Superconducting Thin Film in a Magnetic Field—Theory of Nonlocal and Nonlinear Effects. I. Specular Reflection

Abstract

A self-consistent perturbation solution of the Gor'kov equations for a superconducting film in a strong parallel magnetic field is presented. The order parameter is assumed constant in space, but proper allowance is made for nonlocal effects, important when the thickness $d$ is comparable to or smaller than the pair-correlation distance $\ensuremath{\xi}$, as is often realized in practice. Specular reflection simplifies the mathematical analysis: The usual momentum-space representation and impurity averaging procedures for Green's functions are shown to apply without essential modifications. Size quantization can be ignored as long as $\frac{{p}_{F}{d}^{2}}{\ensuremath{\xi}}\ensuremath{\gg}1$. The Ginzburg-Landau and Maki local theories, valid in the vicinity of the transition temperature and in the dirty limit, respectively, are generalized. The modified equations for the current density, order parameter, free energy, and critical fields are studied in detail and compared with previous semiphenomenological extensions by Bardeen, Toxen, and others. An expression for the critical field ${H}_{c}$ valid in the intermediate temperature and purity range is also obtained. The convergence of our perturbation expansion and the assumed constancy of the order parameter are shown to require $(\frac{e}{c}){H}_{c}d\ensuremath{\xi}$ and $(\frac{e}{c}){H}_{c}{d}^{2}\ensuremath{\ll}1$, respectively. An inconsistency in Rickayzen's alternative to Maki's theory is pointed out. The most important results, together with their ranges of validity, are indicated in the last section.