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.
Access optionsAccess options
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Pan, S. H. et al. Microscopic electronic inhomogeneity in the high-Tc superconductor Bi2Sr2CaCu2O8+x . Nature 413, 282–285 (2001)
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)
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)
Lang, K. M. et al. Imaging the granular structure of high-Tc superconductivity in underdoped Bi2Sr2CaCu2O8+δ . Nature 415, 412–416 (2002)
Hoffman, J. E. et al. Imaging quasiparticle interference in Bi2Sr2CaCu2O8 . Science 297, 1148–1151 (2002)
Campuzano, J. C. et al. Fermi surfaces of YBa2Cu3O6.9 as seen by angle-resolved photoemission. Phys. Rev. Lett. 64, 2308–2311 (1990)
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)
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)
Shen, Z.-X. et al. Anomalously large gap in the a-b plane of Bi2Sr2CaCu2O8+δ . Phys. Rev. Lett. 70, 1553–1556 (1993)
Ding, H. et al. Angle-resolved photoemission spectroscopy study of the superconducting gap anisotropy in Bi2Sr2CaCu2O8+x . Phys. Rev. B. 54, R9678–R9681 (1996)
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)
Damascelli, A., Shen, Z.-X. & Hussain, Z. Angle-resolved photoemission spectroscopy of the cuprate superconductors. Preprint at 〈http://arXiV.org/cond-mat/0208504〉 (2002).
Campuzano, J. C., Norman, M. R. & Randeria, M. Photoemission in the high Tc superconductors. Preprint at 〈http://arXiV.org/cond-mat/0209476〉 (2002).
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)
Wang, Q. & Lee, D.-H. Quasiparticle scattering interference in high temperature superconductors. Phys. Rev. B 67, 020511 (2003)
Zhang, D. & Ting, C. S. Energy-dependent LDOS modulation in cuprate superconductors. Phys. Rev. B 67, 020511 (2003)
Fedorov, A. V. et al. Temperature dependent photoemission studies of optimally doped Bi2Sr2CaCu2O8+δ . Phys. Rev. Lett. 82, 2179–2183 (1999)
Feng, D. L. et al. Signature of superfluid density in the single-particle excitation spectrum of Bi2Sr2CaCu2O8+δ . Science 289, 277–281 (2000)
Ding, H. et al. Coherent quasiparticle weight and its connection to high-Tc superconductivity from angle-resolved photoemission. Phys. Rev. Lett. 87, 227001 (2001)
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)
Lanzara, A. et al. Evidence for ubiquitous strong electron-phonon coupling in high-temperature superconductors. Nature 412, 510–514 (2001)
Valla, T. et al. Evidence for quantum critical behavior in the optimally doped cuprate Bi2Sr2CaCu2O8+δ . Science 285, 2110–2113 (1999)
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)
Crommie, M. F., Lutz, C. P. & Eigler, D. M. Imaging standing waves in a two-dimensional electron gas. Nature 363, 524 (1993)
Howald, C., Eisaki, H., Kaneko, N. & Kapitulnik, A. Coexistence of charged stripes and superconductivity in Bi2Sr2CaCu2O8 + d. Preprint at 〈http://arXiV.org/cond-mat/0201546〉 (2002).
Polkovnikov, A., Sachdev, S. & Vojta, M. Spin collective mode and quasiparticle contributions to STM spectra of d-wave superconductors with pinning. Preprint at 〈http://arXiV.org/cond-mat/0208334〉 (2002).
Podolsky, D. et al. Translational symmetry breaking in the superconducting state of the cuprates: Analysis of the quasiparticle density of states. Preprint at 〈http://arXiV.org/cond-mat/0204011〉 (2002).
Howald, C. et al. Periodic density of states modulations in superconducting Bi2Sr2CaCu2O8+δ . Phys. Rev. B 67, 014533 (2003)
Kivelson, S. A. et al. How to detect fluctuating order in the high-temperature superconductors. Preprint at 〈http://arXiV.org/cond-mat/0210683〉 (2002).
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)
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.
About this article
Directly visualizing the sign change of d-wave superconducting gap in Bi2Sr2CaCu2O8+δ by phase-referenced quasiparticle interference
Nature Communications (2019)
Electronic effect of doped oxygen atoms in Bi2201 superconductors determined by scanning tunneling microscopy
Science China Physics, Mechanics & Astronomy (2018)
Quantum Studies: Mathematics and Foundations (2018)
Anomalous Electron Spectrum and Its Relation to Peak Structure of Electron Scattering Rate in Cuprate Superconductors
Journal of Low Temperature Physics (2018)
Journal of Superconductivity and Novel Magnetism (2018)