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.
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