholar 

  21. Choubey, P., Tu, W.-L., Lee, T. K. & Hirschfeld, P. J. Incommensurate charge ordered states in the t–t′–J model. New J. Phys. 19, 013028 (2017).

    Article  ADS  Google Scholar 

  22. Verret, S., Charlebois, M., Sénéchal, D. & Tremblay, A.-M. S. Subgap structures and pseudogap in cuprate superconductors: role of density waves. Phys. Rev. B 95, 054518 (2017).

    Article  ADS  Google Scholar 

  23. Edkins, S. D. et al. Magnetic field–induced pair density wave state in the cuprate vortex halo. Science 364, 976–980 (2019).

    Article  ADS  CAS  Google Scholar 

  24. Rajasekaran, S. et al. Probing optically silent superfluid stripes in cuprates. Science 359, 575–579 (2018).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  25. Mesaros, A. et al. Topological defects coupling smectic modulations to intra-unit-cell nematicity in cuprates. Science 333, 426–430 (2011).

    Article  ADS  CAS  Google Scholar 

  26. Fujita, K. et al. in Strongly Correlated Systems: Experimental Techniques (eds Avella, A. & Mancini, F.) Ch. 3 (Springer, 2015).

  27. Fujita, K. et al. Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates. Proc. Natl Acad. Sci. USA 111, E3026–E3032 (2014).

    Article  ADS  CAS  Google Scholar 

  28. Nie, L., Tarjus, G. & Kivelson, S. A. Quenched disorder and vestigial nematicity in the pseudogap regime of the cuprates. Proc. Natl Acad. Sci. USA 111, 7980–7985 (2014).

    Article  ADS  CAS  Google Scholar 

  29. Agterberg, D. F. & Garaud, J. Checkerboard order in vortex cores from pair-density-wave superconductivity. Phys. Rev. B 91, 104512 (2015).

    Article  ADS  Google Scholar 

  30. Dai, Z., Zhang, Y.-H., Senthil, T. & Lee, P. A. Pair-density waves, charge-density waves, and vortices in high-T c cuprates. Phys. Rev. B 97, 174511 (2018).

    Article  ADS  CAS  Google Scholar 

  31. Wang, Y. et al. Pair density waves in superconducting vortex halos. Phys. Rev. B 97, 174510 (2018).

    Article  ADS  CAS  Google Scholar 

  32. Cho, D. et al. A strongly inhomogeneous superfluid in an iron-based superconductor. Nature 571, 541–545 (2016).

    Article  ADS  Google Scholar 

  33. Norman, M. R. & Davis, J. C. Quantum oscillations in a biaxial pair density wave state. Proc. Natl Acad. Sci. USA 115, 5389–5391 (2018).

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

We thank S. D. Edkins, E. Fradkin, M. H. Hamidian, S. A. Kivelson, P. A. Lee and J. M. Tranquada, for discussions and advice. Z.D., H.L., K.F. G.G. and P.D.J. acknowledge support from the US Department of Energy, Office of Basic Energy Sciences, under contract number DEAC02-98CH10886. S.H.J. and J.L. acknowledge support from the Institute for Basic Science in Korea (grant number IBS-R009-G2), the Institute of Applied Physics of Seoul National University and a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (number 2017R1A2B3009576). E.P.D. was supported by the Brookhaven National Laboratory Supplemental Undergraduate Research Program (SURP). J.C.S.D. acknowledges support from the Science Foundation of Ireland under award SFI 17/RP/5445 and from the European Research Council (ERC) under award DLV-788932.

Author information

Author notes

  1. These authors contributed equally: Zengyi Du, Hui Li

Authors and Affiliations

  1. CMPMS Department, Brookhaven National Laboratory, Upton, NY, USA

    Zengyi Du, Hui Li, Elizabeth P. Donoway, Genda Gu, Peter D. Johnson & Kazuhiro Fujita

  2. Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, USA

    Hui Li

  3. Department of Physics and Astronomy, Seoul National University, Seoul, Republic of Korea

    Sang Hyun Joo & Jinho Lee

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

    Elizabeth P. Donoway

  5. Department of Physics, University College Cork, Cork, Ireland

    J. C. Séamus Davis

  6. Clarendon Laboratory, University of Oxford, Oxford, UK

    J. C. Séamus Davis

Authors
  1. Zengyi Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sang Hyun Joo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Elizabeth P. Donoway
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jinho Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. J. C. Séamus Davis
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Genda Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Peter D. Johnson
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Kazuhiro Fujita
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

K.F. designed and led the project. Z.D., H.L., S.H.J., E.P.D. and K.F. carried out experiments at the MIRAGE STM of the OASIS complex at Brookhaven National Laboratory; G.G. synthesized and characterized the samples; Z.D., H.L. and K.F. developed and carried out analysis. K.F. wrote the paper with key contributions from J.C.S.D, Z.D., H.L., J.L. and P.D.J. The manuscript reflects the contributions and ideas of all authors.

