Suppression of Spectral Gap and Flat Bands on a Cuprate Superconductor Side-Surface

Side surfaces of cuprate superconductors are expected to display a suppressed $d$-wave order parameter and zero-energy topological flat bands with a large density of states, making them susceptible to symmetry broken orders. Yet such surfaces have never been investigated with momentum-resolved, surface-sensitive probes, because high-temperature superconductors rarely cleave along them. Using focused-ion-beam milling to define a controlled breaking point, we expose pristine (110) side surfaces of overdoped La$_{2-x}$Sr$_x$CuO$_4$ ($x=0.22$) suitable for angle-resolved photoemission. We observe the suppression of the superconducting spectral gap within our energy resolution ($\sim 4~\mathrm{meV}$), and surprisingly, the expected zero-energy flat band peak is also suppressed, despite the high topographic quality of the surface. Self-consistent Bogoliubov--de~Gennes calculations show that the measured geometric roughness of the cleaved surface is too weak to eliminate these modes. The calculations further demonstrate that bulk inhomogeneities characteristic of high-temperature superconductors, modelled as moderate Anderson-type disorder, can broaden the flat-band states beyond detectability. Our results provide the first momentum-resolved view of the electronic structure on a cuprate side surface and reveal disorder as the key factor currently preventing appearance of flat bands and their associated correlated orders.