Open Access

Daniel Guterding

• The term superconductivity describes the phenomenon of vanishing electrical resistivity in a certain material, then called a superconductor, below a critical typically very low temperature. Since the discovery of superconductivity in mercury in 1911 many other superconductors have been found and the critical temperature below which superconductivity occurs could recently be raised to the temperatures encountered in a cold antarctic winter. Superconductors are promising materials for applications. They can serve as nearly loss-free cables for energy transmission, in coils for the generation of high magnetic fields or in various electronic devices, such as detectors for magnetic fields. Despite their obvious advantages, the cost for using superconductors, however, depends a lot on the cooling effort needed to realize the superconducting state. Therefore, the search for a superconductor with critical temperature above room-temperature, which would avoid the need for any specialized cooling system, is one of the main projects of contemporary research in condensed matter physics. While a theory of superconductivity in simple metals has already been developed in the 1950s, it has meanwhile been recognized that many superconductors are unconventional in the sense that their behavior does not follow the aforementioned theory. Unconventional superconductors differ from conventional superconductors mainly by the momentum- and real-space symmetry of the order parameter, which is associated with the superconducting state. While conventional superconductors have a uniform order parameter, unconventional superconductors can have an order parameter that bears structure. Of course, alternative theoretical descriptions have been suggested, but the discussion on the right theory for unconventional superconductivity has not yet been settled. Ultimately, this lack of a general theory of superconductivity prevents a targeted search for the room-temperature superconductor. Any new theoretical approach must, however, prove its value by correctly predicting the structure of the superconducting order parameter and further material properties. In this work we participate in the search for a theory of unconventional superconductivity. We discuss the theory of superconductivity mediated by electron-electron interactions, which has been popular in the last few decades due to its success in explaining various properties of the copper-based superconductors that emerged in the 1980s. We give a detailed derivation of the so-called random phase approximation for the Hubbard model in terms of a diagrammatic many-body theory and apply it in conjunction with low-energy kinetic Hamiltonians, which we construct from first principles calculations in the framework of density functional theory. Density functional theory is an established technique for calculating the electronic and magnetic properties of materials solely based on their crystal structure. Its practical implementations in computer codes, however, do for example not describe complicated many-electron phenomena like the superconducting state that we are interested in here. Nevertheless, it can provide important information about the properties of the normal state of the material, which superconductivity emerges from. In our theory we use these information and approach the superconducting state from the normal state. Such an interfacing of different calculational techniques requires a lot of implementation work in the form of computer code. Inclusion of the computer code into this work would consume by far too much space, but since some of the decisions on approximations in the calculational formalism are guided by the feasibility of the associated computer calculations, we discuss the numerical implementation in great detail. We apply the developed methods to quasi-two-dimensional organic charge transfer salts and iron-based superconductors. Finally, we discuss implications of our findings for the interpretation of various experiments.