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 $H_{c3\mathrm{}}$ at $T\simeq 0°$K. Here we study the region $T\simeq T_c$ and find the first three nontrivial terms in an expansion of the ratio $\frac{H_{c3\mathrm{}}\mathrm{}\left(T\right)\mathrm{}}{H_{c2\mathrm{}}\mathrm{}\left(T\right)\mathrm{}}$ to be $1.695[1+0.614(1-t)-0.577\mathrm{}{\mathrm{}(1-t)\mathrm{}}^{\frac{3}{2}}]$, where $t\equiv \frac{T}{T_c}$. The term linear in $1-t$ 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 $H_{c3\mathrm{}}$ 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.