  Synthesis of monolayer FeSe on SrTiO3, with greatly enhanced
superconductivity compared to bulk FeSe, remains difficult. Lengthy annealing
within a certain temperature window is always required to achieve
superconducting samples as reported by different groups around the world, but
the mechanism of annealing in inducing superconductivity has not been
elucidated. We grow FeSe films on SrTiO3 by molecular beam epitaxy and adjust
the stoichiometry by depositing additional small amounts of Fe atoms. The
monolayer films become superconducting after the Fe deposition without any
annealing, and show similar superconducting transition temperatures as those of
the annealed films in transport measurements. We also demonstrate on the
5-unit-cell films that the FeSe multilayer can be reversibly tuned between the
non-superconducting $\sqrt{5} \times \sqrt{5}$ phase with Fe-vacancies and
superconducting $1 \times 1$ phase. Our results reveal that the traditional
anneal process in essence removes Fe vacancies and the additional Fe deposition
serves as a more efficient way to achieve superconductivity. This work
highlights the significance of stoichiometry in the superconductivity of FeSe
thin films and provides an easy path for superconducting samples.
