High density superconductor-semiconductor-superconductor junctions have a small induced superconducting gap due to the quasiparticle trajectories with a large momentum parallel to the junction having a very long flight time. Because a large induced gap protects Majorana modes, these long trajectories constrain Majorana devices to a low electron density. We show that a zigzag-shaped geometry eliminates these trajectories, allowing the robust creation of Majorana states with both the induced gap E_{gap} and the Majorana size ξ_{M} improved by more than an order of magnitude for realistic parameters. In addition to the improved robustness of Majoranas, this new zigzag geometry is insensitive to the geometric details and the device tuning.

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By exploiting quantum mechanics, quantum computers are able to solve problems---for example in quantum chemistry---that are far out of reach of classical computers. Conventional approaches to quantum computers use qubits (quantum bits) that can either be made out of superconducting circuits (the currently leading approach used to demonstrate quantum supremacy), store quantum information in the state of single electrons, or in several other ways. However, all these designs share the same limitation: noise easily spoils the quantum states and therefore makes quantum error correction necessary. A fundamentally different approach is to use exotic particles: Majorana bound states, or simply Majoranas. These particles do not exist in nature but were predicted to exist in engineered devices. Majorana bound states offer the advantage of being resilient against small energy fluctuations or noise due to being protected by the system's fundamental symmetry. There is an active search across many groups worldwide trying different pathways to experimentally create and detect these particles. One promising approach to create Majoranas relies on a theoretical prediction that a one-dimensional nanowire combining superconductivity, spin-orbit coupling, a tunable chemical potential, and a magnetic field should support Majoranas. Whenever the device parameters are in the right regime, the Majoranas should appear at the edges of the nanowire. The experimental results diverge from the predictions made by the minimal theoretical model due to it missing several physical phenomena. To model the Majorana nanowires more realistically, we include previously neglected physical effects by considering the full three-dimensional geometry of the nanowire. These more complex models exceed the reach of analytical theories and require intensive numerical calculations instead. To cope with the increasing computational complexity, we develop adaptive parallel sampling algorithms (discussed in the Chapter 7), which in our research typically speeds up simulations by at least an order of magnitude. Our approach is illustrated on the cover of this thesis, which shows the conductance of a Majorana nanowire with the interesting regions sampled more accurately. The improved models and efficient sampling, allow us to unveil potential challenges that were not present in the simple model. In Chapter 2, we study how the electrons are influenced by the magnetic field while moving across the nanowire---an effect completely neglected in the minimal model. We observe that this effect has a stronger impact on the Majoranas than what is included in the minimal model. Specifically, we observe that the protection of the Majoranas nearly vanishes when the electron density in the nanowire is high, and find that the magnetic field has to be precisely aligned with the nanowire to guarantee the presence of Majoranas. Bringing two Majorana-carrying nanowires in contact and allowing a supercurrent flow between them is required for making a Majorana qubit. In Chapter 3, we apply our numerical model to analyze experimentally observed behavior of these nanowire junctions. While our results agree with the experimental observations, we observe that the supercurrent decreases by an order of magnitude when Majoranas appear. This suppression poses a new challenge in creating a Majorana qubit. Our findings may seem like bad news for the creation of Majoranas---unexpected pitfalls overlooked by the simplified models. The detailed simulations, however, bring new opportunities as well. Not being constrained to analyzing the devices that are easy to solve, we are able to design a new zigzag device geometry, that improves the robustness of Majoranas by an order of magnitude. In Chapter 6, we show that using a zigzag device geometry (instead of a straight nanowire) eliminates the long electron trajectories that are responsible for the degradation of the Majorana properties. In addition to the improved robustness of the Majoranas, this new zigzag geometry is insensitive to the geometric details and device tuning. This proposal is now the topic of active experimental investigations by several groups. @en

Andreev bound states in hybrid superconductor-semiconductor devices can have near-zero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. These quasi-Majorana states show zero-bias conductance features in a topologically trivial phase, mimicking spatially separated topological Majorana states. We show that in addition to the suppressed coupling between the quasi-Majorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different for increasing magnetic field. As a consequence, quasi-Majorana states mimic most of the proposed Majorana signatures: quantized zero-bias peaks, the 4π Josephson effect, and the tunneling spectrum in presence of a normal quantum dot. We identify a quantized conductance dip instead of a peak in the open regime as a distinguishing feature of true Majorana states in addition to having a bulk topological transition. Because braiding schemes rely only on the ability to couple to individual Majorana states, the exponential control over coupling strengths allows to also use quasi-Majorana states for braiding. Therefore, while the appearance of quasi-Majorana states complicates the observation of topological Majorana states, it opens an alternative route towards braiding of non-Abelian anyons and protected quantum computation.

