Skip to main content

Thank you for visiting 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.

Relating atomic-scale electronic phenomena to wave-like quasiparticle states in superconducting Bi2Sr2CaCu2O8+δ


The electronic structure of simple crystalline solids can be completely described in terms either of local quantum states in real space (r-space), or of wave-like states defined in momentum-space (k-space). However, in the copper oxide superconductors, neither of these descriptions alone may be sufficient. Indeed, comparisons between r-space1,2,3,4,5 and k-space6,7,8,9,10,11,12,13 studies of Bi2Sr2CaCu2O8+δ (Bi-2212) reveal numerous unexplained phenomena and apparent contradictions. Here, to explore these issues, we report Fourier transform studies of atomic-scale spatial modulations in the Bi-2212 density of states. When analysed as arising from quasiparticle interference14,15,16, the modulations yield elements of the Fermi-surface and energy gap in agreement with photoemission experiments12,13. The consistency of numerous sets of dispersing modulations with the quasiparticle interference model shows that no additional order parameter is required. We also explore the momentum-space structure of the unoccupied states that are inaccessible to photoemission, and find strong similarities to the structure of the occupied states. The copper oxide quasiparticles therefore apparently exhibit particle–hole mixing similar to that of conventional superconductors. Near the energy gap maximum, the modulations become intense, commensurate with the crystal, and bounded by nanometre-scale domains4. Scattering of the antinodal quasiparticles is therefore strongly influenced by nanometre-scale disorder.

This is a preview of subscription content

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: The expected wavevectors of quasiparticle interference patterns in a superconductor with electronic band structure like that of Bi-2212.
Figure 2: Atomic resolution images of the LDOS and the resulting Fourier-space images of the wavevectors making up the LDOS modulations.
Figure 3: The measured dispersion of all sets of q-vectors, the resulting FT-STS-derived locus of scattering, the anisotropic energy gap, and the relevant ARPES data for comparison.
Figure 4: The electronic density of states modulations associated with antinodal quasiparticles at energies near the gap maximum.


  1. Pan, S. H. et al. Microscopic electronic inhomogeneity in the high-Tc superconductor Bi2Sr2CaCu2O8+x . Nature 413, 282–285 (2001)

    ADS  CAS  Article  Google Scholar 

  2. Cren, T., Roditchev, D., Sacks, W. & Klein, J. Nanometer scale mapping of the density of states in an inhomogeneous superconductor. Euro. Phys. Lett. 54, 84–90 (2001)

    ADS  CAS  Article  Google Scholar 

  3. Howald, C., Fournier, P. & Kapitulnik, A. Inherent inhomogeneities in the tunneling spectra of Bi2Sr2CaCu2O8-x crystals in the superconducting state. Phys. Rev. B 64, 100504 (2001)

    ADS  Article  Google Scholar 

  4. Lang, K. M. et al. Imaging the granular structure of high-Tc superconductivity in underdoped Bi2Sr2CaCu2O8+δ . Nature 415, 412–416 (2002)

    ADS  CAS  Article  Google Scholar 

  5. Hoffman, J. E. et al. Imaging quasiparticle interference in Bi2Sr2CaCu2O8 . Science 297, 1148–1151 (2002)

    ADS  CAS  Article  Google Scholar 

  6. Campuzano, J. C. et al. Fermi surfaces of YBa2Cu3O6.9 as seen by angle-resolved photoemission. Phys. Rev. Lett. 64, 2308–2311 (1990)

    ADS  CAS  Article  Google Scholar 

  7. Dessau, D. S. et al. Key features in the measured band structure of Bi2Sr2CaCu2O8+x: Flat bands at EF and Fermi surface nesting. Phys. Rev. Lett. 71, 2781–2784 (1993)

    ADS  CAS  Article  Google Scholar 

  8. Aebi, P. et al. Complete Fermi surface mapping of Bi2Sr2CaCu2O8+x(001): Coexistence of short range antiferromagnetic correlations and metallicity in the same phase. Phys. Rev. Lett. 72, 2757–2760 (1994)

    ADS  CAS  Article  Google Scholar 

  9. Shen, Z.-X. et al. Anomalously large gap in the a-b plane of Bi2Sr2CaCu2O8+δ . Phys. Rev. Lett. 70, 1553–1556 (1993)

    ADS  CAS  Article  Google Scholar 

  10. Ding, H. et al. Angle-resolved photoemission spectroscopy study of the superconducting gap anisotropy in Bi2Sr2CaCu2O8+x . Phys. Rev. B. 54, R9678–R9681 (1996)

    ADS  CAS  Article  Google Scholar 

  11. Shen, Z.-X., Spicer, W. E., King, D. M., Dessau, D. S. & Wells, B. O. Photoemission studies of high-Tc superconductors: The superconducting gap. Science 267, 343–350 (1995)

    ADS  CAS  Article  Google Scholar 

  12. Damascelli, A., Shen, Z.-X. & Hussain, Z. Angle-resolved photoemission spectroscopy of the cuprate superconductors. Preprint at 〈〉 (2002).

