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Scanning tunneling microscopy (STM) measurements have observed modulations inthe density of states (DOS) of a number of high-T$_c$ cuprates. Thesemodulations have been interpreted in terms of electron-boson coupling analogousto the dispersion "kinks" observed by angle-resolved photoemission spectroscopy(ARPES). However, a direct a reconciliation of the energy scales and featuresobserved by the two probes is presently lacking. In this paper we examine thegeneral features of el-boson coupling in a $d-$wave superconductor usingEliashberg theory, focusing on the structure of the modulations and the role ofself energy contributions $\lambda_z$ and $\lambda_{\phi}$. We identify thefeatures in the DOS that correspond to the gap-shifted bosonic mode energiesand discuss how the structure of the modulations provides information about anunderlying pairing mechanism and the pairing nature of the boson. We argue thatthe scenario most consistent with the STM data is that of a low-energy bosonmode renormalizing over a second dominant pairing interaction and we identifythis low-energy mode as the out-of-phase bond buckling oxygen phonon. Theinfluence of inelastic damping on the phonon-modulated DOS is also examined forthe case of Bi$_2$Si$_2$CaCu$_2$O$_{8+\delta}$. Using this simplified frameworkwe are able to account for the observed isotope shift and anti-correlationbetween the local gap and mode energies. Combined, this work provides a directreconciliation of the bandstructure renormalizations observed by both ARPES andSTM in terms of coupling to optical oxygen phonons.

physical review. b, condensed matter and materials physics

Density of States Modulations from Oxygen Phonons in d-wave  Superconductors: Reconciling Angle-Resolved Photoemission Spectroscopy and  Scanning Tunneling Microscopy