Corresponding author

Correspondence to Kazuhiro Fujita.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Ting-Kuo Lee, Yi Yin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Fig. 1 Analysis of the tip isotropy.

a, Topography T(r) within 40 nm × 40 nm FOV. b, Real part of Fourier transform of T(r). c, Line profile ReT(q) along the line in the middle panel, representing nearly equal Bragg peak height (difference is less than 7%).

Extended Data Fig. 2 Estimation of the nanoflake tip geometry.

a, Autocorrelation of Δ(r). b, Line profile measured from centre in a is azimuthal-angle averaged. The size of the nanoflake on the tip is estimated from the full-width at half-maximum and is around 3.3 nm.

Extended Data Fig. 3 Possible process of the Josephson tunnelling.

Schematic representation of planar Josephson tunnelling in the presence of two order parameters (OPs): homogeneous SC and PDW.

Extended Data Fig. 4 Preliminary experimental data analysis.

a, c, Preliminary Δ(r) independently measured at 4.2 K on different pieces of  nearly optimally doped Bi2Sr2CaCu2O8+δ. b, d, The magnitude of Fourier transform of Δ(r) in a and c, respectively, representing early observations of 1/8 peaks marked by the red circles.

Extended Data Fig. 5 Differential conductance map and its ratio.

a, g(r, E = 54 meV) map. The eight-unit-cell CDW modulation, that is, the PDW induced N(r) modulation at Q, can be seen. b, Z(r, E = 54 meV) calculated by Z(r, E) = g(r, E)/g(r, −E).

Extended Data Fig. 6 Cut-off-length dependence of \(|{{\bf{D}}}_{2{{\bf{Q}}}_{{\boldsymbol{x}}}}^{{\boldsymbol{Z}}}({\bf{r}})|\) and \(|{{\boldsymbol{\Delta }}}_{{{\bf{Q}}}_{{\boldsymbol{x}}}}({\bf{r}})|\).

The left column shows \(|{D}_{2{{\bf{Q}}}_{{\boldsymbol{x}}}}^{Z}({\bf{r}})|\) at different cut-off lengths, similarly for the right column for \(|{\varDelta }_{{{\bf{Q}}}_{{\boldsymbol{x}}}}({\bf{r}})|\).

Extended Data Fig. 7 Distance analysis.

a, A count distribution sorted by distances between the topological defects in the induced N(r) modulation at 2Q from Fig. 4b and the nearest point on the yellow strings in the PDW phase map from Fig. 4c. b, Average distribution of 100 configurations, within each configuration 25 points are randomly generated in the same FOV and distances to the same yellow strings are calculated and sorted.

Extended Data Fig. 8 Spatial evolution of the PDW phase.

a, A phase map \({\varPhi }_{{{\bf{Q}}}_{{\boldsymbol{x}}}}^{\varDelta }({\bf{r}})\) of the PDW order. Three representative contours surrounding the 2π topological defects from \({\varPhi }_{2{{\bf{Q}}}_{{\boldsymbol{x}}}}^{Z}({\bf{r}})\) across the yellow strings. b, An evolution of the phase along each contour in a. The upside-down black triangle marks the starting point of winding and the upright black triangle marks the ending point, in correspondence with the winding directions in a. π phase windings are clearly seen in the PDW phase surrounding the 2π topological defects from \({\varPhi }_{2{{\bf{Q}}}_{{\boldsymbol{x}}}}^{Z}({\bf{r}})\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, Z., Li, H., Joo, S.H. et al. Imaging the energy gap modulations of the cuprate pair-density-wave state. Nature 580, 65–70 (2020). https://doi.org/10.1038/s41586-020-2143-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-020-2143-x

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing