Microscopic derivation of superconductor-insulator boundary conditions for Ginzburg-Landau theory revisited: Enhanced superconductivity at boundaries with and without magnetic field

Abstract

Using the standard Bardeen-Cooper-Schrieffer (BCS) theory, we revise microscopic derivation of the superconductor-insulator boundary conditions for the Ginzburg-Landau (GL) model. We obtain a negative contribution to free energy in the form of surface integral. Boundary conditions for the conventional superconductor have the form $\mathbf{n}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\nabla}\ensuremath{\psi}=\text{const}\ensuremath{\psi}$. These are shown to follow from considering the order parameter reflected in the boundary. The boundary conditions are also derived for more general GL models with higher-order derivatives and pair-density-wave states. It shows that the boundary states with higher critical temperature and the boundary gap enhancement, found recently in BCS theory, are also present in microscopically derived GL theory. In the case of an applied external field, we show that the third critical magnetic-field value ${H}_{c3}$ is higher than what follows from the de Gennes boundary conditions and is also significant in type-I regime.