Proximity Effect of Superconductors in High Magnetic Fields

The nucleation field $H_0$ for superconductivity at the boundary between a normal and a superconducting semi-infinite half-space is calculated. At this field, for temperatures smaller than the transition temperature $T_{\mathrm{es}}$ of the superconductor, a second-order phase transition occurs from the normal to the superconducting state as the magnetic field is decreased, and superconductivity is nucleated near the boundary between the superconducting and the normal metal. The calculation is general and is applied to clean and dirty superconductors. The normal metal also becomes a superconductor at a transition temperature $T_{\mathrm{cn}}<T_{\mathrm{cs}}$, and the above results apply to temperatures $T_{\mathrm{cn}}\le T\le T_{\mathrm{cs}}$ as well as $T<\mathrm{}{T}_{\mathrm{cn}}$, provided the Ginzburg-Landau equations apply. $H_0$ is temperature-dependent, and lies between the bulk nucleation field $H_{c2\mathrm{}}$ and the surface nucleation field $H_{c3\mathrm{}}$. The value of $H_0$ depends on the Bardeen-Cooper-Schrieffer coherence lengths ${\xi }_0$, the mean free paths $l$ in the normal states, the effective masses $m$, the electron densities $n$, and the transition temperatures $T_c$ of both metals. For example, one finds, in the limit when both metals are dirty ($l\ll {\xi }_0$), that $H_{c2\mathrm{}}\le H_0\le H_{c3\mathrm{}}$ for $0\le \frac{{\sigma }_s}{{\sigma }_n}\le \infty$, where the $\sigma \text{'}\mathrm{s}$ are the normal-state conductivities of the superconducting and the normal metal. This is not in agreement with previous calculations by Hurault, who concludes that when $\frac{{\sigma }_s}{{\sigma }_n}=1\mathrm{}$, the value of $H_0$ is $H_{c2\mathrm{}}$ for $T_{\mathrm{cn}}\le T\le T_{\mathrm{cs}}$. When $T<\mathrm{}{T}_{\mathrm{cn}}$, the value $H_0$ is strongly field- and temperature-dependent, particularly when $\frac{{\sigma }_s}{{\sigma }_n}\lesssim 1$. The above concept is applicable to internal boundary nucleation of superconductivity in the bulk of a superconductor.