Research ArticleCONDENSED MATTER PHYSICS

Quantitative determination of pairing interactions for high-temperature superconductivity in cuprates

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Science Advances  04 Mar 2016:
Vol. 2, no. 3, e1501329
DOI: 10.1126/sciadv.1501329
  • Fig. 1 Color representation of the measured photoemission intensity of the UD89 sample along the θ = 35° direction.

    (A) T = 16 K. (B) T = 107 K. (C) Progression of energy-momentum dispersions at temperatures of 16, 70, 80, 97, and 107 K. The inset to (C) presents, on an expanded scale, an illustration of the consistency of the data up to an accuracy of 5 × 10−3(π/a) in the region at high energy where no temperature-dependent corrections to the dispersion are necessary. In section SII, we show the systematic errors in the data when such accuracy is not met and how we correct them.

  • Fig. 2 Measured MDCs and their fits at five different energies at 97 and 16 K in the UD89 sample along the dark-green trajectory (θ = 20°).

    Such data were taken at 1-meV intervals and at all of the trajectories shown in (C). The vertical scale in (A) is the measured photoelectron current for a fixed photon flux (in arbitrary units). It is crucial that this scale remain within the error bars discussed in the text at all temperatures, energies, and momenta, and that any systematic errors be corrected. (B) Normalized difference in measurements taken at the two temperatures in (A) and at the same quantity calculated from the fits in (A). The fits to the MDCs were made according to the procedure described in fig. S1. The normal and pairing self-energies are extracted from such fits, also as described in fig. S1. (D) intends to show the consistency of the fits to the MDCs at different energies; it shows the EDC generated from the MDC within the complete range of energies measured at an angle θ = 20° at the Fermi surface at 16 K and compares direct measurements of the EDC at the same point and same temperature.

  • Fig. 3 Normal and pairing self-energies.

    (A and B) The evolution of the extracted normal self-energy (A) and the evolution of the pairing self-energy (B) as a function of temperature directly from the fits to the MDCs in OD82. The normal and pairing self-energies show superconducting gap–induced features at low energies up to about 3Δ and smoothly vary in energy thereafter up to a cutoff energy. (C) The pairing self-energies in UD89 at 16 K divided by cos(2θ). Determination of the pairing self-energy has acceptable signal-to-noise ratios up to about 0.2 eV only. The data fall together at the angles shown (up to an accuracy better than 10%) up to about 0.2 eV. (D) The self-energies smoothed over ±5 meV (as discussed in the text) and, after removing the impurity, induced features for OD82 at T = 17 K (solid lines) and T = 70 K (dashed lines).

  • Fig. 4 Eliashberg functions: Normal [εN(θ, ω)] and scaled pairing [εP(θ, ω) ≡ εP(θ, ω)/cos(2θ)].

    These are calculated using the solution for the Eliashberg equations from the measured self-energies. (A) and (B) compare εP(20°, ω) and εN(20°, ω) deep in the superconducting state for the two samples, and the latter also above Tc. At low temperatures, they are of the same accuracy over the whole frequency range, with a large superconductivity-induced enhanced low-energy peak. (C) Closer to Tc, the low-energy peak in εP(ω) disappears. This trend is shown more directly in (F). (D) and (E) give εN(θ, ω) for T above Tc, showing the increase in cutoff energy with increasing θ. The gentle waviness in all of the results represents artifacts of the maximum entropy method for the solution of the Eliashberg equations.

  • Fig. 5 Calculation of pairing self-energies assuming that εP = εN.

    (A and B) The experimentally deduced real and imaginary parts of self-energy (in absolute units) are compared with a calculation of the same quantity from the Eliashberg equations, assuming that εP = εN for (A) T = 70 K and (B) T = 35 K in the OD82 sample. This agreement occurs only if the Eliashberg equations are applicable for the analysis of the data and for the relation of the two Eliashberg functions.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/3/e1501329/DC1

    SI. Extraction of normal and pairing self-energies from ARPES in the superconducting state

    SI. 1. Single-particle spectral function

    SI. 2. Procedure for extracting self-energy

    SII. Correction of systematic errors and renormalization of ARPES data

    SII. 1. Limits of the validity of results

    SIII. Equations for self-energy

    SIII. 1. Exact representation of self-energy

    SIII. 2. Familiar Eliashberg integral equations for d-wave superconductors

    SIV. Comparison of theories with experimental results

    Fig. S1. Data corrections that are necessary because of systematic errors attributable to the movement of the sample with change in temperature.

    Fig. S2. Exact representation of the normal and pairing self-energies.

    Fig. S3. Imaginary part of the pairing self-energy at various angles across the Fermi surface calculated by Hong and Choi (28) from the spin-fluctuation spectra in La2−xSrxCuO4 measured by Vignolle et al. (29) at optimal doping.

    Fig. S4. Imaginary part of the gap function for the Hubbard model calculated by Gull and Millis (30) for various dopings indicated in the plot.

    References (3648)

  • Supplementary Materials

    This PDF file includes:

    • SI. Extraction of normal and pairing self-energies from ARPES in the superconducting state
    • SI. 1. Single-particle spectral function
    • SI. 2. Procedure for extracting self-energy
    • SII. Correction of systematic errors and renormalization of ARPES data
    • SII. 1. Limits of the validity of results
    • SIII. Equations for self-energy
    • SIII. 1. Exact representation of self-energy
    • SIII. 2. Familiar Eliashberg integral equations for d-wave superconductors
    • SIV. Comparison of theories with experimental results
    • Fig. S1. Data corrections that are necessary because of systematic errors attributable to the movement of the sample with change in temperature.
    • Fig. S2. Exact representation of the normal and pairing self-energies.
    • Fig. S3. Imaginary part of the pairing self-energy at various angles across the Fermi surface calculated by Hong and Choi (28) from the spin-fluctuation spectra in La2−xSrxCuO4 measured by Vignolle et al. (29) at optimal doping.
    • Fig. S4. Imaginary part of the gap function for the Hubbard model calculated by Gull and Millis (30) for various dopings indicated in the plot.
    • References (36–48)

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