Considerations on the Magnetic Field Problem in Superconducting Thin Films

We adapt the results obtained in a previous paper on the magnetic field dependence of the energy gap in superconductivity for bulk specimen to thin-film superconductors, using the model of discrete quantization in momentum space. Only the case of parallel and constant external magnetic field along the film surfaces is considered. A series of elementary theorems and some specific calculations lead to the conclusions: (1) A second-order phase transition should be observed at all temperatures for thin film thicknesses. (2) A simple scaling rule exists concerning the field and temperature dependence on the energy gap. (3) The critical field $H_c$ depends on thickness $L$ and reduced temperature $t$ like $H_c\sim L^{-\frac{3}{2}}{\left[\ln \right(\frac{1}{t}\left)\right]}^{\frac{1}{2}}$ for not too thin films ($L\gtrsim 0.5\times {10}^{-5\mathrm{}}$ cm). The behavior changes as the film becomes very thin or as the temperature becomes moderately low. A crude comparison with available experimental data seems to bear out our conclusions qualitatively.