Enhanced coherence of antinodal quasiparticles in a dirty $d$-wave superconductor

Recent angle-resolved photoelectron spectroscopy experiments show a narrow quasiparticle peak at the gap edge along the antinodal [1,0] direction for the overdoped cuprate superconductors. We show that within weak coupling BCS theory for a $d$-wave superconductor, the $s$-wave single-impurity scattering cross section vanishes for energies $\omega =\Delta$ (gap edge). This coherence effect occurs through multiple scattering off the impurity. For small impurity concentrations the spectral function has a pronounced increase of the (scattering) lifetime for antinodal quasiparticles but shows a very broad peak in the nodal direction, in qualitative agreement with the above-mentioned experiment and in strong contrast to the behavior observed in underdoped cuprates.

Figure 1

(a) DOS (upper panel) and the lifetime (lower panel) for $\Gamma ∕\Delta$ in the Born limit. (b) The same plot for the unitarity limit.

Figure 2

(Color online) Variation of spectral density $A({\xi }_{\mathbf{k}},\omega )$ for $-2<\xi ∕\Delta <2$ and $\Gamma ∕\Delta =0.05$ in the unitarity limit, (a) antinodal $({\varphi }_{\mathbf{k}}=0)$ (b) nodal direction $({\varphi }_{\mathbf{k}}=\pi ∕4)$. A cut of the spectral density along ${\xi }_{\mathbf{k}}=0$ for $0.05<\Gamma ∕\Delta <0.5$, (c) antinodal (d) nodal direction.

Figure 3

Phase shift dependence of the lifetime at (a) $\omega ∕\Delta =0$ and (b) $\omega ∕\Delta =1$.