Superconductivity amid phase inhomogeneity: the case of K$_{x}$Fe$_{2-y}$Se$_{2}$

The recently discovered Fe-based superconductors, K$_{x}$Fe$_{2-y}$Se$_{2}$, is studied using neutron diffraction and the pair density function analysis to investigate the nature of the atomic disorder induced by the K and Fe site vacancies. In this system, both superconductivity and magnetic ordering can coexist, while superconductivity is observed in a narrow range of potassium concentration, between 0.6 $<$ x $<$ 0.8. While no crystal transition occurs across with x, the Fe site vacancies are ordered in the $\sqrt{5} \times \sqrt{5}$ structure. At high temperatures, the Fe vacancies are not ordered. Why does superconductivity appear in the vicinity of the 0.8 composition? To provide a clue towards the answer, instead of probing the periodic structure, we probed the local atomic structure that provides information regarding the short-range correlations in real space. The results suggest a strong dependence of the Fe-Fe bond lengths to the K concentration. What is unique to this system is that a double-well bond distribution of short and long Fe - Fe bonds exists, originating from the fully occupied Fe site. As the K concentration increases to x=1, the distribution shifts weight from the short to the long while in the superconducting case, it is equal between the two.