# Magnetic properties of superconducting films

## Abstract

The general Gor'kov equations are solved for a superconducting film in a parallel magnetic field. The method determines the best pairing in the superconducting state without the need for ad hoc assumptions about pairing such as are used in other theories. The critical field Hc(T) and energy gap (H, T) are determined for temperatures near the transition temperature at zero field Tc. The energy gap in the quasiparticle excitation spectrum is shown to be approximately equal to the spatial average of the order parameter. For films whose thickness d is less than the coherence length T the Gor'kov equations are nonlocal and differ from the Ginzburg-Landau (GL) equations. In this range we find Hcd-32 in agreement with experiment. For films with d>T the solution of the Gor'kov equations are the same as the GL results, as expected, since this is a local regime. We find that for all d (excepting ultrathin films) and in the temperature range (1-TTc)1 the field dependence of the energy gap is the same as that given by the GL equations, i.e, (H)(0)={1-(HHc)2}12. Thus, nonlocal effects do not change the field dependence of the gap. Most of the experimental data are in accord with this equation. However, some recent results for aluminum films show deviations which we interpret as probably being due to the important role played by energy-level quantization of single-particle states in ultrathin films. The extension of the method to lower temperatures and higher fields is also discussed. © 1964 The American Physical Society.