  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 self-consistently the associated
three-dimensional Schr\"odinger-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 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.
