Semiconductor-superconductor Josephson junctions in the presence of Zeeman and spin-orbit fields

Epitaxially grown Al-InAs hybrids have a great potential for future applications. The most prominent incentive in this regard are potential Majorana zero modes, which are to be believed ideal candidates for fault-tolerant quantum computers. However, with the recent access to these novel materials, it is furthermore possible to conduct experiments on a wide range of generic phenomena. With the help of top-down fabrication, individual designed Josephson junctions offer an unprecedented playground for experimentalists due to the unique combination of the two-dimensional electron gas (2DEG) and the superconductor. This dissertation is about examining of the fundamental building blocks of single Josephson junctions built on such a heterostructure. For this purpose, we elaborated a fabrication process and installed a measurement technique based on a cold RLC resonator in the low MHz regime that is placed in series to the sample. In contrast to the normal resistance, the resonator is a tool which allows us to access the inductance of a superconducting system and thus to probe the supercurrent-carrying Andreev bound states (ABS). The main discoveries of this work include a complete picture of the ABS dependency on various parameters, such as the charge carrier density, the dc current, the magnetic fields, the temperature, or the transparency of the junction, which is close to unity. In the heterostructure, we can break inversion and time-reversal symmetry simultaneously with the interaction of spin-orbit and Zeeman fields. This, in combination with the ballistic character of the Josephson device, leads to a non-reciprocal current that depends on the cross product of current and Zeeman field. Furthermore, we report a rectification effect of the supercurrent even far below the critical temperature of the superconductor. The observed non-reciprocal current is a consequence of a distorted current-phase relation (CPR). Using the inductance, we can display this distortion and derive the novel magnetochiral anisotropy (MCA) coefficient ${\gamma }_L$ for supercurrents. Moreover, with the MCA coefficient ${\gamma }_L$ we extract the Dresselhaus component and witness furthermore a peculiar sign change of the MCA at the point where the Zeeman energy is as large as the induced gap. Finally, with the gained understanding and experience of single superconductor-semiconductor Josephson junctions, we create the basis for more complex devices, e.g. multiterminal Josephson junctions (MTJJs). Such junctions with multiple superconducting leads are predicted to host synthetic Weyl singularities in their ABS spectrum. In this work, we present first results of this new topic and show that it is possible to fabricate such MTJJs and to measure their inductance.