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Spin-orbit interaction (SOI) plays a key role in creating Majorana zero modes in semiconductor nanowires proximity coupled to a superconductor. We track the evolution of the induced superconducting gap in InSb nanowires coupled to a NbTiN superconductor in a large range of magnetic field strengths and orientations. Based on realistic simulations of our devices, we reveal SOI with a strength of 0.15-0.35 eV Å. Our approach identifies the direction of the spin-orbit field, which is strongly affected by the superconductor geometry and electrostatic gates.

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We study the effects of strong coupling between a superconductor and a semiconductor nanowire on the creation of the Majorana bound states, when the quasiparticle dwell time in the normal part of the nanowire is much shorter than the inverse superconducting gap. This "short-junction" limit is relevant for the recent experiments using the epitaxially grown aluminum characterized by a transparent interface with the semiconductor and a small superconducting gap. We find that the small superconducting gap does not have a strong detrimental effect on the Majorana properties. Specifically, both the critical magnetic field required for creating a topological phase and the size of the Majorana bound states are independent of the superconducting gap. The critical magnetic field scales with the wire cross section, while the relative importance of the orbital and Zeeman effects of the magnetic field is controlled by the material parameters only: g factor, effective electron mass, and the semiconductor-superconductor interface transparency.

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Junctions created by coupling two superconductors via a semiconductor nanowire in the presence of high magnetic fields are the basis for the potential detection, fusion, and braiding of Majorana bound states. We study NbTiN/InSb nanowire/NbTiN Josephson junctions and find that the dependence of the critical current on the magnetic field exhibits gate-tunable nodes. This is in contrast with a well-known Fraunhofer effect, under which critical current nodes form a regular pattern with a period fixed by the junction area. Based on a realistic numerical model we conclude that the Zeeman effect induced by the magnetic field and the spin-orbit interaction in the nanowire are insufficient to explain the observed evolution of the Josephson effect. We find the interference between the few occupied one-dimensional modes in the nanowire to be the dominant mechanism responsible for the critical current behavior. We also report a strong suppression of critical currents at finite magnetic fields that should be taken into account when designing circuits based on Majorana bound states.

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We investigate the mechanistic nature of the Snf2 family protein HARP, mutations of which are responsible for Schimke immuno-osseous dysplasia. Using a single-molecule magnetic tweezers assay, we construct RPA-stabilized DNA bubbles within torsionally constrained DNA to investigate the annealing action of HARP on a physiologically relevant substrate. We find that HARP closes RPA-stabilized bubbles in a slow reaction, taking on the order of tens of minutes for ∼600 bp of DNA to be re-annealed. The data indicate that DNA re-anneals through the removal of RPA, which is observed as clear steps in the bubbleclosing traces. The dependence of the closing rate on both ionic strength and HARP concentration indicates that removal of RPA occurs via an association-dissociation mechanism where HARP does not remain associated with the DNA. The enzyme exhibits classical Michaelis-Menten kinetics and acts cooperatively with a Hill coefficient of 3 ± 1. Our work also allows the determination of some important features of RPA-bubble structures at low supercoiling, including the existence of multiple bubbles and that RPA molecules are mis-registered on the two strands.

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Ballistic electron transport is a key requirement for existence of a topological phase transition in proximitized InSb nanowires. However, measurements of quantized conductance as direct evidence of ballistic transport have so far been obscured due to the increased chance of backscattering in one-dimensional nanowires. We show that by improving the nanowire-metal interface as well as the dielectric environment we can consistently achieve conductance quantization at zero magnetic field. Additionally we study the contribution of orbital effects to the sub-band dispersion for different orientation of the magnetic field, observing a near-degeneracy between the second and third sub-bands.

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Studies of Majorana bound states in semiconducting nanowires frequently neglect the orbital effect of a magnetic field. Systematically studying its role leads us to several conclusions for designing Majoranas in this system. Specifically, we show that for experimentally relevant parameter values the orbital effect of a magnetic field has a stronger impact on the dispersion relation than the Zeeman effect. While Majoranas do not require the presence of only one dispersion subband, we observe that the size of the Majoranas becomes unpractically large, and the band gap unpractically small, when more than one subband is filled. Since the orbital effect of a magnetic field breaks several symmetries of the Hamiltonian, it leads to the appearance of large regions in parameter space with no band gap whenever the magnetic field is not aligned with the wire axis. The reflection symmetry of the Hamiltonian with respect to the plane perpendicular to the wire axis guarantees that the wire stays gapped in the topologically nontrivial region as long as the field is aligned with the wire.

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