Anomalous effects in Quantum Spin Hall-based Josephson junctions

The main subject of this PhD Thesis is Quantum Spin Hall-based Josephson junctions. A Quantum Spin Hall Insulator is a topological phase of matter featuring a two-dimensional insulating bulk and symmetry-protected counter-propagating helical edge states on the boundary. Proximitizing it by two superconducting leads on opposite sides gives rise to the aforementioned junctions. At energies lower than the superconducting gap, transport across the junction is mediated by pairs of correlated electrons, namely Cooper pairs, traveling from one lead to the other through the helical edge states, and originating the dissipationless Josephson current. An important lengthscale at play is the so-called superconducting coherence length, which represents effectively the "size" of Cooper pairs and, in turn, the distance over which the two bounded electrons can split. Previous works already showed that a junction's width comparable to this value allows the two electrons to be not only injected into a same edge, but also to split and propagate over opposite ones. This Thesis investigates further phenomena having lengthscales comparable or less than the superconducting coherence length, such that Cooper pair splitting, in different ways, plays a key role. In particular, a constriction between the edge channels is included. The implementation of a constriction and the proximization of Quantum Spin Hall Insulators have been experimentally achieved, therefore the assembly of the entire structure appears as a feasible goal, and its theoretical analysis holds significant relevance. The first consequence of the constriction is that each electron of a pair can independently undergo inter-edge tunneling. If the confinement potential for the edge states is smooth, the two tunneling amplitudes for electrons moving in opposite directions can unbalance. This effect, known as edge reconstruction, is also taken into account. Moreover, if the edge channels have a spatial extent comparable to the superconducting coherence length, the two electrons can even propagate over different trajectories within a same edge. This range of possibilities is described in detail, and shown to lead to clear-cut signatures in transport measurements, in particular at the level of the interference pattern and the current-phase relation of the junction. Apart from the characterization of this experimentally conceivable system, the main outcomes of the Thesis are two. From a fundamental point of view, a better understanding of the effects of single-electron physics in a superconducting context, usually dominated by the physics of Cooper pairs. For practical purposes, it is demonstrated that this platform is naturally entitled to interesting functionalities in the realm of Josephson junctions. Indeed, it can host the anomalous Josephson effect, that can be used to design phase batteries and to drive superconducting circuits and superconducting memories, and the superconducting diode effect, which inspires great technological perspectives as well, without the need of external biases or magnetic fields.