  Scanning tunneling microscopy (STM) is perhaps the most promising way to
detect the superconducting gap size and structure in the canonical
unconventional superconductor Sr$_2$RuO$_4$ directly. However, in many cases,
researchers have reported being unable to detect the gap at all in simple STM
conductance measurements. Recently, an investigation of this issue on various
local topographic structures on a Sr-terminated surface found that
superconducting spectra appeared only in the region of small nanoscale canyons,
corresponding to the removal of one RuO surface layer. Here, we analyze the
electronic structure of various possible surface structures using first
principles methods, and argue that bulk conditions favorable for
superconductivity can be achieved when removal of the RuO layer suppresses the
RuO$_4$ octahedral rotation locally. We further propose alternative
terminations to the most frequently reported Sr termination where
superconductivity surfaces should be observed.
