Critical-Field RatioHc3Hc2for Pure Superconductors Outside the Landau-Ginzburg Region. I.T≃0°K

Abstract

In this paper and a subsequent one (Paper II) we study the nucleation of superconductivity near a sample surface at temperatures outside the Landau-Ginzburg region. We develop a generalized image method to solve for the normal electron temperature Green's function for a semi-infinite sample with a specularly reflective plane boundary in an external magnetic field. Gor'kov's linearized gap equation is then obtained and studied for such a sample geometry. The pair wave function $\ensuremath{\Delta}$ is found to obey the Landau-Ginzburg boundary condition at all $T<{T}_{c}$, even though this boundary condition was originally suggested only for the Landau-Ginzburg region (i.e., when ${T}_{c}\ensuremath{-}T\ensuremath{\ll}{T}_{c}$). However, we also find that merely adding the boundary condition to the differential equation appropriate to the bulk case does not give the correct solution to the problem, except when ${T}_{c}\ensuremath{-}T\ensuremath{\ll}{T}_{c}$. At $T=0$\ifmmode^\circ\else\textdegree\fi{}K, the integral gap equation is solved by a variational approach, yielding the critical-field ratio $\frac{{H}_{c3}}{{H}_{c2}}\ensuremath{\ge}1.925$. This should be compared with Saint-James and de Gennes's result, \ensuremath{\sim}1.7, for $T$ in the Landau-Ginzburg region. The small-$T$ correction to the ratio near $T=0$\ifmmode^\circ\else\textdegree\fi{}K is found to be proportional to ${T}^{2}\mathrm{ln}T$ with a small coefficient. An upper bound is also found for the $T=0$\ifmmode^\circ\else\textdegree\fi{}K ratio to be 5.22, which is useful mainly in proving the existence of a ground state, so as to help justify the use of a variational approach.