  The interfacial FeSe/TiO$_{2-\delta}$ coupling induces high-temperature
superconductivity in monolayer FeSe films. Using cryogenic atomically resolved
scanning tunneling microscopy/spectroscopy, we obtained atomic-site dependent
surface density of states, work function, and pairing gap in the monolayer FeSe
on SrTiO$_3$(001)-($\sqrt{13} \times \sqrt{13}$)-33.7${\deg}$ surface. Our
results disclosed the out-of-plane Se-Fe-Se triple layer gradient variation,
switched DOS for Fe sites on and off TiO$_{5\square}$, and inequivalent Fe
sublattices, which gives global spatial modulation of pairing gap contaminant
with the ($\sqrt{13} \times \sqrt{13}$) pattern. Moreover, the coherent lattice
coupling induces strong inversion asymmetry and in-plane anisotropy in the
monolayer FeSe, which is demonstrated to correlate with the particle-hole
asymmetry in coherence peaks. The strong atomic-scale correlations in lattice
and electronic structure, and pairing gap in particular, put constraints on
exploring the unconventional high-temperature superconductivity emerging from
interface coupling, e.g., strong demand for atomic-scale interface engineering
and characterization.
