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

Scanning tunneling microscopy (STM) measurements have observed modulations in the density of states (DOS) of a number of high-T$_c$ cuprates. These modulations have been interpreted in terms of electron-boson coupling analogous to the dispersion "kinks" observed by angle-resolved photoemission spectroscopy (ARPES). However, a direct a reconciliation of the energy scales and features observed by the two probes is presently lacking. In this paper we examine the general features of el-boson coupling in a $d-$wave superconductor using Eliashberg theory, focusing on the structure of the modulations and the role of self energy contributions $λ_z$ and $λ_ϕ$. We identify the features in the DOS that correspond to the gap-shifted bosonic mode energies and discuss how the structure of the modulations provides information about an underlying pairing mechanism and the pairing nature of the boson. We argue that the scenario most consistent with the STM data is that of a low-energy boson mode renormalizing over a second dominant pairing interaction and we identify this low-energy mode as the out-of-phase bond buckling oxygen phonon. The influence of inelastic damping on the phonon-modulated DOS is also examined for the case of Bi$_2$Si$_2$CaCu$_2$O$_{8+δ}$. Using this simplified framework we are able to account for the observed isotope shift and anti-correlation between the local gap and mode energies. Combined, this work provides a direct reconciliation of the bandstructure renormalizations observed by both ARPES and STM in terms of coupling to optical oxygen phonons.

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