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Analysis of Laser ARPES from Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ in superconductive state: angle resolved self-energy and fluctuation spectrum

We analyze the ultra high resolution laser angle resolved photo-emission spectroscopy (ARPES) intensity from the slightly underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ in the superconductive (SC) state. The momentum distribution curves (MDC) were fitted at each energy $\w$ employing the SC Green's function along several cuts perpendicular to the Fermi surface with the tilt angle $θ$ 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 $ω$ is directly obtained from the fitting. The obtained angle resolved self-energy is then used to deduce the Eliashberg function $α^2 F^{(+)}(þ,\w)$ 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 $ω_c$, the deduced $α^2 F$ exhibits two peaks around 0.05 eV 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 $þ$ moves away from the nodal direction. The implication of these findings are discussed.