Recent observation of unusual vortex patterns in ${\mathrm{MgB}}_2$ single crystals raised speculations about possible “type-1.5” superconductivity in two-band materials, mixing the properties of both type-I and type-II superconductors. However, the strict application of the standard two-band Ginzburg-Landau (GL) theory results in simply proportional order parameters of the two bands—and does not support the “type-1.5” behavior. Here we derive the extended GL formalism (accounting all terms of the next order over the small $\tau =1-T/T_c$ parameter) for a two-band clean $s$-wave superconductor and show that the two condensates generally have different spatial scales, with the difference disappearing only in the limit $T\to T_c$. The extended version of the two-band GL formalism improves the validity of GL theory below $T_c$ and suggests revisiting the earlier calculations based on the standard model.

• Received 18 October 2010

DOI:https://doi.org/10.1103/PhysRevLett.106.047005