Healing Length of the Superconducting Order Parameter

Using the Bardeen-Kümmel-Jacobs-Tewordt approach to the BCS theory of a nonuniform superconductor, we study the problem of a semi-infinite superconductor with a rigid potential barrier at the interface. Very close to $T_c$, the spatial variation of the order parameter is given by the Ginzburg-Landau formula $\frac{\Delta \mathrm{}(z,T)\mathrm{}}{{\Delta }_{\infty }\mathrm{}\left(T\right)\mathrm{}}=tanh\left[\frac{z}{\sqrt{2}}{\xi }_{\mathrm{GL}}\mathrm{}\right(T\left)\right]$. At decreasing temperatures, however, the order parameter heals much more rapidly than $\xi \mathrm{}\left(T\right)=\frac{v_F}{\pi {\Delta }_{\infty }\mathrm{}\left(T\right)\mathrm{}}$, where ${\xi }_{\mathrm{GL}}\mathrm{}\left(T\right)=\lim 0.74 \xi \mathrm{}\left(T\right)\mathrm{}$ as $T\to T_c$; and, at low and intermediate temperatures, does so over atomic distances.