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We investigate the impact of electrostatics on the proximity effect between a magnetic insulator and a semiconductor wire in semiconductor–superconductor–magnetic-insulator hybrid structures. By performing self-consistent Schrödinger-Poisson calculations using an effective model of the hybrid system, we find that large effective Zeeman fields consistent with the emergence of topological superconductivity emerge within a large parameter window in wires with overlapping layers of magnetic insulator and superconductor, but not in nonoverlapping structures. We show that this behavior is essentially the result of electrostatic effects controlling the amplitude of the low-energy wave functions near the semiconductor–magnetic-insulator interface. 

We address an outstanding problem that represents a critical roadblock in the development of the Majorana-based topological qubit using semiconductor-superconductor hybrid structures: the quantitative characterization of disorder effects generated by the unintentional presence of charge impurities within the hybrid device. Given that disorder can have far-reaching consequences for the Majorana physics but is intrinsically difficult to probe experimentally in a hybrid structure, providing a quantitative theoretical description of disorder effects becomes essential. To accomplish this task, we develop a microscopic theory that (i) provides a quantitative characterization of the effective potential generated by a charge impurity embedded inside a semiconductor wire proximity coupled to a superconductor layer by solving selfconsistently the associated three-dimensional Schrödinger-Poisson problem, (ii) describes the low-energy physics of the hybrid structure in the presence of s-wave superconductivity, spin-orbit coupling, Zeeman splitting, and disorder arising from multiple charge impurities by using the results of (i) within a standard free-fermion approach, and (iii) links the microscopic results to experimentally observable features by generating tunneling differential-conductance maps as a function of the control parameters (e.g., Zeeman field and chemical potential). We find that charge impurities lead to serious complications regarding the realization and observation of Majorana zero modes, which have direct implications for the development of Majorana-based qubits. More importantly, our work provides a clear direction regarding what needs to be done for progress in the field, including specific materials-quality and semiconductor-purity targets that must be achieved to create a topological qubit. 

We use the available transportmeasurements in the literature to develop a dataset for the likely amount of disorder in semiconductor (InAs and InSb) materials which are used in fabricating the superconductor-semiconductor nanowire samples in the experimental search for Majorana zero modes. Using the estimated disorder in direct Majorana simulations, we conclude that the current level of disorder in semiconductor Majorana nanowires is at least an order of magnitude higher than that necessary for the emergence of topological Majorana zero modes. In agreement with existing results, we find that our estimated disorder leads to the occasional emergence of trivial zero modes, which can be post-selected and then further fine-tuned by varying system parameters (e.g., tunnel barrier), leading to trivial zero-bias conductance peaks in tunneling spectroscopy with ∼2e2/h magnitude. Most calculated tunnel spectra in these disordered systems, however, manifest essentially no significant features, which is also consistent with the current experimental status, where zero-bias peaks are found only occasionally in some samples under careful fine-tuning. 

We propose a semiconductor-superconductor hybrid device for realizing topological superconductivity and Majorana zero modes consisting of a planar Josephson junction structure with periodically modulated junction width. By performing a numerical analysis of the effective model describing the low-energy physics of the hybrid structure, we demonstrate that the modulation of the junction width results in a substantial enhancement of the topological gap and, consequently, of the robustness of the topological superconducting phase and associated Majorana zero modes. This enhancement is due to the formation of minibands with strongly renormalized effective parameters, including stronger spin-orbit coupling, generated by the effective periodic potential induced by the modulated structure. In addition to a larger topological gap, the proposed device supports a topological superconducting phase that covers a significant fraction of the parameter space, including the low Zeeman field regime, in the absence of a superconducting phase difference across the junction. Furthermore, the optimal regime for operating the device can be conveniently accessed by tuning the potential in the junction region using, for example, a top gate.