What are the different types of superconductors and how do they work?

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As I understand it there are two types of superconductors, conventional and non-conventional. Conventional superconductors are explained by the BCS theory, my understanding of which being electrons pair up because the bending fields from their movement hold them together and there is not enough other movement to break these forces, but non-conventional ones are not. How do the non-conventional superconductors work then?
 
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Short version, nobody knows. There are several models. None of them explain high temperature type II superconductors.
 


I thought Ginzburg–Landau theory somewhat predicted observed features of type I and type II?
 


I think that the OP is asking about conventional versus e.g. high-temperature oxide superconductors.
 


Feldoh said:
I thought Ginzburg–Landau theory somewhat predicted observed features of type I and type II?
There are several models that explain this and that. There just isn't one that explains everything, even qualitatively, and gives a prediction for Tc that's anywhere close to the Tc of high-temperature superconductors, which really means that we don't understand what's going on with these.