Scaling Laws for Thin Films near the Superconducting-to-Insulating Transition

We propose a Lagrangian function, which combines Landau-Ginzburg term and Chern-Simons term, for describing the competition between disorder and superconductivity. To describe the normal-to-superconducting transition in the thin superconducting films, we apply Wilson's renormalization group methods into this Lagrangian function. Finally, we obtain a scaling law between critical temperature (T_c), film thickness (d), sheet resistance of the film at the normal state (R_s), and number density of the electrons at the normal state (N). Such a scaling law is in agreement with recent experimental investigations [Ivry, Y. et al, Physical Review B 90, 214515 (2014)]. Our finding may have potential benefits for improving transition temperature T_c.