Coulomb interaction and phonons in doped semiconducting and metallic two-dimensional materials

Two-dimensional (2D) materials present a rapidly developing field of research with sometimes highly unusual and uniquely two-dimensional physics. Starting with graphene, many recent studies have investigated 2D materials, with results for properties encompassing such different topics as Dirac electrons, charge ordering, and superconductivity. To get closer to a predictive theory of the phases of 2D materials, this thesis systematically tackles the previously unclear problem of the Coulomb interaction and its influence on the electronic and many-body properties, with the focus on transition metal dichalcogenides (TMDCs). While this influence can be quite strong due to the low dimensionality and the corresponding reduced screening, there is so far no comprehensive understanding of the Coulomb interaction in 2D, its effects, and the possibilities for engineering it. Furthermore, the interplay between electron-electron interaction, electron-phonon interaction, and screening is far from being fully understood. The goal of this thesis is to improve on this by providing a description of the Coulomb interaction for the example of the TMDCs as well as to develop a material-systematic database on the basis of ab-initio calculations for electrons and phonons. We use Density Functional Theory to describe the electronic structure, Density Functional Perturbation Theory for the phonons and the electron-phonon interaction, and the Random Phase Approximation to obtain Coulomb matrix elements. In addition to both semiconducting and metallic TMDCs, we look at functionalized graphene C8H2. The first step is a quantification of the Coulomb interaction and the screening in the TMDCs along with calculations for the plasmonic spectra, which turn out to be highly susceptible to environment and doping. Secondly, we discuss the influence of the interaction on different many-body instabilities and find a small suppression of superconducting order in semiconducting TMDCs, depending again on doping and the dielectric environment, while the magnetic order in metallic TMDCs is enhanced. If we further include the phonons, we see that superconductivity is predicted to be a global phenomenon in the doped semiconducting TMDCs and C8H2, and that charge density waves at different wave vectors are supposedly occurring in all TMDCs.