Generalization of the Eliashberg equations and Density Functional Theory applied to the analysis of the fundamental properties of iron-based superconductors.

Density Functional Theory (DFT) allows a fully ab-initio treatment of almost all the quantities that enter in the Eliashberg theory and in many other approaches used to study both the superconducting and the normal phase. A complete description from first principles of real materials is, al least in principle, possible. Here DFT and Eliashberg theory are applied to the study of some members of the new family of superconductors discovered in 2008, the iron-compounds. Superconductivity here is unconventional and unlikely mediated by phonons. When electronic mechanisms are involved and the properties of the compounds are more complex, as in the case of Fe-based superconductors, also the ab-initio treatment may require some reasonable ad hoc approximations, derived from experimental evidences or theoretical argumentations, introducing some phenomenological aspects in the formulation. In this thesis, Eliashberg theory and DFT are applied to study the properties of some iron compounds, in particular the properties of LiFeAs and Co-doped Ba-122 are discussed both in the normal and in the superconducting state. In order to unveil the the properties of the superconducting state, in particular the symmetry and the amplitude of the order parameter and the coupling mechanism, a four bands Eliashberg model is discussed for LiFeAs suggesting that the specific electronic structure of this peculiar compound may lead to the breakdown of the Migdal's theorem forcing the model to be phenomenological". The relation between the topology of the Fermi surface and the presence of nodes is studied in Point-contact Andreev-reflection spectra of Ba(Fe1-xCox)2As2 (both thin fims and single crystals) and Ca(Fe1-xCox)2As2. The curves presented are fitted within the multiband 3D-BTK model that allows the inclusion of the real shape of the Fermi surface evaluated within DFT. Thanks to the inclusion of the results obtained within the Eliashberg theory in the 2D-BTK model some additional structures due to the strong electron-boson interaction, that appear at energy higher than the amplitude of the gaps in some AR spectra, can be studied and some guesses about the nature of the superconducting mechanism can be made. This technique is here applied to the case of Ba(Fe1-xCox)2As2 thin films with x = 0.08. Finally, as concern the normal state, the temperature dependence of resistivity is reproduced both in LiFeAs and Ba(Fe1-xCox)2As2 with a model that contains two different kind of carriers. For both compounds spin fluctuations play an important role also in the normal state in addition to being the main bosons that mediate the Cooper pairing and in both compound the transport coupling constant results to be sensibly smaller than the superconducting one, suggesting a way to find a unifying principle for HTCS.