Title: Stabilizing Topological Superconductivity in Disordered Spin-Orbit Coupled Semiconductor-Superconductor Heterostructures

We theoretically consider the problem of one-dimensional semiconductor-superconductor (SM and SC) heterostructure with Rashba spin-orbit coupling and a parallel Zeeman field in the presence of short-ranged disorder from random charged impurities. With no disorder, this system was proposed as a model platform for realizing bulk topological superconductivity (TS) characterized by zero energy Majorana excitations localized at the wire ends. With disorder, however, it has been shown that disorder-induced trivial low-energy states can render the detection of topological superconductivity and zero-energy Majorana states experimentally very challenging, and, for strong disorder may even lead to the disappearance of the TS state from the experimentally accessible part of the phase diagram. Starting with the Hamiltonians of the SM and the SC and using the formalism of an effective SM Green's function by integrating out the SC, we show in this paper that for a strongly disordered SM, strong coupling to the SC is generically beneficial for stabilizing a robust TS state in the semiconductor. Furthermore, we find that, for the phase diagram defined by the chemical potential (μ) and Zeeman field (Γ), with increasing strength of disorder the robust topological regions move to the parts of the phase diagram defined by larger values of Γ. These results lead us to propose that (a) stronger SM and SC coupling by interface engineering, (b) SM systems with a larger gyromagnetic ratio, or (c) a stronger proximity effect allowing the application of larger values of the Zeeman field may help defeat the dominance of disorder-induced low-energy states in the semiconductor, revealing and expanding the underlying robust topological superconducting state in the phase diagram.