Sign change of surface energy and stability of composite vortex in two-component superconductivity

Existence of thermodynamically stable composite vortices in a two-component superconductor may form distinctive vortex patterns, and may lead to type-1.5 superconductivity. Here we study the surface energy of the two-component superconductor and show that the sign of surface energy is determined not only by the Ginzburg-Landau parameters ${\kappa }_i(i=1,2)$ of two superconducting components, but also by a temperature independent parameter ${\kappa }_{\xi }$, which is defined as the ratio of the coherence lengths of two components. Since the negative surface energy conduces to the invasion of thermodynamically stable composite vortices into a superconductor, the criterions for stability of composite vortex are these three independent dimensionless parameters. We find that there can exist thermodynamically stable composite vortex in a type-1+type-2 or type-2+type-2 material. We also predict that unusual vortex patterns like those observed in $Mg{B}_2$ (V. Moshchalkov et al, Phys. Rev. Lett., 102, 117001 (2009)) can occur in some type-2+type-2 superconductors.