  As one of the simplest systems for realizing Majorana fermions, topological
superconductor plays an important role in both condensed matter physics and
quantum computations. Based on \emph{ab~initio} calculations and the analysis
of an effective 8-band model with the superconducting pairing, we demonstrate
that the three dimensional extended $s$-wave Fe-based superconductors such as
Fe$_{1+\text{y}}$Se$_{0.5}$Te$_{0.5}$ have a metallic topologically nontrivial
band structure, and exhibit a normal-topological-normal superconductivity phase
transition on the ($001$) surface by tuning the bulk carrier doping level. In
the topological superconductivity (TSC) phase, a Majorana zero mode is trapped
at the end of a magnetic vortex line. We further show that, the surface TSC
phase only exists up to a certain bulk pairing gap, and there is a
normal-topological phase transition driven by the temperature, which has not
been discussed before. These results pave an effective way to realize the TSC
and Majorana fermions in a large class of superconductors.
