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 $\mathrm{\Phi }$ enclosed by all one-electron trajectories in the normal state during a prescribed time interval $\mathrm{t}$. We show that (1) if $\mathrm{\Phi }$ 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 $\mathrm{N}\left(0\right)\mathrm{V}$ $\eta \left(\mathrm{H}\right)$. (2) if $\mathrm{\Phi }$ 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 $\left(\mathrm{HT}\right)$ plane.