Type-1.5 superconductor

For multicomponent superconductors with so called U(1)xU(1) symmetry the Ginzburg-Landau model is a sum of two single-component Ginzburg-Landau model which are coupled by a vector potential ${\displaystyle A}$ :

${\displaystyle F=\sum _{i,j=1,2}{\frac {1}{2m}}|(\nabla -ieA)\psi _{i}|^{2}+\alpha _{i}|\psi _{i}|^{2}+\beta _{i}|\psi _{i}|^{4}+{\frac {1}{2}}(\nabla \times A)^{2}}$

where ${\displaystyle \psi _{i}=|\psi _{i}|e^{i\phi _{i}},i=1,2}$ are two superconducting condensates. In case if the condensates are coupled only electromagnetically, i.e. by ${\displaystyle A}$ the model has three length scales: the London penetration length ${\displaystyle \lambda ={\frac {1}{e{\sqrt {|\psi _{1}|^{2}+|\psi _{2}|^{2}}}}}}$ and two coherence lengths ${\displaystyle \xi _{1}={\frac {1}{\sqrt {2\alpha _{1}}}},\xi _{2}={\frac {1}{\sqrt {2\alpha _{2}}}}}$. The vortex excitations in that case have cores in both components which are co-centered because of electromagnetic coupling mediated by the field ${\displaystyle A}$. The necessary but not sufficient condition for occurrence of type-1.5 regime is ${\displaystyle \xi _{1}>\lambda >\xi _{2}}$.[2] Additional condition of thermodynamic stability is satisfied for a range of parameters. These vortices have a nonmonotonic interaction: they attract each other at large distances and repel each other at short distances.[2][3][4] It was shown that there is a range of parameters where these vortices are energetically favorable enough to be excitable by an external field, attractive interaction notwithstanding. This results in the formation of a special superconducting phase in low magnetic fields dubbed "Semi-Meissner" state.[2] The vortices, whose density is controlled by applied magnetic flux density, do not form a regular structure. Instead, they should have a tendency to form vortex "droplets" because of the long-range attractive interaction caused by condensate density suppression in the area around the vortex. Such vortex clusters should coexist with the areas of vortex-less two-component Meissner domains. Inside such vortex cluster the component with larger coherence length is suppressed: so that component has appreciable current only at the boundary of the cluster.

In a two-band superconductor the electrons in different bands are not independently conserved thus the definition of two superconducting components is different. A two-band superconductor is described by the following Ginzburg-Landau model.[5]

${\displaystyle F=\sum _{i,j=1,2}{\frac {1}{2m}}|(\nabla -ieA)\psi _{i}|^{2}+\alpha _{i}|\psi _{i}|^{2}+\beta _{i}|\psi _{i}|^{4}-\eta (\psi _{1}\psi _{2}^{*}+\psi _{1}^{*}\psi _{2})+\gamma [(\nabla -ieA)\psi _{1}\cdot (\nabla +ieA)\psi _{2}^{*}+(\nabla +ieA)\psi _{1}^{*}\cdot (\nabla -ieA)\psi _{2}]+\nu |\psi _{1}|^{2}|\psi _{2}|^{2}+{\frac {1}{2}}(\nabla \times A)^{2}}$

where again ${\displaystyle \psi _{i}=|\psi _{i}|e^{i\phi _{i}},i=1,2}$ are two superconducting condensates. In multiband superconductors quite generically ${\displaystyle \eta \neq 0,\gamma \neq 0}$. When ${\displaystyle \eta \neq 0,\gamma \neq 0,\nu \neq 0}$ three length scales of the problem are again the London penetration length and two coherence lengths. However, in this case the coherence lengths ${\displaystyle {\tilde {\xi }}_{1}(\alpha _{1},\beta _{1},\alpha _{2},\beta _{2},\eta ,\gamma ,\nu ),{\tilde {\xi }}_{2}(\alpha _{1},\beta _{1},\alpha _{2},\beta _{2},\eta ,\gamma ,\nu )}$ are associated with "mixed" combinations of density fields.[3][4][6]