Hard gap in a normal layer coupled to a superconductor

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

Figure 1

(a) A quasiclassical result for the density of states $N\left(E\right)$ in a normal layer coupled to a superconductor. In this approximation, one obtains a gapless density of states that vanishes linearly at the Fermi energy. (b) A tunneling-Hamiltonian result for the density of states in a 2D normal layer coupled to a superconductor. A BCS-like gap is induced, with the size of the gap determined by the transparency of the SN interface (shown here for a highly transparent interface).

Figure 2

(a) Geometry under consideration in this paper, where a normal layer of finite thickness $d$ is placed in contact with a semi-infinite superconductor (both materials are infinite in the $yz$ plane). A sharp potential barrier is included at the SN interface, which is located at $x=0$.

Figure 3

Plot of $f_0\left(\theta \right)$ at $E=0$ [Eq. (13)] for several values of ${\theta }_0=2\sqrt{\pi }(d/d_c)$. Because no solution to $f_0\left(\theta \right)=0$ exists, the normal layer is gapped for each ${\theta }_0$.

Figure 4

Numerical solution for the proximity-induced gap $E_g$ as a function of thickness in the absence of Fermi surface mismatch, plotted for various values of barrier strength $w$. Fermi energy was chosen so that $E_F/\mathrm{\Delta }={10}^3$.

Figure 5

Numerical solution for proximity-induced gap $E_g$ as a function of thickness with strong Fermi surface mismatch, plotted for various values of barrier strength $w$. To enable a comparison with Fig. 4, we keep $E_{FN}/\mathrm{\Delta }={10}^3$ and $s=1$; we also choose $E_{FS}/E_{FN}=m_S/m_N=10$.