Superconductivity at the vacancy disorder boundary in K$_x$Fe$_{2-y}$Se$_2$

The role of phase separation in the emergence of superconductivity in alkali metal doped iron selenides A$_{x}$Fe$_{2-y}$Se$_{2}$ (A = K, Rb, Cs) is revisited. High energy X-ray diffraction and Monte Carlo simulation were used to investigate the crystal structure of quenched superconducting (SC) and as-grown non-superconducting (NSC) K$_{x}$Fe$_{2-y}$Se$_{2}$ single crystals. The coexistence of superlattice structures with the in-plane $\sqrt{2}\times\sqrt{2}$ K-vacancy ordering and the $\sqrt{5}\times\sqrt{5}$ Fe-vacancy ordering were observed in SC and NSC crystals along side the \textit{I4/mmm} Fe-vacancy free phase. Moreover, in the SC crystal an Fe-vacancy disordered phase is additionally present. It appears at the boundary between the \textit{I4/mmm} vacancy free phase and the \textit{I4/m} vacancy ordered phase ($\sqrt{5}\times\sqrt{5}$). The vacancy disordered phase is most likely the host of superconductivity.