The topological aspects of superconductivity on doped Weyl semimetals are investigated. / Topological phases of matter have sparked significant experimental and theoretical interest due to the topologically robust edge modes they host, as well as their classification through rich mathematics. An interesting example of a topological phase in three dimensions, the Weyl semimetal, can exhibit topological ordering through the existence of Fermi arcs on the surfaces of the material. For the doped Weyl semimetal, we investigate possible resulting Weyl superconducting states --- both the inter-Fermi surface pairing state following Li and Haldane, and the intra-Fermi surface pairing state following Burkov --- in this thesis, and study their topological properties by computing the gapless Weyl-Majorana edge modes they host. The results obtained in Ref. \cite{LH} for the inter-Fermi surface superconducting state are reproduced, and the bulk and edge properties of the intra-Fermi surface pairing superconducting state are studied in detail. / Thesis / Master of Science (MSc) / In this thesis, we study an interesting class of topological materials called the Weyl semimetal as well as its associated superconducting phases. A description of the Fermi arcs on Weyl semimetals are given, and the topological properties of the inter-Fermi surface and intra-Fermi surface pairing states are studied in detail.

In recent years, Weyl semi-metals have attracted a lot of interest in topological condensed matter, for instance for their significant potential application in quantum electronics, as the coupling between Weyl semi-metals and superconductivity, either intrinsically in the material or at the interface of a heterostructure, gives rise to a new type of topological superconductivity, which could be used to perform quantum computation operations free from decoherence. In this thesis, we investigate the low temperature magneto-transport properties of trigonal-PtBi2, a layered material, both in the presence of quantum confinement (exfoliated nano-structures) and in its absence (macro-structures). We report band structure calculations showing that trigonal-PtBi2 is a type-I Weyl semi-metal with multiple bands at the Fermi level. Shubnikov-de-Haas oscillations in macrostructures confirm the contribution to transport of carriers from multiple pockets, and magneto-transport measurements show an unusual angular dependence of the magnetoresistance with the field, which might be a manifestation of the large anisotropy of the topological band. We also report the existence of a large planar Hall effect in nano-structures, which is one of the predicted manifestations of Weyl physics. At very low temperature, below 1K, trigonal-PtBi2 becomes superconducting. We investigate the superconducting state in both macro- and nano-structures, and find that quantum confinement in nano-structures makes the superconductivity become two-dimensional. This result is confirmed by the characterization of a Berezinskii–Kosterlitz–Thouless (BKT) transition in nano-structures. This transition is very robust, as it occurs in nano-structures five times thicker than what had previously been reported for any BKT transitions in the literature. We also report on the impact of inhomogeneities on the superconducting transition.