The superconductor-insulator transition is an archetypal example of a quantum phase transition that plays a crucial role in physics and statistical mechanics. Belonging to different universality classes, the classical and quantum percolative transitions usually manifest in different systems and are studied separately. Here, we fabricate thickness-gradient ${\mathrm{Fe}}_y{\mathrm{Te}}_{1-x}{\mathrm{Se}}_x$ (FTS) films and observe the existence of these two transitions. Variation in the film thickness, which controls substrate misfit and leads to a modification of Fe concentration, results in a superconducting quantum percolative transition. By exposing FTS films to air, we observe a shift in electrical resistivity behavior and a superconducting classical percolative transition. FTS films subjected to the two treatments exhibit very similar phase diagrams, except for different critical exponents. Remarkably, we identify a two-step superconductor-insulator transition at low temperature, where the appearance of superconductive zero resistivity coincides with a sign change in the temperature derivative of electrical resistivity in the normal state. Our analysis can provide a universal scheme to categorize the superconductor-insulator transition in various copper-oxide and iron-based superconductors.

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• Quantum phase transitions
• Superconductor-insulator transition
• Chalcogenides
• Iron-based superconductors
• Superconductivity
• Thin films
• Resistivity measurements
• X-ray diffraction
• X-ray reflectivity