Electronic Inhomogeneity and Vortex Disorder in Superconducting Sr$_{0.75}$K$_{0.25}$Fe$_{2}$As$_{2}$

We characterize the surface structure, superconducting, and vortex properties in the hole-doped superconductor Sr$_{0.75}$K$_{0.25}$Fe$_{2}$As$_{2}$ (underdoped, $T_{c}=$32 K) by scanning tunneling microscopy. A 1 $\times$ 2 surface reconstruction and inhomogeneous superconducting gap with clear coherence peaks are universally found on the dominant Sr/K-terminated surfaces. Rarer patches of As termination show no reconstruction and no gap. The superconducting gap energy $\Delta $ anti-correlates with both the zero bias conductance and coherence peak strength with a characteristic length scale of $\sim$ 3 nm. Isotropic single-quantum vortices with short-range hexagonal order are imaged at 9 T magnetic field. By fitting the vortex-induced subgap density of states, the coherence length $\xi \sim$ 2.8 nm is found to be comparable to the length scale of $\Delta $ variations. We suggest that the vortices are strongly pinned by nanoscale electronic inhomogeneity arising from K clustering.