We study the proximity effect in a one-dimensional nanowire strongly coupledto a finite superconductor with a characteristic size which is much shorterthan its coherence length. Such geometries have become increasingly relevant inrecent years in the experimental search for Majorana fermions with thedevelopment of thin epitaxial Al shells which form a very strong contact witheither InAs or InSb nanowires. So far, however, no theoretical treatment of theproximity effect in these systems has accounted for the finite size of thesuperconducting film. We show that the finite-size effects become verydetrimental when the level spacing of the superconductor greatly exceeds itsenergy gap. Without any fine-tuning of the size of the superconductor (on thescale of the Fermi wavelength), the tunneling energy scale must be larger thanthe level spacing in order to reach the hard gap regime which is seenubiquitously in the experiments. However, in this regime, the large tunnelingenergy scale induces a large shift in the effective chemical potential of thenanowire and pushes the topological phase transition to magnetic fieldstrengths which exceed the critical field of Al.
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