Josephson junctions form a two-level system which is used as a building block for many types of superconducting qubits. Junctions fabricated from semiconducting nanowires are gate-tunable and offer electrostatically adjustable Josephson energy, highly desirable in qubit architecture. Studying nanowire weak links is therefore important for future quantum computing applications. The inherent spin-orbit interaction and high g-factor of InSb nanowires promise rich physics when combined with superconductivity, especially when an external magnetic field is applied. In particular, it can give rise to topological state of matter including Majorana bound states, paving the way for a novel type of fault-tolerant topological qubit. Such quantum computation can be realized when Majorana bound states are braided through a network of topological wires. Probing the magnitude and phase of the supercurrent through InSb nanowires provides insight on the feasibility of realizing topological states in these wires. This thesis describes experiments measuring the critical current and density of states of InSb nanowire Josephson junctions which are either voltage- current- or phase-biased, as the chemical potential or magnetic field inside the wire is changed. In Chapter 3, the critical current through an InSb nanowire with NbTiN electrodes is measured. The critical current can be as high as ∼100 nA but decays rapidly with magnetic field followed by an aperiodic oscillation. Numerical simulations of the supercurrent through the nanowire show that this supercurrent profile is caused mostly by the interference between the transverse modes carrying the supercurrent inside the nanowire. This so--called orbital effect becomes significant beyond 100 mT, while the spin--orbit and Zeeman interactions become substantial at magnetic field of the order 1 T. The Josephson energy through cross-shaped nanowires, grown by merging individual InSb nanowires, is investigated in Chapter 4. A finite Josephson coupling is measured through all branches of the nanocross, even when the length of the weak link extends beyond 1 μm. This is a requirement for braiding Majorana bound states hosted in such nanowire networks. In Chapter 5 we build a quantum dot with two superconducting and a normal contact using the three legs of a nanowire cross. The superconducting terminals are joined in a loop such that superconducting interference can be probed by threading a flux. The density of states as a function of voltage bias, dot chemical potential and flux is probed through the quantum dot via the normal lead acting as a tunnel probe. It is revealed that the proximity effect can be turned on and off via both the bias and gate voltage. The pairing amplitude on the dot remains finite for in-plane magnetic field values up to 600 mT, suggesting that the nanowire cross platform is suitable for braiding, since a topological state can be reached at 100-200 mT. As the conductance through the dot is sensitive to the flux through the loop, the device may also be used as a mangetometer converting flux to current with a sensitivity of $1 nA/Φ0. The superconducting phase across a nanowire quantum dot as a function of the magnitude and direction of a large in-plane magnetic field is investigated in Chapter 6. The nanowire is embedded in a DC-SQUID where one arm consists of a gate-defined quantum dot in the nanowire and the other is a nanowire reference junction, also gate-tunable. By measuring the critical current through the SQUID as a function of the flux and the chemical potential of the dot, we can detect the change of phase through the ground state of the dot. At zero-field we measure the 0-$\pi$ transition of the quantum dot Josephson junction as the ground state parity of the dot changes. When the magnetic field exceeds 100 mT a 0-φ transition is measured indicating the presence of an anomalous supercurrent flow at vanishing phase difference across the quantum dot. This anomalous current is enabled by the breaking of the chiral symmetry due to spin-orbit interaction in the nanowire and the time-reversal symmetry breaking of the magnetic field. The phase of the 0-φ transition, or equivalently the magnitude of the anomalous current, can be tuned continuously via the gate underneath the dot. Such a φ0 junction may serve as a phase bias element and have applications in superconducting spintronics. Chapter 7 focuses on future experiments aiming to detect and control Majorana bound states in a superconducting InSb nanowire. Such devices can be expanded to a braiding circuit, realizing a topological quantum computer.
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