Sharp Low-Energy Feature in Single-Particle Spectra due to Forward Scattering in $d$-Wave Cuprate Superconductors

There is an enormous interest in the renormalization of the quasiparticle (qp) dispersion relation of cuprate superconductors both below and above the critical temperature $T_c$ because it enables the determination of the fluctuation spectrum to which the qp’s are coupled. A remarkable discovery by angle-resolved photoemission spectroscopy (ARPES) is a sharp low-energy feature (LEF) in qp spectra well below the superconducting energy gap but with its energy increasing in proportion to $T_c$ and its intensity increasing sharply below $T_c$. This unexpected feature needs to be reconciled with $d$-wave superconductivity. Here, we present a quantitative analysis of ARPES data from ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212) using Eliashberg equations to show that the qp scattering rate due to the forward scattering impurities far from the Cu-O planes is modified by the energy gap below $T_c$ and shows up as the LEF. This is also a necessary step to analyze ARPES data to reveal the spectrum of fluctuations promoting superconductivity.

Figure 1

(a) The impurity self-energy from the model calculation of Eq. (6). The real and imaginary parts are shown along several cuts with the dashed and solid curves, respectively. [(b),(c)] The extracted self-energy from MDC analysis of UD89K at $T=16\mathrm{K}$ and of OD82K at $T=17\mathrm{K}$.

Figure 2

The imaginary parts of the extracted self-energy from UD89K [(a),(b)] and OD82K [(c),(d)] data both above and below $T_c$ along $\theta =0°$ and 15°. The dashed curves are the fittings from Eq. (7). Notice that the LEF is sharply enhanced below $T_c$ especially along off-nodal cuts.

Figure 3

The real parts of the self-energy of UD89K at 16 K. The blue and green curves represent the extracted and calculated [Eq. (8)] self-energy, and the red curves show the calculated self-energy with the impurity term removed.

Figure 4

Comparison of the imaginary parts of the calculated impurity self-energy (blue curves) with the extracted one (black curves) for UD89K. The red curves represent their difference, that is, the intrinsic self-energy due to coupling to bosons. (a) Angle varies at $T=16\mathrm{K}$, and (b) temperature varies at $\theta =15°$.