The cuprate high-temperature superconductors are known to host a wide array of effects due to interactions and disorder. In this paper, we look at some of the consequences of these effects which can be visualized by scanning tunneling spectroscopy in order to provide a guide for future experiments. These interaction and disorder effects can be incorporated into a mean-field description by means of a self-energy appearing in the Green's function. We first examine the quasiparticle scattering interference (QPI) spectra in the superconducting state at optimal doping as temperature is increased. Assuming agreement with angle-resolved photoemission experiments which suggest that the scattering rate depends on temperature, resulting in the filling of the $d$-wave gap, we find that the peaks predicted by the octet model become progressively smeared as temperature is increased. When the scattering rate is of the same order of magnitude as the superconducting gap, the spectral function shows Fermi-arc-like patterns, while the power spectrum of the local density of states shows the destruction of the octet-model peaks. We next consider the normal state properties of the optimally-doped cuprates. We model this by adding a marginal Fermi liquid (MFL) self-energy to the normal-state propagator and consider the dependence of the QPI spectra on frequency, temperature, and doping. We demonstrate that the MFL self-energy leads to a smearing of the caustics appearing in the normal-state QPI power spectrum as either temperature or frequency is increased at fixed doping. The smearing is found to be more prominent in the MFL case than in an ordinary Fermi liquid. We also consider the case of a marginal Fermi liquid with a strongly momentum-dependent self-energy which gives rise to a visible “nodal-antinodal” dichotomy at the normal state and discuss how the spectra as seen in ARPES and STS differ from both an isotropic metal and a broadened $d$-wave superconductor. Finally, we discuss how these results become modified in the presence of weak distributed disorder and finite-temperature smearing.

• Received 8 June 2018
• Revised 17 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.094518