Studies of Topological Superconductors via Defects

Defects from impurities, edges, junctions or domain walls have local detrimental effects on the superconducting state. Defects are also important in topological superconductors, and useful in studying the properties of the superconducting order parameter. In the former case, gapless edge states appear as a consequence of the bulk-edge correspondence, which relates the edge states to the change in bulk topological invariants across the edge. In the latter case impurity scattering of electrons modifies the density of states locally and may (or not) lead to the appear-ance of subgap states depending on the nature of the impurity and the superconductor. Due to these telltale effects of defects in topological superconductors, this Thesis uses defects to probe topological superconductors. The studies are divided into two parts. In the first part Majo-rana bound states at the ends of one-dimensional topological superconductors are studied. The Majorana bound states at the ends of magnetic chains are found very stable in a disordered su-perconducting medium, as long as the superconducting order parameter does not vanish locally. Also junction formed in a nanowire in proximity to two superconductors is found to give rise spontaneously to trivial zero-energy states that imitate Majorana bound states, due to finite size effect of the superconductors. However, there is a sign reversal in the supercurrent when trivial zero-energy states are present at the junction, as magnetic field is tuned, whereas the supercur-rent does not exhibit such sign reversal when Majorana bound states are present at the junction. Thus, supercurrent serves as a tool for distinguishing between Majorana bound states and trivial zero-energy states. The second part is devoted to unconventional and topological superconduc-tivity in graphene. Domain wall states formed between topological chiral dx2−y2 ± idxy-wave states on the honeycomb lattice are studied. The results find four domain wall states and that domain wall configurations with the lowest width in the order parameter is favorable over other configurations. Finally, by using a single potential impurity spin-singlet gap symmetries of superconducting graphene are explored. Investigation of the subgap states, due to the potential impurity, of each superconducting state and their corresponding quasi-particle interference shows that superconducting states of graphene can be identified by defects