Bridging atomic and mesoscopic length scales with Replica Scanning Tunneling Microscopy: Visualizing the atomic lattice of UTe$_2$ and the atomic scale superconducting gap modulations of FeSe close to micron length scales

Scanning Tunneling Microscopy is a cornerstone technique for visualizing the electronic density of states with atomic resolution (typically below 0.1 nm). While the field of view of most STM setups extends up to a few microns, obtaining atomic resolution over these large areas is often impractical and excessively time-consuming. This is due to the need to acquire maps with a point number reaching $10^7$ or more with a full current or conductance vs voltage curve at each point. The standard procedure is to make large scale maps and then select small regions to zoom-in for high-resolution atomic scale analysis. However, this approach fails to address a question which is often critical: Does a specific atomic-scale modulation of the electronic density of states persist over much larger, mesoscopic length scales? Here we present a new method: Replica STM (R-STM), that overcomes this limitation, allowing the study of atomic-scale phenomena up to micron length scales. We obtained new large-area STM tunneling conductance maps in UTe$_2$ and FeSe, spanning areas over 200 nm in size. In these large scale maps we discovered periodic signals with wavelengths significantly exceeding interatomic distances. We show that these large-wavelength periodic signals are replicas of the underlying atomic-scale density of states modulations. R-STM leverages these replica signals to efficiently track atomic-scale features over large areas. We discuss the influence of phase slips, disorder and defects in the replicas. Our results suggest that atomic scale modulations of the superconducting density of states could persist over large length scales in FeSe. R-STM provides a new capability for STM to compare atomic scale with micrometer scale phenomena. The proof of principle of R-STM can be extended to any other scanning probe microscopy experiment where a periodic signal is traced as a function of position.