Features and Transport Properties of 2D Topological-Insulator Polygonal Flakes

Conventional topological insulators in d-dimensions are characterised by gapless states localised at their boundaries with topologically trivial insulators, such as the vacuum. These gapless states exist in (d−1)-dimensions. Higher-order topological insulators host gapless boundary states in fewer dimensions, along hinges (d= 3) or at the corners (d= 3, 2) of a system. This thesis focuses on two-dimensional second- order topological insulators with localised states at their corners. Two models are considered, where the topological character is protected by i) a combined fourfold rotation and time-reversal symmetry or ii) inversion symmetry. In both cases a low-energy theory along an edge is used to derive conditions on the features, locations and total number of these states in polygonal flakes.

Multi-terminal transport through a rectangular flake is studied, where the leads are first-order topological insulators supporting helical edge states. It is demonstrated that these setups function as topological switches, where the transmission between neighbouring contacts is controlled by an in-plane magnetic field. This functionality is shown to be remarkably robust to the presence of strong disorder due to the topological nature of the states contributing to transport. Introducing a proximity-induced s-wave pairing in the leads of a two-terminal setup provides a new perspective to Fu and Kane’s study of a superconductor/quantum-spin-Hall/superconductor junction: the edge states in the leads resemble Kitaev’s one-dimensional p-wave superconducting wire, in which Majorana zero-modes are predicted to localise at the ends (near the flake). The current phase relation has 4π rather than 2π periodicity due to the hybridisation of these modes. This is a well known Majorana signature, and many proposals have been made to realise it in other systems. The connection between the work in this thesis and the literature is discussed.