Critical-Field RatioHc3Hc2for Pure Superconductors outside the Ginzburg-Landau Region. II.T≃Tc

Abstract

In Paper I we found the appropriate generalization of Gor'kov's linearized gap equation for a pure, semiinfinite weak-coupling superconductor in a magnetic field, separated from vacuum or an insulator by a specularly reflecting surface. In that paper we used the gap equation to study the surface-nucleation critical field ${H}_{c3}$ at $T\ensuremath{\simeq}0\ifmmode^\circ\else\textdegree\fi{}$K. Here we study the region $T\ensuremath{\simeq}{T}_{c}$ and find the first three nontrivial terms in an expansion of the ratio $\frac{{H}_{c3}(T)}{{H}_{c2}(T)}$ to be $1.695[1+0.614(1\ensuremath{-}t)\ensuremath{-}0.577{(1\ensuremath{-}t)}^{\frac{3}{2}}]$, where $t\ensuremath{\equiv}\frac{T}{{T}_{c}}$. The term linear in $1\ensuremath{-}t$ has been found previously, but the last term is new. For $T$ close enough to ${T}_{c}$ we show that the system is accurately described by the linearized Ginzburg-Landau equation with the usual boundary condition and thus regain the results of Saint-James and de Gennes. At lower temperatures the pair wave function has a slowly varying component which satisfies a finite-order differential equation and a surface component which does not. An analysis of the surface component gives an effective boundary condition on the slowly varying part; from this condition the field ${H}_{c3}$ is derived. In combination with the results of Paper I we propose an interpolation formula for the entire temperature range below ${T}_{c}$. A comparison with the available experimental data is encouraging.