Energy gaps in high-transition-temperature cuprate superconductors

Journal name:
Nature Physics
Volume:
10,
Pages:
483–495
Year published:
DOI:
doi:10.1038/nphys3009
Received
Accepted
Published online

Abstract

The spectral energy gap is an important signature that defines states of quantum matter: insulators, density waves and superconductors have very different gap structures. The momentum-resolved nature of angle-resolved photoemission spectroscopy (ARPES) makes it a powerful tool to characterize spectral gaps. ARPES has been instrumental in establishing the anisotropic d-wave structure of the superconducting gap in high-transition-temperature (Tc) cuprates, which is different from the conventional isotropic s-wave superconducting gap. Shortly afterwards, ARPES demonstrated that an anomalous gap above Tc, often termed the pseudogap, follows a similar anisotropy. The nature of this poorly understood pseudogap and its relationship with superconductivity has since become the focal point of research in the field. To address this issue, the momentum, temperature, doping and materials dependence of spectral gaps have been extensively examined with significantly improved instrumentation and carefully matched experiments in recent years. This article overviews the current understanding and unresolved issues of the basic phenomenology of gap hierarchy. We show how ARPES has been sensitive to phase transitions, has distinguished between orders having distinct broken electronic symmetries, and has uncovered rich momentum- and temperature-dependent fingerprints reflecting an intertwined and competing relationship between the ordered states and superconductivity that results in multiple phenomenologically distinct ground states inside the superconducting dome. These results provide us with microscopic insights into the cuprate phase diagram.

At a glance

Figures

  1. High-Tc cuprate superconductors.
    Figure 1: High-Tc cuprate superconductors.

    a, Schematic phase diagram. The inset shows the crystal structure of the CuO2 planes, which are of central relevance to superconductivity and the pseudogap. b, Schematic band dispersion in reciprocal space for cuprates along the high-symmetry cuts, as shown in blue in c. c, Fermi surface, where the nodal and antinodal momenta and the Fermi angle θ are defined.

  2. d-wave superconducting gap symmetry in cuprates witnessed by ARPES and the improvement of ARPES data quality.
    Figure 2: d-wave superconducting gap symmetry in cuprates witnessed by ARPES and the improvement of ARPES data quality.

    a, Superconducting gap anisotropy first observed in 1993 (reproduced from ref. 6). The lower spectra (A) are taken at the node and the upper spectra (B) at the antinode. b, Schematic of a d-wave order parameter on a circular Fermi surface. Gap is zero at the node where the superconducting gap changes sign (A) and maximum at the antinode (B). c, Typical synchrotron gap measurement a decade ago as a function of the Fermi angle θ. Error bars indicate uncertainty of determining EF (±0.5 meV), error from the fitting procedure and an additional 100% margin. d, Near-nodal gaps measured by a modern laser-based ARPES system with superior resolution and high photon flux. Error bars reflect 3σ error in the fitting procedure and an additional 100% margin. e, Three-dimensional ARPES data set, showing the quasi-particle dispersions both perpendicular and parallel to the Fermi surface near the node, reproduced from ref. 16.

  3. Nodal-antinodal dichotomy in Bi2212.
    Figure 3: Nodal–antinodal dichotomy in Bi2212.

    a,b, Momentum dependence of the symmetrized ARPES spectra for UD92 at T less double Tc and T > Tc, respectively. Red spectra denote the ungapped region (Fermi arc). c, Gap function for UD92K at T less double Tc and T > Tc plotted as a function of the Fermi angle θ. Gaps near the node close around Tc, forming a Fermi arc, whereas the gap magnitude near the antinode does not diminish across Tc. d,e, Temperature dependence of the symmetrized spectra near the node and at the antinode, respectively, for UD92. Inset of d shows raw spectra that clearly show the upper Bogoliubov peak (right). f, Temperature dependence of the gap size at various momenta. The gap near the node closes at Tc, following BCS-like behaviour, whereas the gap around the antinode does not close across Tc. The intermediate region shows an intermediate behaviour. g,h, Doping dependence of the symmetrized spectra near the node and at the antinode, respectively. Dashed curves in d,e,g,h are guides to the eye tracking the peak position, corresponding to the gap size. i, Doping dependence of the gap function as a function of the d-wave form factor |cos(kx) − cos(ky)|/2. Results at each doping level are shown with a vertical offset of 20 meV for clarity. Dashed lines, indicative of a d-wave form of the gap, are guides to the eye which emphasize the deviation of the gap function from the d-wave form in the underdoped region. The slopes of the dashed lines for UD50, UD65, UD72 and UD92 are fixed. The red dashed line on UD65 indicates vΔ, and its doping dependence is plotted in Fig. 6e. Part of the data is reproduced from refs 14, 28, 52. Error bars indicate uncertainty of determining EF (±0.5 meV), error from the fitting procedure and an additional 100% margin.

