Nucleation of Superconductivity at a Tunneling Barrier of High Transmissivity

We present a calculation of the parallel critical field of a plane tunneling barrier in an otherwise homogeneous superconductor, valid for $T_c-T\ll T_c$. We have found a solution of the Ginzburg-Landau equation which satisfies the semiphenomenological boundary conditions proposed by de Gennes. The critical field is significantly lower than $H_{c3\mathrm{}}$ when the tunneling current is of the same order of magnitude as the sheath current. When the ratio (tunneling current)/(sheath current) is small, it is proportional to $\xi \mathrm{}\left(T\right)\mathrm{}$. Since $\xi \mathrm{}\left(T\right)\mathrm{}\to \infty$ at $T_c$, in principle the critical field of a tunneling barrier of any transmissivity must go to $H_{c2\mathrm{}}$ as $T\to T_c$, although in practice this may be impossible to observe for small transmissivities because of the finite transition width at $T_c$. For the pure-metal, specular-barrier model at a fixed temperature $T$ not too close to $T_c$, most of the drop in critical field from $H_{c3\mathrm{}}$ to $H_{c2\mathrm{}}$ occurs for transmissivities in the range 0.01 to 0.1.