The Mixed State in Superconducting Thin Films

Critical magnetic fields of superconducting vacuum-evaporated tin films were measured for all angles $\theta$ between the magnetic field and the film surface at temperatures from $0.4T_c$ to $T_c$. The experimental results were found to be consistent with Ginzburg-Landau theory provided that the original theory was modified to account for the extended temperature range and the nonlocal nature of superconducting electrodynamics. Measurements of the temperature dependence of the parallel and perpendicular critical fields supported the extensions of Ginzburg-Landau theory developed, most notably, by Maki and de Gennes. The angular dependence of the critical field for $0<\mathrm{}\theta <\frac{1}{2}\pi$ at different temperatures was found to be in agreement with Tinkham's original formula for several sufficiently thin films, $d\ll {\xi }_T$; a detailed derivation of that formula is given here, with discussion of its range of validity. For thicker films the corrections found by Yamafuji et al. were necessary to obtain agreement with the data. The critical fields were carefully measured for $\theta \approx 0$, and evidence of surface superconductivity was found in the thickest film studied, but not in thinner films. This is consistent with the numerical calculations of St. James and de Gennes.