Boundary Effect and Critical Temperature of Two-Band Superconducting Films: Application to MgB\(_2\)

Based on the two-band Bogoliubov–de Gennes theory, we study the boundary effect of an interface between a two-gap superconductor and an insulator (or vacuum). New boundary terms are introduced into the two-band Ginzburg–Landau free energy, which modifies the boundary conditions for the corresponding order parameters of the superconductor. A microscopic analysis of these terms is also given and the characteristic length scale of the boundary effect can be estimated. The theory allows for a simple calculation of the critical temperature suppression with the decrease in film thickness for the typical two-band superconductor magnesium diboride. Our numerical results are in good agreement with the experimental data observed in this material.