| In this thesis, three works on topological insulators (TIs) and topological superconductors (TSCs) will be reported. First, we study the quasiparticle interference (QPI) patterns caused by scattering off nonmagnetic, magnetic point impurities, and edge impurities, separately, in a two dimensional helical liquid, which describes the surface states of a topological insulator. The unique features associated with hexagonal warping effects are identified in the QPI patterns of charge density with nonmagnetic impurities and spin density with magnetic impurities. The symmetry properties of the QPI patterns can be used to determine the symmetry of microscopic models. The Friedel oscillation is calculated for edge impurities and the decay of the oscillation is not universal, strongly depending on Fermi energy. Some discrepancies between our theoretical results and current experimental observations are discussed. Second, we theoretically propose a manipulation scheme for the one-dimensional Majorana fermions emerged in a topological insulator-ferromagnet-superconductor (TI-FM-SC) heterostructure by engineering its magnetic configurations. We demonstrate that the existence of the Majorana edge modes critically depends on the combination of the directions and magnitudes of spin polarization on all surfaces of the TI, and that a model describing the states on only one surface is insufficient. In particular, as shown by analytic calculations, the number, the positions and the chirality of these edge modes corresponding to various topological SC phases can be engineered in this way as well. Third, a new propose for the realization of Majorana zero-modes in 1D silicene domain wall is reported. By applying inhomogeneous perpendicular electric fields to the silicene sheet, 1D domain wall -- which supports midgap states -- can be generated. Plus the appreciable spin-orbit couplings (SOC) due to the buckled structure of silicene, the domain wall can be a good analog of a 1D wire made of semiconductors with strong SOC, which provides us a new access to the realization of Majorana zero-modes in 1D systems. |
