Many-body theory of electron-phonon coupling in polar semiconductors and superconductors

In polar semiconductors and insulators, thermal vibrations of the ionic lattice give rise to long-ranged electric fields that can scatter electrons and holes very efficiently. This interaction is known as Fröhlich coupling, and it is the source for a range of intriguing many-body effects in these materials. We consider two scenarios: Firstly, the normal state, in which Fröhlich coupling can give rise to a coupled electron-phonon state, or "polaron", which causes a strong renormalisation of the electronic energy with major implications for excitation energies and charge-carrier mobilities in a material. A characteristic feature of polaron spectra, e.g. as observed in photoemission experiments, are low-energy sidebands near the main photoemission peak whose energetics are of the order of the most strongly coupled phonon modes. These sidebands are often referred to as "polaron satellites". Secondly, the superconducting state, in which Fröhlich coupling provides the attractive pairing potential required for the formation of Cooper pairs. A paradigmatic example for the large class of polar semiconductors is SrTiO₃, which hosts a strongly renormalised polaronic state in the conduction band, and whose phase diagram features a superconducting dome as a function of electron doping. In this thesis, we employ an analytical model of Fröhlich-type coupling between a parabolic electron band with finite electron occupation and a dispersionless LO phonon to study the coupled electron-phonon state in doped SrTiO₃. Our analytical approach allows us to model all the relevant interactions with arbitrary resolution and provides insights into the intriguing many-body effects in Fröhlich-type semiconductors. In the normal state, we calculate the interacting electron spectral function from a Green's function theory of electron phonon coupling for a range of doping levels and coupling strengths. Electron-phonon coupling is taken into account at the level of the Fan-Migdal self-energy. We compare the spectral functions obtained from two approximations to the interacting Green's function for a wide range of Fermi energies and coupling strengths, derive analytical expressions for the quasiparticle renormalisation as a function of electron doping, and discuss limiting cases of the second-order self-energy approximation. In the superconducting state, we employ Migdal-Eliashberg theory to calculate the critical temperature TC of doped SrTiO₃ as a function of doping level. In particular, our analytical approach allows us to explicitly model the attractive Fröhlich interaction and repulsive Coulomb interaction at generalised frequencies and momenta. Very good qualitative agreement of our results with experimental data is obtained. We emphasise the major contribution of free-carrier screening to the effective pairing potential, and we investigate promising avenues for improving the predictive power of ab initio calculations of the superconducting TC.