Vortex states and the phase diagram of a multiple-component Ginzburg-Landau theory with competing repulsive and attractive vortex interactions

We investigate the behavior of vortices of multicomponent superconductivity with intercomponent coupling, realized in ${\mathrm{MgB}}_2$ and Fe-based superconductors, within the framework of Ginzburg-Landau (GL) theory in terms of numerical calculations of the time-dependent GL equations and the variational method. It is revealed that close to the critical point of the composite system the intercomponent coupling makes the system behave as a single component superconductivity. However, when the bare mean-field critical points of the two components coincide with each other, and furthermore the interband coupling disappears at the same temperature, interesting phenomena occur as follows. Vortices interact attractively at large separation and repulsively at short distance in certain parameter space. Because of the nonmonotonic interaction profile, phase separations between vortex clusters of triangular order and the Meissner state take place, which indicates a first-order phase transition associated with the penetration of the magnetic field into a superconductor sample. Phase diagrams of vortex states are then constructed with the associated magnetization curve. It is found that the system shares some features of both type-I and -II superconductors.

Figure 1

Left: schematic view of two vortices in a superconducting square disk. Right: numerical discretization scheme. ${\Psi }^{\left(k\right)}$ is defined on nodes, $\mathbf{J}$, $\mathbf{A}$, and $\mathbf{U}$ are defined on bonds, and $B_z$ is defined inside the plaquette.

Figure 2

(a) Profiles of order parameters, magnetic field, and supercurrent for a single vortex. (b) A stable two-vortex solution in a finite sample derived numerically using the approach of TDGL equations. The vortex separation is $r_{\mathrm{m}}=2.7{\lambda }_1$.

Figure 3

Vortex structure obtained by the TDGL equations (symbols) and the variational calculations (lines).

Figure 4

Distribution of the order parameters (purple and blue lines) and magnetic field (yellow line) at vortex separation (a) $d=8{\lambda }_1$, (b) $d=6{\lambda }_1$,(c) $d=3{\lambda }_1$, and (d) $d=0{\lambda }_1$. Distribution of the supercurrent at vortex separation (e) $d=8{\lambda }_1$, (f) $d=3{\lambda }_1$, and (g) $d=0$.

Figure 5

Dependence of the total energy $E$ per unit length on the separation $r$ between two vortices. The distance $r_m$ corresponds to the energy minimum of the two-vortex system. Another length denoted by $r_0$ is the lattice constant when the ordered vortex patterns are formed, as discussed later.

Figure 6

Vortex configurations at several typical values of magnetic field: (a) vortex droplet, (b) vortex stripe, (c) vortex void, and (d) triangular vortex lattice. Black dots denote the centers of vortices and open region is the superconducting region. Red lines are Voronoi polygons. The size of the simulation box is (a) $L=100{\lambda }_1$, (b) $L=70{\lambda }_1$, (c) $L=50{\lambda }_1$, and (d) $L=42{\lambda }_1$, with the number of vortices $N_v=400$ fixed.

Figure 7

Mean-field $H-T$ (a) and $B-T$ (b) phase diagrams of superconductors with competing intervortex interaction. The red (thin) line in (a) indicates the first-order phase transition, while that in (b) indicates the upper boundary for the phase separation (the lower boundary is the horizontal axis), and the blue (thick) lines are for the second-order phase transition. Inset is for the dependence of magnetic induction on the applied field, and the dashed curve represents the hysteresis associated with the first-order phase transition.

Figure 8

Distribution of the order parameters and magnetic field at several typical temperatures. The results are obtained with $T_{c1}=0.6$, $T_{c2}=0.8$ and $T_c=0.87$.