The BCS theory of superconductivity of phonon mediated pairing is one of the most remarkable intellectual achievements of the twentieth century. It was so successful that by the early 70's superconductivity was considered by many a completely understood subject, with the maximum achievable critical temperatures believed to have been reached in the A15 compounds. The field of superconductivity research then took a dramatic turn. Completely new classes of materials such as organics, heavy fermions and transition metal oxides were studied and shown to exhibit unconventional superconductivity at surprisingly high transition temperatures. This flurry of discoveries continues till today, the latest addition to this class of unconventional superconductors provided by the iron pnictides and chalcogenides.
In this set of lectures, I will present an overview of this active field of research. We will start with a modern derivation of the BCS Migdal Eliashberg theory of phonon mediated superconductivity, and the surprising discovery of high temperature superconductivity in MgB2 in the light of this theory. We will also examine the problem of strong electron phonon coupling in other systems such as the high temperature superconductor families based on Ba1-xKxBiO3.
We will then continue with a survey the materials landscape of strongly correlated high temperature superconductors searching for the ideas and organizing principles that can guide the search for new materials. Drawing examples from He3, heavy fermions, organics, ruthenates, copper oxides and iron pnictides based materials we will discuss common aspects such as the proximity of superconductivity to magnetic phases, and its signatures in neutron scattering as well as systems specific properties of these classes of materials. We will comment on some of the prevalent theoreticalideas and methods to treat superconductivity in correlated materials. Dynamical Mean Field Theory (DMFT) methods will be used to formulate the problem of strongly correlated superconductivity, and its connection with strong coupling and weak coupling approaches will be presented. The lectures will describe a panoramic survey of the landscape of interesting classes of superconducting materials which are now subject of intensive experimental and theoretical investigations.