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Full-shell nanowires have been proposed as an alternative nanowire design inthe search of topological superconductivity and Majorana zero modes (MZMs).They are hybrid nanostructures consisting of a semiconductor core fully coveredby a thin superconductor shell and subject to a magnetic flux. In this work wecritically examine this proposal, finding a very rich spectral phenomenologythat combines the Little-Parks modulation of the parent-gap superconductor withflux, the presence of flux-dispersing Caroli-de Gennes-Matricon (CdGM) analogsubgap states, and the emergence of MZMs across finite flux intervals thatdepend on the transverse wavefunction profile of the charge density in the coresection. Through microscopic simulations and analytical derivations, we studydifferent regimes for the semiconductor core, ranging from the hollow-coreapproximation, to the tubular-core nanowire appropriate for a semiconductortube with an insulating core, to the solid-core nanowire. We compute the phasediagrams for the different models in cylindrical nanowires and find that MZMstypically coexist with CdGM analogs at zero energy, rendering them gapless.However, we also find topologically protected parameter regions, or islands,with gapped MZMs. In this sense, the most promising candidate to obtaintopologically protected MZMs in a full-shell geometry is the nanowire with atubular-shaped core. Moving beyond pristine nanowires, we study the effect ofmode mixing perturbations. Strikingly, mode mixing can act like a topological$p$-wave pairing between particle-hole Bogoliubov partners, and is thereforeable to create new topologically protected MZMs in regions of the phase diagramthat were originally trivial. As a result, the phase diagram is utterlytransformed and exhibits protected MZMs in around half of the parameter space.

Phenomenology of Majorana zero modes in full-shell hybrid nanowires