Modeling scanning tunneling spectra of $Bi_2 Sr_2 CaCu_2 O_{8+\delta}$
  Recent angle-resolved photoemission and neutron scattering data have provided new ingredients for the interpretation of scanning tunneling spectra on Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. We analyze the low-temperature tunneling spectra, from oxygen overdoped to underdoped samples, including details about the bilayer splitting and the neutron resonance peak. Two van Hove singularities are identified: the first is integrated in the coherence peaks, the second is heavily broadened at higher binding energy. The shape of the tunneling spectra suggests a strong coupling of the quasiparticles with a collective mode, and a comparison with photoemission shows that the scattering rate in tunneling is an order of magnitude smaller than in ARPES. Finally, the theoretical spectra calculated with an isotropic tunneling matrix element are in better agreement with the experimental data than those obtained with anisotropic matrix elements. 
B. W. Hoogenboom, C. Berthod, M. Peter, \O. Fischer, and A. A. Kordyuk