Crystal growth and detailed structural characterization of superconducting and non-superconducting phases in the K$_{1-x}$Fe$_{2-y}$Se$_2$ system

Amid the flurry of activity on K$_{1-x}$Fe$_{2-y}$Se$_2$ superconductors, it remains established that the stoichiometric compound K$_2$Fe$_4$Se$_5$ is an antiferromagnetic semiconductor. This raises the question of whether subtle Fe$^{1+/3+}$ doping causes K$_{1-x}$Fe$_{2-y}$Se$_2$ to become a bulk superconductor, and if so, is there a structural distinction between superconducting and non-superconducting phases? We have grown K$_{1-x}$Fe$_{2-y}$Se$_2$ samples that show superconductivity with $T_C$ = 31 K, even when growth conditions are starkly different from those reported in the literature. Here we present high-resolution synchrotron X-ray diffraction measurements, alongside single-crystal x-ray and electron diffraction, to elucidate the phase space in this system. Combined with magnetometry, heat capacity, and transport measurements, our structure-property relations help prescribe how chemical composition and heat treatment induce superconductivity and vacancy ordering in the K$_{1-x}$Fe$_{2-y}$Se$_2$ system.