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

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 $\Delta$ is found to obey the Landau-Ginzburg boundary condition at all $T<\mathrm{}{T}_c$, even though this boundary condition was originally suggested only for the Landau-Ginzburg region (i.e., when $T_c-T\ll T_c$). 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 $T_c-T\ll T_c$. At $T=0\mathrm{}$°K, the integral gap equation is solved by a variational approach, yielding the critical-field ratio $\frac{H_{c3\mathrm{}}}{H_{c2\mathrm{}}}\ge 1.925$. This should be compared with Saint-James and de Gennes's result, ∼1.7, for $T$ in the Landau-Ginzburg region. The small-$T$ correction to the ratio near $T=0\mathrm{}$°K is found to be proportional to $T^2\ln T$ with a small coefficient. An upper bound is also found for the $T=0\mathrm{}$°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.