Ginzburg-Landau theory of the upper critical field in filamentary superconductors

The upper critical field for coupled filamentary superconductors is analyzed within the context of a Ginzburg-Landau theory similar to the Lawrence-Doniach theory for coupled layered superconductors. Upward curvature in the critical field as the temperature is lowered results from the decreased coupling of the filaments, and an ultimate divergence in the critical field at all angles occurs below a decoupling temperature $T^{*}$. Unusual anomalies are predicted to occur in the $H_{c2\mathrm{}\parallel }\mathrm{}\left(T\right)\mathrm{}$ curve, corresponding to a commensurate fitting of the vortices into the filament lattice. The behavior of $H_{c2\mathrm{}}$ for coupled filaments is contrasted with that of an isolated fiber of finite diameter. The model is applied to ${\left(\mathrm{SN}\right)}_x$, to the transition-metal trichalcogenides Nb${\mathrm{Se}}_3$ and Ta${\mathrm{Se}}_3$, and to mercury embedded in asbestos.