  Using a field-effect transistor (FET) configuration with solid Li-ion
conductor (SIC) as gate dielectric, we have successfully tuned carrier density
in FeSe$_{0.5}$Te$_{0.5}$ thin flakes, and the electronic phase diagram has
been mapped out. It is found that electron doping controlled by SIC-FET leads
to a suppression of the superconducting phase, and eventually gives rise to an
insulating state in FeSe$_{0.5}$Te$_{0.5}$. During the gating process, the
(001) peak in XRD patterns stays at the same position and no new diffraction
peak emerges, indicating no evident Li$^+$ ions intercalation into the
FeSe$_{0.5}$Te$_{0.5}$. It indicates that a systematic change of electronic
properties in FeSe$_{0.5}$Te$_{0.5}$ arises from the electrostatic doping
induced by the accumulation of Li$^+$ ions at the interface between
FeSe$_{0.5}$Te$_{0.5}$ and solid ion conductor in the devices. It is striking
that these findings are drastically different from the observation in FeSe thin
flakes using the same SIC-FET, in which $T_c$ is enhanced from 8 K to larger
than 40 K, then the system goes into an insulating phase accompanied by
structural transitions.
