Research ArticleCONDENSED MATTER PHYSICS

Unusual behavior of cuprates explained by heterogeneous charge localization

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Science Advances  25 Jan 2019:
Vol. 5, no. 1, eaau4538
DOI: 10.1126/sciadv.aau4538
  • Fig. 1 Gap inhomogeneity and phase diagram of the cuprates.

    (A) Gaussian gap distribution function g at several doping levels, shown as a function of energy. The parameters are pc = 0.2, Δ0 = 4000 K, and δ = 800 K. The fraction of the distribution that has reached the Fermi level (the portion above zero energy indicated with a dashed line) is added to the p delocalized doped charge carriers at temperature T = 0. (B) Effective density of delocalized carriers per CuO2 unit at T = 0, obtained as the sum of the doped hole concentration p and delocalized holes from the distributions shown in (A) (full line). The dashed line corresponds to the (skewed Gaussian) gap distribution parameters used for comparison with experiments on La2−xSrxCuO4 [LSCO; see Figs. 2 (C and D) and 4, Materials and Methods, and Supplementary Materials], with pc = 0.22, Δ0 = 3900 K, δ = 800 K, and skew parameter α = 2. (C) Second derivative of the normal-state resistivity, multiplied by peff at T = 0. This result is obtained by combining the effective carrier density from Eq. 1, obtained with the gap distributions in (A), with the experimentally determined Fermi liquid scattering rate of itinerant carriers (5). The characteristic features of the phase diagram are apparent: a quadratic resistive regime at both low and high doping, the temperatures T** and T*, and the linear-T-like regime around optimal doping.

  • Fig. 2 Comparison of the model to experiments.

    (A) Resistivity and (B) Hall constant for underdoped Hg1201 [p ≈ 0.1; blue circles; data from (5)] compared with the gap disorder model (red lines). A skewed Gaussian gap distribution and linear mean gap doping dependence are used, with pc = 0.2, Δ0 = 4000 K, δ = 600 K, and α = 2 (see Materials and Methods). The asymptotic value of the measured RH at T = 0 corresponds to ~90% of the nominal concentration of itinerant carriers, and the calculated curve is thus multiplied by the same factor. Note that this value is within the experimental uncertainty due to sample size and shape uncertainty (2, 5). (C) Resistivity curvature map of LSCO normalized at each doping level to ρ(300 K), from (7). In addition to the characteristic temperatures T** and T*, LSCO features a structural transition, which is detected in the resistivity measurement (red downward sloping line) (5). (D) Calculated resistivity curvature map, with added percolative superconducting precursor regime (14) close to Tc and parameters pc = 0.22, Δ0 = 3900 K, δ = 800 K, and α = 2 (same as for the dashed line in Fig. 1B). Resistivity is normalized by peff(T = 0), which is nearly equivalent to the experimental normalization and multiplied by a factor of 4·10−6 to have the same absolute color scale as (C).

  • Fig. 3 Mean localization gap.

    Comparison of the characteristic high-energy scale for different compounds: high-energy pseudogap scale in photoemission [angle-resolved photoemission spectroscopy (ARPES)], superconductor-insulator-superconductor (SIS) tunneling spectra, and mid-infrared peak in optical conductivity data. The solid green line is our generic parameterization of the localization gap, and the shaded green band indicates the gap distribution width (Fig. 1). The dashed line is an alternative phenomenological form of the doping dependence (see Materials and Methods), which features an upward curvature and extrapolates to the transport charge-transfer gap of ~1 eV at zero doping (24). ARPES and SIS data for LSCO and Bi2Sr2CaCu2O8+δ (Bi2212) are adapted from (11); the data are from multiple experiments [see (11) for original references]. For LSCO optical conductivity peak extraction, see the Supplementary Materials. YBCO data are from (18).

  • Fig. 4 Superfluid density, inhomogeneous localization, and superconducting percolation.

    (A) Zero-temperature superfluid density ρs0 across the cuprate phase diagram calculated from Eq. 4 (orange line) and compared with experiments. The same parameters as in Figs. 1B and 2D are used (pc = 0.22, Δ0 = 3900 K, δ = 800 K, and α = 2). At low doping, ρs0 is limited by the density of mobile holes (red circles) and by pair-breaking impurities, whereas at high doping, the limiting factor is the density of localized holes (green squares). Diamonds are data from (17), whereas the other symbols are data for La2−xBaxCuO4 (at p = x = 0.095) and LSCO from optical conductivity [empty circles (97)], penetration depth [full circles (98)], and muon spin rotation [squares (51)]. The insets are schematic, and one green square represents approximately four planar CuO2 units. A percolation regime is expected on both ends of the superconducting (SC) dome due to spatial inhomogeneity of superconducting gaps (1416). (B) ρs0 for overdoped LSCO (17) in units of holes CuO2 units cell (blue diamonds). Between optimal doping and Tc ~ 12 K (0.17 < p < 0.24), ρs0 is determined by the density of localized holes ploc (dash-dotted line), whereas at lower Tc superconducting gap inhomogeneity causes percolation [dash-dotted green line; see Materials and Methods for the form of f(Tc)]. The product of the two (full line) gives a reasonable description of the entire curve. Inset: Log-log plot of ρs0 at low Tc demonstrates good agreement with percolative scaling ρs0 ~ Tc1.6 (see the Supplementary Materials) (dashed dotted line) compared with the previously used quadratic scaling (full line) (17). A power-law fit ρs0 ~ Tcγ below 9 K gives γ = 1.64 ± 0.07.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/1/eaau4538/DC1

    Supplementary Text

    Fig. S1. Normal-state phase diagram for different gap distributions.

    Fig. S2. Normal-state phase diagrams for two doping dependences of the gap distribution parameters.

    Fig. S3. Local probes of disorder in cuprates.

    Fig. S4. High-energy gap scale in cuprates.

    Fig. S5. Characteristic temperature and localization.

    Fig. S6. Temperature and doping dependence of Hall constant for LSCO.

    Fig. S7. Doping dependence of linear and quadratic resistivity coefficients of LSCO.

    Fig. S8. Doping dependence of sheet resistance coefficients.

    References (6196)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text
    • Fig. S1. Normal-state phase diagram for different gap distributions.
    • Fig. S2. Normal-state phase diagrams for two doping dependences of the gap distribution parameters.
    • Fig. S3. Local probes of disorder in cuprates.
    • Fig. S4. High-energy gap scale in cuprates.
    • Fig. S5. Characteristic temperature and localization.
    • Fig. S6. Temperature and doping dependence of Hall constant for LSCO.
    • Fig. S7. Doping dependence of linear and quadratic resistivity coefficients of LSCO.
    • Fig. S8. Doping dependence of sheet resistance coefficients.
    • References (6196)

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