We analyze the ultrahigh-resolution laser angle-resolved photoemission spectroscopy intensity from the slightly underdoped Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ in the superconductive (SC) state. The momentum distribution curves were fitted at each energy $\omega$ employing the SC Green’s function along several cuts perpendicular to the Fermi surface with the tilt angle $\theta$ with respect to the nodal cut. The clear observation of particle-hole mixing was utilized such that the complex self-energy as a function of $\omega$ is directly obtained from the fitting. The obtained angle-resolved self-energy is then used to deduce the Eliashberg function ${\alpha }^2{F}^{(+)}(\theta ,\omega )$ in the diagonal channel by inverting the $d$-wave Eliashberg equation using the maximum entropy method. Besides a broad featureless spectrum up to the cutoff energy ${\omega }_c$, the deduced ${\alpha }^{2}F$ exhibits two peaks around 0.05 and 0.015 eV. The former and the broad feature are already present in the normal state, while the latter emerges only below $T_c$. Both peaks become enhanced as $T$ is lowered or the angle $\theta$ moves away from the nodal direction. The implication of these findings is discussed.

• Received 2 June 2011

DOI:https://doi.org/10.1103/PhysRevB.84.104521