Thephenomenon of superconductivity was first observed about 90 years ago by the group of Heike Kamerlingh-Onnes at the University of Leyden. They discovered that various metals, such as mercury or lead, become perfect conductors when cooled down below a critical temperature which depends on the specific properties of the material. Furthermore, these compounds were found to expel magnetic fields in this special phase. Onnes was awarded the Nobel Prize in physics in 1913. However, a theoretical explanation of this intriguing effect had to wait for about half a century. It was ultimately given by John Bardeen, Leon Cooper, and Robert Schrieffer who explained that an effective attraction between electrons is responsible for the unusual low temperature superconducting state of these materials. For this achievement, they jointly received the Nobel Prize in 1972. While their theory could have been the culminating point for the field of superconductivity, the discovery of the entirely new class of high-temperature superconductors stirred up the physics community in 1986. These materials are complex ceramic compounds, containing layers of copper and oxygen as well as less common elements, such as ytrium, barium, and lanthanum. So far, superconductivity had only been observed at ultra-low temperatures in the range of liquefied helium, a few degrees Kelvin above absolute zero temperature. In contrast, the new ceramic high-temperature superconductors, discovered by Georg Bednorz and Alexander Müller (Nobel prize 1987), have transition temperatures in the range of liquid air. Although this may still seem very cold, it is a relatively high temperature compared to liquid helium, putting it closer towards the range of technological applications, such as magnetically levitated trains and electrical transmission cables.

During the last ten years more novel types of unconventional superconducting compounds with high transition temperatures and exotic properties have been found, challenging our theoretical understanding of the origin, nature, and restrictions of superconductivity in these materials. Experiments strongly suggest that there are fundamental differences between the physical properties of the conventional low-temperature materials, discovered almost a century ago, and these new classes of superconductors. Apart from the fact that the superconducting transition temperatures dramatically differ by orders of magnitude between these categories, probably the most significant distinction between conventional and unconventional superconductors is found in the symmetry of their order parameters. Conventional superconductors are well described by a so-called s-wave order parameter which implies isotropic attractive forces between electrons in all spatial directions. The situation in unconventional superconductors appears to be much more complex. For example, the high-temperature superconductors have a d_x² _{- y}² -wave order parameter, implying a strong directional dependence of their electron-electron interactions. Other newly discovered classes of superconductors with unconventional order parameters include heavy-fermion materials, organic compounds, and most recently MgB2. Recently, this very common material has been found to become superconducting at 39 Kelvin. The plethora of new materials with exotic properties makes one wonder how many more interesting superconductors are waiting to be discovered.

Our condensed matter theory group at USC has contributed to this field on many different levels, including the construction and analysis of theories for mechanisms leading to unconventional superconductivity, the numerical simulation and testing of microscopic models, and the development of a comprehensive phenomenological description of anisotropic superconductors. Let us highlight our activities in this field by discussing a prominent example of the past year.

Imaging impurity bound states of unconventional superconductors. In a pure superconductor, pairs of electrons propagate freely without scattering effects. However, impurities introduced by the replacement of atoms in the crystal lattice can break these pairs and localize electrons around the impurity sites. The local density distribution of these electrons in the vicinity of the impurity atoms can be measured by a scanning tunneling microscope. For anisotropic superconductors these experiments reveal exotic patterns, such as those shown in seen in the figures to the left, which serve as fingerprints for the underlying unconventional order parameter. For a conventional isotropic superconductor, one would expect concentric rings centered around the impurity site. In contrast, for the case of an anisotropic high-temperature superconductor with a d_x² _{- y}² -wave order parameter, one obtains the fourfold symmetric lobe patterns seen in these figures. The first experimental image of such a localized bound state around a zinc impurity atom in the high-temperature superconductor Bi₂Sr₂CaCu₂O₈ was obtained by the group of J.C. Davis at UC Berkeley (upper figure). Using a variational solution of the corresponding Bogoliubov-de Gennes equations for d_x² _{- y}² -wave superconductors (lower figure), we were able to reproduce the generic features which were observed by scanning tunneling microscopy. Furthermore we predicted the bound state patterns around other impurity atoms such as nickel. These predictions were verified experimentally.