  4. Dichotomy between the antinodal and nodal regions in Bi2201.
    Figure 4: Dichotomy between the antinodal and nodal regions in Bi2201.

    a,b, ARPES spectra divided by the Fermi–Dirac function for an antinodal cut a and a near-nodal cut b at T > T and T < Tc. c, Gap function at T < Tc. The energy position of the lowest energy feature (green circles in a) deviates from the simple d-wave form near the antinode (Δsc = 15.5 meV, black dashed curve), and the intensity maximum is found at higher energy in the antinodal region. Error bars are estimated based on the sharpness of features, based on different EDC analyses13. Red dashed curve is from ref. 40. Inset: (schematic Fermi surface) shows the cuts A and B used in panels a and b, respectively. d, Dispersion along the antinodal cut at T >T and T < Tc.kF and back-bending momenta (kG) are indicated by red dashed lines and green arrows, respectively. e, Renormalized band dispersion produced by simulations assuming coexistence of d-wave superconductivity (order parameter 35 meV) and bond direction q1 = (0.15π, 0) and q2 = (0, 0.15π) chequerboard density wave (order parameter 20 meV) along the antinodal cut. Dashed curve is the bare band dispersion from a global tight-binding fit to the experimental dispersions of the intensity maximum at 172 K. All the figures are reproduced from ref. 13.

  5. Broken-symmetry nature of the pseudogap in Bi2201.
    Figure 5: Broken-symmetry nature of the pseudogap in Bi2201.

    a, Temperature dependence of the ARPES spectra at antinodal kF. Blue and red circles indicate the intensity maxima of the spectra at 10 K and 160 K, respectively. b, Temperature dependence of the energy position of the intensity maximum at kF given by ARPES, in comparison with the Kerr rotation angle (θK) measured by the PKE. The inset shows the temperature dependence of the transient reflectivity change measured by TRR (left axis). The dashed black curves (right axis) in the main panel and inset are guides to the eye for the PKE data, showing a mean-field-like critical behaviour close to T. Error bars are estimated based on the sharpness of features, based on different EDC analyses13. c, Dispersions above T (red circles) and well below T (blue circles). kF and back-bending momenta are misaligned. d,e, Simulated dispersions for d-wave superconductivity (order parameter 30 meV) and antiferromagnetic order (order parameter 60 meV) with a short correlation length (ten-unit cell), respectively. In ce the cuts are along (π, −π)–(π, 0)–(π, π), respectively. kF and back-bending momenta (kG) in are indicated by red dashed lines and green arrows, respectively. The red (blue) curve is for the true normal (gapped) state. The spectral weight is proportional to the curve thickness. a and ce are reproduced from ref. 12 and b from ref. 13.

  6. Proposed phase diagram of Bi2212.
    Figure 6: Proposed phase diagram of Bi2212.

    ac, Phase competition in the superconducting and pseudogap order (SC + PG) region. Gaps in UD40, UD65 and UD92 at 10 K, 0.9 Tc and 12 K above Tc. Synchrotron and laser data are shown with open and filled symbols, respectively. Error bars in laser ARPES reflect 3σ error in the fitting procedure and an additional 100% margin. Error bars in synchrotron data reflect uncertainty of determining EF (±0.5 meV), error from fitting procedure and an additional 100% margin. Dashed lines are guides to the eye. Doping-independent or doping-dependent gaps are indicated by pink or blue shading, respectively. Dashed box marks momenta where gaps are doping dependent in b and c but doping independent in a. d, Proposed phase diagram. The superconducting dome is divided into three phenomenologically distinct regions (ref. 14): the green-shaded region, characterized by a fully gapped Fermi surface, the blue-shaded region, where pseudogap order coexists with superconductivity (SC + PG), and the red-shaded region, where the pseudogap order is absent below Tc (SC). T is determined from ARPES measurements at the antinode14, 58, STS (ref. 146) and superconductor–insulator–superconductor tunnelling144, 152. T which is higher than the measurement temperature is estimated from an extrapolation of the antinodal gap size. Error bars in T are 3σ in a linear fit. For T which is accessible by ARPES, error bars are the temperature interval between data points14. e, Doping dependence of the symmetrized antinodal spectra slightly above Tc, indicating the existence of the pseudogap at least up to p  0.22. f, Doping dependence of vΔ (see a) at T less double Tc.vΔ shows an abrupt change at p  0.19 (indicated by an arrow), which is interpreted as the T = 0 endpoint of the pseudogap order. vΔ is from a fit over the linear portion of the gap function, as shown by a solid line in a. Error bar in vΔ is the 3σ confidence interval for slope. Figures are adapted from ref. 14 with some data points from more recent experiments added.

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Affiliations

  1. Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

    • Makoto Hashimoto
  2. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

    • Inna M. Vishik,
    • Rui-Hua He,
    • Thomas P. Devereaux &
    • Zhi-Xun Shen
  3. Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA

    • Inna M. Vishik,
    • Rui-Hua He,
    • Thomas P. Devereaux &
    • Zhi-Xun Shen
  4. Departments of Physics and Applied Physics, Stanford University, Stanford, California 94305, USA

    • Inna M. Vishik,
    • Rui-Hua He &
    • Zhi-Xun Shen
  5. Present address: Department Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.

    • Inna M. Vishik
  6. Present address: Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA.

    • Rui-Hua He

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