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.
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