Semi-Meissner state and nonpairwise intervortex interactions in type-1.5 superconductors

We demonstrate the existence of nonpairwise interaction forces between vortices in multicomponent and layered superconducting systems. That is, in contrast to most common models, the interaction in a group of such vortices is not a universal superposition of Coulomb or Yukawa forces. Next, we consider the properties of vortex clusters in a semi-Meissner state of type-1.5 two-component superconductors. We show that under certain conditions nonpairwise forces can contribute to the formation of very complex vortex states in type-1.5 regimes.

Figure 1

Schematic picture of magnetization curves of type-1, type-2, and type-1.5 superconductors. A type-1.5 superconductor has a first-order phase transition in low magnetic fields. In a low magnetic field it can support a phase separation into vortex droplets and Meissner domains (the “semi-Meissner” state).

Figure 2

Schematic picture of a collection of Josephson-coupled layers of type-1 and type-2 superconductors. Vortices induced by a magnetic field are kept in stacks by interlayer Josephson and electromagnetic coupling. If, in the type-1 layers the cores will extend beyond the average magnetic-field penetration length, these extended cores should cause attractive interaction between these vortex lines. The system can be used to model the type-1.5 behavior.

Figure 3

Interaction energy between a single vortex and a fixed vortex pair (the positions of the fixed vortices are shown by two black lines). GL parameters are $({\alpha }_1,{\beta }_1)=(-1.0,1.0)$ and $e=1.3$; $({\alpha }_2,{\beta }_2)=(3.0,0.5)$, $\eta =3$ for (a)–(c); $({\alpha }_2,{\beta }_2)=(3.0,0.5)$, $\eta =7$ for (d)–(f); and $({\alpha }_2,{\beta }_2)=(-0.0625,0.25)$, $\eta =0.5$ for (g)–(i). The top row [(a), (d), and (g)] gives the total interaction energy, while the second one [(b), (e), and (h)] gives the sum of pairwise interactions. The third row [(c), (f), and (i)] displays the difference between these energies, which demonstrates the existence of strong nontrivial, nonpairwise (three-body) interactions. The regimes shown in the left, middle, and right columns are type-2, type-1.5, and type-1, correspondingly.

Figure 4

Ground state of a $N_v=9$ flux quanta configuration in a type-1.5 $U\left(1\right)\times U\left(1\right)$ superconductor (i.e., $\eta =0$). The parameters of the potential are $({\alpha }_1,{\beta }_1)=(-1.00,1.00)$ and $({\alpha }_2,{\beta }_2)=(-0.60,1.00)$, while the electric charge is $e=1.48$. The physical quantities displayed here are (a) the magnetic-flux density and (b) and (c) the density of the first and second condensates $|{\psi }_{1,2}{|}^2$, respectively; (d) and (e) show the norm of the supercurrent in the first and second components, respectively; (f) shows $\mathrm{Im}\left({\psi }_1^{*}{\psi }_2\right),$ which is nonzero when there appears a difference between the phases of ${\psi }_1^{*}$ and ${\psi }_2$. The second component has a type-1-like behavior: its density is depleted in the vortex cluster and its current is mostly concentrated on the boundary of the cluster.

Figure 5

Ground state of a $N_v=12$ flux quanta configuration. The same parameter set as in Fig. 4. Here the global eight-folded discrete symmetry of the cluster has been broken down to the three-folded symmetry. There is a competition between type-1-like (a normal circular cluster with a boundary current) and type-2-like tendencies (a vortex lattice).

Figure 6

Ground state of nine vortices in a $U\left(1\right)\times U\left(1\right)$ superconductor with two active bands $({\alpha }_2,{\beta }_2)=(-0.6,1)$ (without interband coupling). The parameter set here is as in Fig. 4 except $e=1.55$, which places the system closer to the transition with the type-1 regime. This is manifest in the more circular boundary of the cluster compared to Fig. 4.

Figure 7

Ground state of an $N_v=9$ vortex configuration with the parameter set given by Fig. 4, but with $e=1.59$ and added Josephson coupling $\eta =0.1$. Although the Josephson term introduced an energy penalty for phase difference, it has little effect on the vortex cluster boundary. At the boundary, the competing magnetic and density-density interactions win over phase-locking terms and generates phase difference gradients. Also, in this case, the system is close to a type-1 regime: i.e., most of the current is concentrated on the boundary of the cluster.

Figure 8

Ground state of nine vortices in a superconductor with two active bands. Parameters of the interacting potential are $({\alpha }_1,{\beta }_1)=(-1.00,1.00)$, $({\alpha }_2,{\beta }_2)=(-0.0625,0.25)$ while the interband coupling is $\eta =0.5$, which is substantially larger than in Fig. 7. The electric charge, parameterizing the penetration depth of the magnetic field, is $e=1.30$, so that the well in the nonmonotonic interacting potential is very small. In this case there is a visible admixure of the current in the second component in the vortices inside the cluster.

Figure 9

Ground state of $N_v=12$ vortices, for a superconductor with broken $U\left(1\right)\times U\left(1\right)$ symmetry, with the parameters as in Fig. 8. The system again compromises between both type-1-like and type-2-like tendencies in the behaviors of the different superconducting components.

Figure 10

Elongated cluster of vortices in the same superconductor as in Fig. 8, but with $N_v=18$ vortices.

Figure 11

Bound state of $N_v=28$ vortices, for a superconductor with broken $U\left(1\right)\times U\left(1\right)$ symmetry, with the parameters as in Fig. 8. This irregularly shaped cluster represents a local minimum of the free energy. The local minima originate from the competing interactions, yielding a complicated free-energy landscape.

Figure 12

Bound state of an $N_v=25$ vortex configuration in the case when superconductivity in the second band is due to the interband proximity effect and the Josephson coupling is very strong at $\eta =7.0$. The initial configuration in this simulation was a giant vortex. Other parameters are $({\alpha }_1,{\beta }_1)=(-1.00,1.00)$, $({\alpha }_2,{\beta }_2)=(3.00,0.50)$, $e=1.30$.

Figure 13

Bound state of an $N_v=30$ vortex configuration for the system with parameters as in Fig. 12, which was obtained using a dilute initial configuration of vortices.[15]