Superconductivity is a crucial properties of certain materials which characterized by zero-resistivity and Meissner effect. Below the critical temperature T_C, the resistivity will drop to zero abruptly, and inside the superconductor satisfies B=0 regardless of the existence of external magnetic field H. These two properties are essential in observation of superconductivity, and have many important applications such as the superconducting maglev train, superconducting coils, and so on.

In general, superconductors can be divided into conventional superconductors and unconventional superconductors. Conventional superconductors consist of the most metals, compounds and alloys, and it can be interpreted by the BCS theory successfully, which is presented by J. Bardeen, L. Cooper and J. R. Schrieffer in 1957, and they won the Nobel Prize in physics in 1972 for this theory. Based on the assumption of weak coupling, it claims that superconductivity is a novel state of electrons and protected by the superconducting gap \(2\Delta\). In the framework of BCS theory, the superconductivity is a macroscopic quantum coherent effect and arises from the condensation of Cooper pairs. Around the Fermi Surface, the BCS theory argues that with the participation of phonons the electrons with opposite momentum vector (k, -k) will form Cooper pairs. The BCS theory gives a profound understanding of superconductivity, and in great agreement with most experiments.

However, BCS theory cannot explain the unconventional superconductors perfectly. Some novel pairing mechanisms beyond electron-phonon coupling and the strong correlation effect of electrons have been proposed. In 1986, Bednorz and M\(\ddot{u}\)ller found the cooper oxide superconductor, and its critical temperature is about 30 K，after that more and more cooper-based superconductors have been discovered and their critical temperature is higher than McMillan limitation in BCS theory. Nevertheless, in 2006 the Japanese group discovered the first iron-based superconductor, and they have some similar properties as cooper-based superconductors. In general, the high-temperature superconductors have layered structure such as the CuO₂ planes in cooper-based superconductors and FeAs or FeSe planes in iron-based superconductors, which is crucial for the existence of superconductivity. Currently, there is not a unified interpretation of high-temperature superconductors. Because of the extraordinary ability of detecting energy dispersions, ARPES is a powerful tool to study high-temperature superconductors.

[1] I.I. Mazin, Nat. Mater. 464, 183 (2010).

The electronic structures of topological materials have a property which is invariant under some topological continuous deformation. Topological materials consists of various types such as topological insulators, topological semimetals, topological superconductors and so on.

Topological insulators are materials with conducting surface states and insulating bulk states. Backscattering of the surface states by nonmagnetic impurities is prohibited, which may lead to wide applications in spin engineering. The most definite evidence of topological insulators is the existence of spin-polarized Dirac-bands at the surfaces. In contrast, topological semimetals, such as Dirac semimetals, Weyl semimetals, topological nodal-line semimetals, host Dirac bands in the bulk. Topological semimetals have some interesting properties, such as Fermi arc states, monopoles of Berry curvature in momentum space, resulting in unique electric transports.

When the electronic structure of superconducting Bogoliubov quasiparticles is topological, the material is a topological superconductor. There are Majorana states at the surface/edge of a topological superconductor, which satisfy non-abelian statistics. It is proposed that the proximity effect between a topological insulator and an s-wave superconductor can produce topological superconductivity at the interface. This proposal can be realized in a heterostructure of a topological insulator and an s-wave superconductor. In iron-based superconductors, the topological insulator state and the superconducting state coexist in a single crystal, which is much easier to fabricate and study than a heterostructure. Iron-based superconductors provide a simple and high-Tc platform to study topological superconductors.