Self-consistent temperature dependence of quasiparticle bands in monolayer FeSe on SrTiO$_3$

Abstract: We study the temperature evolution of the quasiparticle bands of the FeSe monolayer on the SrTiO$_3$ (STO) substrate from 10 to 300 K by applying the anisotropic, multiband and full-bandwidth Eliashberg theory. To achieve this, we extend this theory by self-consistently coupling the chemical potential to the full set of Eliashberg equations. In this way, the electron filling can accurately be kept at a constant level at any temperature. Solving the coupled equations self-consistently, and with focus on the interfacial electron-phonon coupling, we compute a nearly constant Fermi surface with respect to temperature and predict a non-trivial temperature evolution of the global chemical potential. This evolution includes a total shift of 5 meV when increasing temperature from 10 to 300 K and a hump-like dependence followed by a kink at the critical temperature T$_c$. We argue that the latter behavior indicates that superconductivity in FeSe/SrTiO$_3$ is near to the BCS-BEC crossover regime. Calculating the temperature dependent Angle Resolved Photoemission Spectroscopy (ARPES) spectra, we suggest a new route to determine the energy scale of the interfacial phonon mode by measuring the energy position of second-order replica bands. Further, we re-examine the often used symmetrization procedure applied to such ARPES curves and demonstrate substantial asymmetric deviations. Lastly, our results reveal important aspects for the experimental determination of the momentum anisotropy of the superconducting gap.