  FeTe, a non-superconducting parent compound in the iron-chalcogenide family,
becomes superconducting after annealing in oxygen. Under the presence of
magnetism, spin-orbit coupling, inhomogeneity and lattice distortion, the
nature of its superconductivity is not well understood. Here, we combined
mutual inductance technique with magneto transport to study the magnetization
and superconductivity of FeTe thin films. We found that the films with the
highest Tc showed non-saturating superfluid density and a strong magnetic
hysteresis distinct from that in a homogeneous superconductor. Such hysteresis
can be well explained by a two-level critical state model and suggested the
importance of granularity to superconductivity in this compound.
