  We investigate the role of the coupling between a spin-orbit coupled
semiconductor nanowire and a conventional $s$-wave superconductor on the
emergence of the topological superconducting phase with Majorana bound states
in an applied magnetic field. We show that when the coupling is strong, the
topological phase transition point is very sensitive to the size of the
superconductor and in order to reach the topological phase a strong magnetic
field is required, which can easily be detrimental for superconductivity.
Moreover, the induced energy gap separating the Majorana bound states and other
quasi-particles in the topological phase is substantially suppressed compared
to the gap at zero field. In contrast, in the weak coupling regime, we find
that the situation is essentially the opposite, with the topological phase
emerging at much lower magnetic fields and a sizable induced energy gap in the
topological phase, that can also be controlled by the chemical potential of the
superconductor. Furthermore, we show that the weak coupling regime does not
generally allow for the formation of topologically trivial zero-energy states
at the wire end points, in stark contrast to the strong coupling regime where
such states are found for a wide range of parameters. Our results thus put
forward the weak coupling regime as a promising route to mitigate the most
unwanted problems present in nanowires for realizing topological
superconductivity and Majorana bound states.