  13. Campuzano, J. C., Norman, M. R. & Randeria, M. Photoemission in the high Tc superconductors. Preprint at 〈〉 (2002).

  14. Byers, J. M., Flatté, M. E. & Scalapino, D. J. Influence of gap extrema on tunneling conductance near an impurity in an anisotropic superconductor. Phys. Rev. Lett. 71, 3363 (1993)

    ADS  CAS  Article  Google Scholar 

  15. Wang, Q. & Lee, D.-H. Quasiparticle scattering interference in high temperature superconductors. Phys. Rev. B 67, 020511 (2003)

    ADS  Article  Google Scholar 

  16. Zhang, D. & Ting, C. S. Energy-dependent LDOS modulation in cuprate superconductors. Phys. Rev. B 67, 020511 (2003)

    Article  Google Scholar 

  17. Fedorov, A. V. et al. Temperature dependent photoemission studies of optimally doped Bi2Sr2CaCu2O8+δ . Phys. Rev. Lett. 82, 2179–2183 (1999)

    ADS  CAS  Article  Google Scholar 

  18. Feng, D. L. et al. Signature of superfluid density in the single-particle excitation spectrum of Bi2Sr2CaCu2O8+δ . Science 289, 277–281 (2000)

    ADS  CAS  Article  Google Scholar 

  19. Ding, H. et al. Coherent quasiparticle weight and its connection to high-Tc superconductivity from angle-resolved photoemission. Phys. Rev. Lett. 87, 227001 (2001)

    ADS  CAS  Article  Google Scholar 

  20. Johnson, P. D. et al. Doping and temperature dependence of the mass enhancement observed in the cuprate Bi2Sr2CaCu2O8+δ . Phys. Rev. Lett. 87, 177007 (2001)

    ADS  CAS  Article  Google Scholar 

  21. Lanzara, A. et al. Evidence for ubiquitous strong electron-phonon coupling in high-temperature superconductors. Nature 412, 510–514 (2001)

    ADS  CAS  Article  Google Scholar 

  22. Valla, T. et al. Evidence for quantum critical behavior in the optimally doped cuprate Bi2Sr2CaCu2O8+δ . Science 285, 2110–2113 (1999)

    CAS  Article  Google Scholar 

  23. Hoffman, J. E. et al. A four unit cell periodic pattern of quasiparticle states surrounding vortex cores in Bi2Sr2CaCu2O8+δ . Science 295, 466–469 (2002)

    ADS  CAS  Article  Google Scholar 

  24. Crommie, M. F., Lutz, C. P. & Eigler, D. M. Imaging standing waves in a two-dimensional electron gas. Nature 363, 524 (1993)

    ADS  CAS  Article  Google Scholar 

  25. Howald, C., Eisaki, H., Kaneko, N. & Kapitulnik, A. Coexistence of charged stripes and superconductivity in Bi2Sr2CaCu2O8 + d. Preprint at 〈〉 (2002).

  26. Polkovnikov, A., Sachdev, S. & Vojta, M. Spin collective mode and quasiparticle contributions to STM spectra of d-wave superconductors with pinning. Preprint at 〈〉 (2002).

  27. Podolsky, D. et al. Translational symmetry breaking in the superconducting state of the cuprates: Analysis of the quasiparticle density of states. Preprint at 〈〉 (2002).

  28. Howald, C. et al. Periodic density of states modulations in superconducting Bi2Sr2CaCu2O8+δ . Phys. Rev. B 67, 014533 (2003)

    ADS  Article  Google Scholar 

  29. Kivelson, S. A. et al. How to detect fluctuating order in the high-temperature superconductors. Preprint at 〈〉 (2002).

  30. Campuzano, J. C. et al. Direct observation of particle-hole mixing in the superconducting state by angle-resolved photoemission. Phys. Rev. B 53, R14737–R14740 (1996)

    ADS  CAS  Article  Google Scholar 

Download references


We thank J. C. Campuzano, M. E. Flatté, P. Johnson, S. A. Kivelson, B. Lake, R. B. Laughlin, J. W. Loram, M. Norman, D. J. Scalapino, Z.-X. Shen, J. Tranquada and J. Zaanen for discussions and communications. This work was supported by an LDRD from the Lawrence Berkeley National Laboratory, the ONR, the NSF, and by Grant-in-Aid for Scientific Research on Priority Area (Japan), a COE grant from the Ministry of Education (Japan), and NEDO (Japan). J.E.H. is grateful for support from a Hertz Fellowship.

Author information

Authors and Affiliations


Corresponding author

Correspondence to J. C. Davis.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

McElroy, K., Simmonds, R., Hoffman, J. et al. Relating atomic-scale electronic phenomena to wave-like quasiparticle states in superconducting Bi2Sr2CaCu2O8+δ. Nature 422, 592–596 (2003).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

Further reading


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.


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