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The ability to induce a sizable gap in the excitation spectrum of a normallayer placed in contact with a conventional superconductor has becomeincreasingly important in recent years in the context of engineering atopological superconductor. The quasiclassical theory of the proximity effectshows that Andreev reflection at the superconductor/normal interface induces anonzero pairing amplitude in the metal but does not endow it with a gap.Conversely, when the normal layer is atomically thin, the tunneling of Cooperpairs induces an excitation gap that can be as large as the bulk gap of thesuperconductor. We study how these two seemingly different views of theproximity effect evolve into one another as the thickness of the normal layeris changed. We show that a fully quantum-mechanical treatment of the problempredicts that the induced gap is always finite but falls off with the thicknessof the normal layer, $d$. If $d$ is less than a certain crossover scale, whichis much larger than the Fermi wavelength, the induced gap is comparable to thebulk gap. As a result, a sizable excitation gap can be induced in normal layersthat are much thicker than the Fermi wavelength.

Hard gap in a normal layer coupled to a superconductor