Exact Solutions of the Superconducting Surface Sheath

The exact solutions of the superconducting surface sheath of a semi-infinite superconducting half-space for various values of the Ginzburg-Landau parameter $\kappa$ were obtained as a function of the applied magnetic field between the critical fields $H_c$ and $H_{c3\mathrm{}}$ (for $0.417<\mathrm{}\kappa <0.707\mathrm{}$; type I) and between $H_{c2\mathrm{}}$ and $H_{c3\mathrm{}}$ (for $0.707<\mathrm{}\kappa <\infty$; type II). For type-II superconductors the solutions were also extended into the mixed state well below $H_{c2\mathrm{}}$. The order parameter, the supercurrents, the internal magnetic field, the vector potential, the thickness of the surface sheath, and the magnetization per unit volume for a "thick" film were determined. These quantities vary strongly with applied magnetic field for $\kappa <3\mathrm{}$. In order to obtain these solutions a boundary condition was derived which relates the absolute value of the order parameter at the surface to the phase of the order parameter and the value of the vector potential at the surface. Two supercurrents of equal magnitude, but opposite direction, flow near the surface for magnetic fields above $H_c$ (type I) or $H_{c2\mathrm{}}$ (type II). Below $H_{c2\mathrm{}}$ these currents are slightly different owing to the bulk solution and give rise to a small net surface current.