Magnetic quantum oscillations in systems of low dimensionality: study of the symmetry of the superconducting state in the compound UPt₃

In a first part, we develop the analytical theory of the de Haas-van Alphen (magnetization oscillations) and of the Shubnikov-de Haas (magnetoresistance oscillations) effects in the quasi-two-dimensional electronic systems, which consists of highly conducting layers with a small inter-plane dispersion for the quasi-particles. We show that the application of a magnetic field perpendicularly to the layers leads to the reduction of the apparent dimensionality of the system. In high magnetic fields, the behavior of the magnetization oscillations is close to that of a two-dimensional electrons gas. We study in details the influence of the chemical potential oscillations on the form of the de Haas-van Alphen oscillations in the presence of one or multiple bands of states at the Fermi level. The magnetization oscillations inside the superconducting mixed state are also investigated. We finally calculate the oscillations of the longitudinal magnetoresistance in the framework of the quantum transport theory in presence of point-like impurities. In high magnetic fields, we find giant oscillations with a thermally activated behavior of the resistance maxima. In a second part, we study the properties of the mixed state in the unconventional superconductor UPt₃ for a magnetic field parallel to the hexagonal crystal axis mainly in the framework of the theoretical models of a two-component superconducting state coupled to a symmetry-breaking field. We investigate the form of the vortex lattice in the A phase of UPt₃ from symmetry considerations and from the London local approximation, and show that only the superconducting state with the symmetry E2u is consistent with experiments. We also point out that the transition between the mixed phases A and B is rather a crossover than a second order phase transition due to the presence of mixing gradient terms in the expression of the Ginzburg-Landau free energy. Finally, we demonstrate that the dependence of the slope of the upper critical field at the critical temperature on the impurity concentration is not sensitive to the pairing symmetry. (author)