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Superconducting state is achieved through quantum condensation of Cooperpairs which are new types of charge carriers other than single electrons innormal metals. The theory established by Bardeen-Cooper-Schrieffer (BCS) in1957 can successfully explain the phenomenon of superconductivity in manysingle-element and alloy superconductors. Within the BCS scheme, the Cooperpairs are formed by exchanging the virtual vibrations of lattice (phonons)between two electrons with opposite momentum near the Fermi surface. The BCStheory has dominated the field of superconductivity over 64 years. Manysuperconductors discovered in past four decades, such as the heavy Fermionsuperconductors, cuprates, iron pnictide/chalcogenide and nickelates seem,however, to strongly violate the BCS picture. The most important issue is that,perhaps the BCS picture based on electron-phonon coupling are the special casefor superconductivity, there are a lot of other reasons or routes for theCooper pairing and superconductivity. In this short overview paper, we willsummarize part of these progresses and try to guide readers to some newpossible schemes of superconductivity after the BCS paradigm. We also proposeseveral empirical rules for the exploration of high-temperature unconventionalsuperconductors.

Unconventional superconductivity after the BCS paradigm and empirical rules for the exploration of high temperature superconductors