Spectromicroscopy of electronic phase separation in K$_x$Fe$_{2-y}$Se$_2$ superconductor

Structural phase separation in A$_x$Fe$_{2-y}$Se$_2$ system has been studied by different experimental techniques, however, it should be important to know how the electronic uniformity is influenced, on which length scale the electronic phases coexist, and what is their spatial distribution. Here, we have used novel scanning photoelectron microscopy (SPEM) to study the electronic phase separation in K$_x$Fe$_{2-y}$Se$_2$, providing a direct measurement of the topological spatial distribution of the different electronic phases. The SPEM results reveal a peculiar interconnected conducting filamentary phase that is embedded in the insulating texture. The filamentary structure with a particular topological geometry could be important for the high T$_c$ superconductivity in the presence of a phase with a large magnetic moment in A$_x$Fe$_{2-y}$Se$_2$ materials.