Magnetic behavior of very small superconducting particles

This paper discusses the critical field of superconducting particles, or films, much smaller in size than the coherence length and the penetration depth; it is restricted to situations where the order parameter may be taken as constant in space, and where the superconducting transition in the presence of the field is of second order. The critical field calculation is then reduced to a study of the magnetic flux $\mathrm{\ensuremath{\Phi}}$ enclosed by all one-electron trajectories in the normal state during a prescribed time interval $\mathrm{t}$. We show that (1) if $\mathrm{\ensuremath{\Phi}}$ does not have a completely ergodic behavior at large times, the equation of state is of the BCS type, but with a renormalized, field dependent coupling constant $\mathrm{N}(0)\mathrm{V}$ $\ensuremath{\eta}(\mathrm{H})$. (2) if $\mathrm{\ensuremath{\Phi}}$ has a certain ergodic property, the effect of the field is comparable to the effect of paramagnetic impurities, as first pointed out in a particular example by Maki. Among other things there is a region of gapless superconductivity in the $(\mathrm{HT})$ plane.A thin film in a parallel field with diffuse boundary scattering but no volume defects belongs to case (1). This surprising result is due to a geometrical cancellation of successive contributions to $\mathrm{\ensuremath{\Phi}}$. However, a rather small amount of scattering in the bulk is enough to restore case (2). Numerical values of the resulting critical field are discussed in detail for various ratios of the bulk mean free path 1 to the film thickness $\mathrm{d}$. In the situations of major physical interest the theoretical values are proportional to ${\mathrm{d}}^{\ensuremath{-}3/2}$ and are in rather good agreement with the experimental data.

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