Semi-Meissner state and non-pairwise intervortex interactions in type-1.5 superconductors

Supplement material



Johan Carlström, Julien Garaud and Egor Babaev
Description of the simulations :

Bound state configurations of nv vortices found in high-precision large-scale numerical minimzation of the two-component Ginzburg-Landau free energy in the type-1.5 regime.
The variational problem is defined using a finite element formulation provided by Freefem++ framework. A nonlinear conjugate gradient algorithm is used to find the minima of the energy. More details about the numerical methods employed to study can be found in a Supplementary Material Letter.

In the following, we present evolution from initial configurations being either a giant vortex, or a diluted collection of vortices.

The displayed  quantities:
From left to right, on the first line are plotted the magnetic field, and condensate densities psi1 and psi2. On the second line,  the supercurrents moduli J1 and J2 and im12 which is
nonzero when there appears the phase difference between both components.

1. Superconductors with two active bands: we first consider the case where both components are below their critical temperature (i.e. they are both active).  The first band is the stronger while the second one is weaker.


Movie 1 : corresponds to Figure 2 of the paper.
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Movie 1 : The ground state of a nv9 flux quanta configuration in type-1.5 u1xu1 superconductor (i.e. eta0). The parameters of the potential are a1b1 and a2b2_2A, while the electric charge is e148

Movie 2 : corresponds to Figure 3 of the paper.
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Movie 2 : The ground state of a nv12 flux quanta configuration using the same parameter set as in Movie 1. Here the global 8-folded discrete symmetry of the cluster has been broken toward a less symmetric configuration (a 3-folded discrete symmetry) in favor of a regular vortex lattice in the cluster. There is again a competition between type-I-like (normal circular cluster with a boundary current) and type-II-like tendencies (vortex lattice).

Movie 3 : corresponds to Figure 4 of the paper.
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Movie 3 : Magnetic ground state of an nv9 vortex configuration with a stronger electric charge coupling e155 (parameters of the potential are the same as in Movie 1). This shows the behavior of the system with respect to a charge increase. Increasing the electric charge decreases penetration length and thus pushes the system towards type-I regime i.e. with more circular boundary.

Movie 4 : corresponds to Figure 3 of the paper.
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Movie 4 : Magnetic ground state of an n9 vortex configuration with the parameter set given by Movie 1, but with e159 and added  Josephson coupling  eta01. Although the Josephson term introduced an energy penalty for phase difference, it has little effect on the vortex cluster boundary where the high magnetic pressure still generates strong phase difference gradients.

Movie 5 :
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Movie 5 : Magnetic ground state of an n12 vortex configuration. parameters of the potential are a1b1 and a2b2_2Aa, while the electric charge is e130 and added  Josephson coupling  eta05. Although the Josephson term introduced an energy penalty for phase difference, it has little effect on the vortex cluster boundary where the high magnetic pressure still generates strong phase difference gradients.

Movie 6 :
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Movie 6 : Magnetic ground state of an n18 vortex configuration. parameters of the potential are a1b1 and a2b2_2Aa, while the electric charge is e130 and added  Josephson coupling  eta05. Although the Josephson term introduced an energy penalty for phase difference, it has little effect on the vortex cluster boundary where the high magnetic pressure still generates strong phase difference gradients.


2. Superconductor with one active and one passive bands and strong interband Josephson coupling: here we consider the case where one band is below its critical temperature (i.e. active) while the second band has superfluid density only because interband proximity effect (i.e. passive).

Movie 7 : corresponds to Figure 6 of the paper.
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Movie 7 : A  bound state of an nv25 vortex configuration in case when  superconductivity in the second band is due to interband proximity effect and the  Josephson coupling is strong eta7. Other parameters are a1b1, a2b2_1A1P,  e13.
Simulations show the increased importance of non-pairwise interactions which lead to formation of  vortex chains rather than a compact cluster.

Movie 8 : corresponds to Figure 7 of the paper.
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Movie 5 : A  bound state of an nv25 vortex configuration in case when  superconductivity in the second band is due to interband proximity effect and the  Josephson coupling is strong eta7. Other parameters are a1b1, a2b2_1A1P,  e13.
We start with a diluted initial configuration where vortices are located with the attractive range of their two-body potential. However instead of contracting, initially the vortex cluster expands, thus showing that the evolution is dominated by nonpairwise inyteraction.
Once the group breaks into several subgroups of vortices and thus gets rid of some of the multibody forces, the subclusters start to shrink. The intervortex distance within each subcluster in the final configuration is smaller
   than that in the initial configuration despite the expansion in the initial stage of the evolution.