Evidence for the Importance of Extended Coulomb Interactions and Forward Scattering in Cuprate Superconductors

S. Johnston, I. M. Vishik, W. S. Lee, F. Schmitt, S. Uchida, K. Fujita, S. Ishida, N. Nagaosa, Z. X. Shen, and T. P. Devereaux

Phys. Rev. Lett. 108, 166404 – Published 20 April 2012

Abstract

The prevalent view of the high-temperature superconducting cuprates is that their essential low-energy physics is captured by local Coulomb interactions. However, this view been challenged recently by studies indicating the importance of longer-range components. Motivated by this, we demonstrate the importance of these components by examining the electron-phonon ( $e$ -ph) interaction with acoustic phonons in connection with the recently discovered renormalization in the near-nodal low-energy ( $\sim 8 - 15 meV$ ) dispersion of ${Bi}_{2} {Sr}_{2} {CaCu}_{2} O_{8 + δ}$ . By studying its nontrivial momentum and doping dependence we conclude a predominance of forward scattering arising from the direct interplay between the $e$ -ph and extended Coulomb interactions. Our results thus demonstrate how the low-energy renormalization can provide a pathway to new insights into how these interactions interplay with one another and influence pairing and dynamics in the cuprates.

Received 6 January 2011

DOI:https://doi.org/10.1103/PhysRevLett.108.166404

Authors & Affiliations

S. Johnston^1,4, I. M. Vishik^1,2,3, W. S. Lee^1,2, F. Schmitt^1,2, S. Uchida⁵, K. Fujita⁶, S. Ishida⁵, N. Nagaosa^7,8, Z. X. Shen^1,2,3, and T. P. Devereaux^1,2

¹Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, California 94305, USA
²Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
³Department of Physics and Applied Physics, Stanford University, Stanford, California 94305, USA
⁴IFW Dresden, P.O. Box 27 01 16, D-01171 Dresden, Germany
⁵Department of Physics, Graduate School of Science, University of Tokyo, Bunkyo-Ku, Tokyo 113-0033, Japan
⁶Laboratory for Atomic and Solid State Physics, Department of Physics, Cornell University, Ithaca, New York 14853, USA
⁷Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
⁸Cross-Correlated Materials Research Group (CMRG) and Correlated Electron Research Group (CERG), RIKEN-ASI, Wako 351-0198, Japan

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Vol. 108, Iss. 16 — 20 April 2012

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Images

Figure 1
Nodal low-energy dispersion renormalization in Bi-2212 UD55 (11 K). (a) Color image plot of the raw data with both the low-energy and 70 meV renormalizations marked by arrows. (b) The band dispersion derived by fitting the momentum distribution curves (MDC) at each energy in (a) with a Lorentzian line shape. The low-energy renormalization is defined by the deviation of the dispersion from $v_{mid}$ , the velocity fit between 30–40 meV (dotted line). (c) The MDC FWHM which shows a more rapid decrease close to $E_{F}$ as a consequence of the renormalization of the low-energy dispersion.Reuse & Permissions
Figure 2
(a) Calculated $A (k, ω)$ along the nodal direction $(0, 0) - (π, π)$ in the superconducting state (25 K). Here, an additional component $\propto ω^{2}$ has been added to the imaginary part of the self-energy to simulate broadening due to $e − e$ interactions. (b) The MDC-derived dispersion and (c) MDC FWHM obtained from (a). The parameters used in the calculation are given in the text.Reuse & Permissions
Figure 3
Low-energy dispersion renormalization away from the node (theory/experiment UD55, $T = 11 K$ ). (a) MDC dispersions at the node (rightmost) and away from the node, offset horizontally for clarity. The black circles indicate the gap energy, determined by fitting symmetrized data to a minimal model [28]. The red circles indicate the approximate scale of the low-energy renormalization, determined from the deviation from $v_{mid}$ . (b) MDC dispersions obtained from the calculated $A (k, ω)$ . The parameters used here are identical to those used in Fig. 2. (c) A sketch of how the energy of the renormalization can be further quantified by subtracting a linear offset from the dispersion (shown here for a nodal cut) between the gap energy and 40 meV. (d) Difference in $k$ between the dispersion and dotted line in (c) for cuts taken at the node (top) and progressively away from the node (towards the bottom). The data has been smoothed over 20 iterations and the curves are offset vertically for clarity. (e) As in (d) but obtained from the calculated $A (k, ω)$ . (f) The momentum dependence of the gap and renormalization energies (from experiment), the latter determined by the method in (c). The error bars on the gap (1.5 meV), reflect uncertainty in determining $E_{F}$ (0.5 meV) and uncertainty in the fit (0.5 meV) plus a 50% margin. Error bars for the renormalization energy reflect variation upon smoothing by different numbers of iterations.Reuse & Permissions
Figure 4
(a) Nodal MDC-derived dispersions obtained from the model $A (k, ω)$ for different $q_{TF}$ . (Any doping dependence of $ϵ$ has been neglected.) (b) Solid line: $λ (k_{node})$ for the acoustic mode as a function of $q_{TF}$ . Symbols: experimental estimates for $λ (k_{node}) = v_{mid} / v_{F} - 1$ as a function of doping (Refs. 6, 9).Reuse & Permissions

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