Ginzburg–Landau theory: the case of two-band superconductors

Recent studies of two-band superconductors using the Ginzburg–Landau (GL) theory are reviewed. The upper and lower critical fields [$H_{c2}(T)$ and $H_{c1}(T)$, respectively], thermodynamic magnetic field $H_{cm}(T)$, critical current density $j_c(T)$, magnetization $M(T)$ near the upper critical field, and the upper critical field $H^{\mathrm{film}}_{c2}(T)$ of thin films are examined from the viewpoint of their temperature dependence at a point $T_c$ using the two-band GL theory. The results are shown to be in good agreement with the experimental data for the bulky samples of superconducting magnesium diboride, $\mathrm{MgB}_2$, and nonmagnetic borocarbides $\mathrm{LuNi}_2\mathrm{B}_2\mathrm{C}$ and $\mathrm{YNi}_2\mathrm{B}_2\mathrm{C}$. The specific heat jump turns out to be smaller than that calculated by single-band GL theory. The upper critical field of thin films of two-band superconductors is calculated and the Little–Parks effect is analyzed. It is shown that magnetic flux quantization and the relationship between the surface critical magnetic field $H_{c3}(T)$ and the upper critical field $H_{c2}(T)$ are the same as in the single-band GL theory. Extension of the two-band GL theory to the case of layered anisotropy is presented. The anisotropy parameter of the upper critical field $H_{c2}$ and the London penetration depth $\lambda$, calculated for $\mathrm{MgB}_2$ single crystals, are in good agreement with the experimental data and show opposite temperature behavior to that in single-band GL theory.