We present transport and scanning SQUID measurements on InAs/GaSb double quantum wells, a system predicted to be a two-dimensional topological insulator. Top and back gates allow independent control of density and band offset, allowing tuning from the trivial to the topological regime. In the trivial regime, bulk conductivity is quenched but transport persists along the edges, superficially resembling the predicted helical edge-channels in the topological regime. We characterize edge conduction in the trivial regime in a wide variety of sample geometries and measurement configurations, as a function of temperature, magnetic field, and edge length. Despite similarities to studies claiming measurements of helical edge channels, our characterization points to a non-topological origin for these observations.

The Josephson effect describes supercurrent flowing through a junction connecting two superconducting leads by a thin barrier. This current is driven by a superconducting phase difference φbetween the leads. In the presence of chiral and time-reversal symmetry of the Cooper pair tunnelling process, the current is strictly zero when φvanishes. Only if these underlying symmetries are broken can the supercurrent for φ= 0 be finite. This corresponds to a ground state of the junction being offset by a phase φ0, different from 0 or π. Here, we report such a Josephson φ0 -junction based on a nanowire quantum dot. We use a quantum interferometer device to investigate phase offsets and demonstrate that φ0 can be controlled by electrostatic gating. Our results may have far-reaching implications for superconducting flux- and phase-defined quantum bits as well as for exploring topological superconductivity in quantum dot systems.