Review of annealing effects and superconductivity in Fe$_{1+y}$Te$_{1-x}$Se$_x$ superconductors

Fe$_{1+y}$Te$_{1-x}$Se$_x$ is unique in their structural simplicity, consisting of only FeTe/Se layers, which is favorable for probing the mechanism of superconductivity. Recently, a topological surface superconductivity as well as the Majorana Fermions has been observed, which makes Fe$_{1+y}$Te$_{1-x}$Se$_x$ the first high temperature topological superconductor. Since large size single crystal of Fe$_{1+y}$Te$_{1-x}$Se$_x$ can be easily grown, many researches have been performed. However, a large part of the reported results are under controversy, including the resistivity, susceptibility, Hall effect, gap structure, phase diagram, etc. These controversies are believed to come from the sample-dependent Fe nonstoichiometries, which originate from the partial occupation of the second Fe site (excess Fe) in the Te/Se layer. The excess Fe with valence near Fe$^+$ will provide electron doping into the system. Meanwhile, the excess Fe is also strongly magnetic, which will act as a paring breaker and also localize the charge carriers. Removing the excess Fe is essential to probe the intrinsic properties and mechanism of superconductivity of Fe$_{1+y}$Te$_{1-x}$Se$_x$ compounds. In this topical review, we propose the effective approaches to remove excess Fe in Fe$_{1+y}$Te$_{1-x}$Se$_x$. Furthermore, we discuss the mechanism of annealing based on the evolutions of structure, composition, and morphology with annealing. Moreover, we also review the annealing effects on the normal state and superconducting properties, including the magnetism, transport properties, band structure, $T_{\rm{c}}$, phase diagram, upper critical field, anisotropy, critical current density, gap structure, and superconducting pairing. This review presents not only the optimal way to prepare crystals without excess Fe, but also the intrinsic properties of Fe$_{1+y}$Te$_{1-x}$Se$_x$ without the influence of excess Fe.