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We theoretically consider the observed soft gap in the proximity-inducedsuperconducting state of semiconductor nanowires in the presence of spin-orbitcoupling, Zeeman splitting, and tunneling leads, but in the absence of anyextrinsic disorder. We critically consider the effects of three distinctintrinsic physical mechanisms (tunnel barrier to normal leads, temperature, anddissipation) on the phenomenology of the gap softness in the tunnelspectroscopy of the normal-superconductor junction as a function of spinsplitting and chemical potential. We find that all three mechanismsindividually can produce a soft gap, leading to calculated conductance spectraqualitatively mimicking experimental results. We also show that thephenomenology of the soft gap is intrinsically tied to the broadening and theheight of the Majorana zero mode(MZM)-induced differential conductance peakabove the topological quantum phase transition point with both the soft gap andthe quality of the MZM being simultaneously affected by tunnel barrier,temperature, and dissipation. We establish that the MZM splitting oscillationscan be suppressed by temperature or dissipation, but not by the tunnel barrier.Since all three mechanisms are likely to be present in any realistic nanowires,discerning the effects of various mechanisms is difficult, necessitatingdetailed experimental data as a function of all the system parameters, some ofwhich (e.g., dissipation, chemical potential, tunnel barrier) may not be knownexperimentally. While the tunneling-induced soft-gap behavior is benign with nodirect adverse effect on the Majorana topological properties with the zero-biaspeak remaining quantized at $2e^2/h$, the soft gap induced by finitetemperature and/or finite dissipation is detrimental to topological propertiesand must be avoided as much as possible.

Phenomenology of soft gap, zero bias peak, and zero mode splitting in ideal Majorana nanowires