“Smile” Gap in the Density of States of a Cavity between Superconductors

The density of Andreev levels in a normal metal ($N$) in contact with two superconductors ($S$) is known to exhibit an induced minigap related to the inverse dwell time. We predict a small secondary gap just below the superconducting gap edge—a feature that has been overlooked so far in numerous microscopic studies of the density of states in $S\text{−}N\text{−}S$ structures. In a generic structure with $N$ being a chaotic cavity, the secondary gap is the widest at zero phase bias. It closes at some finite phase bias, forming the shape of a “smile”. Asymmetric couplings give even richer gap structures near the phase difference $\pi$. All the features found should be amendable to experimental detection in high-resolution low-temperature tunneling spectroscopy.

Figure 1

Upper plot: DOS in the central region at zero phase difference and $E_{\text{Th}}=\mathrm{\Delta }$ showing the usual minigap around $E=0$ and additionally a secondary gap below $E=\mathrm{\Delta }$. On the right: Quantum circuit theory [26] diagram of the system under investigation. A pseudoterminal labeled with $E_{\text{Th}}$ accounts for random phase shifts between electron and hole components of the quasiparticle wave functions (not implying an electric connection to the ground). Lower plot: DOS near $\mathrm{\Delta }$ illustrating the phase dependence of the secondary gap.

Figure 2

Upper plot: Disappearing secondary gap for $E_{\text{Th}}<\mathrm{\Delta }$. The colored regions represent the gapped part of the spectrum for different Thouless energies. Lower plot: Critical parameters of the secondary gap for $G_1=G_2$ in dependence of $E_{\text{Th}}$: Upper gap edge for $\phi =0$ (red), lower gap edge for $\phi =0$ (dashed blue) and critical phase (dotted green). The colored regions denote the gap.

Figure 3

Universal shape of the DOS for $\phi =0$ between the gaps in the limit $E_{\text{Th}}\gg \mathrm{\Delta }$. The curve is obtained from Eq. (3) and the characteristics are given in the main text.

Figure 4

Dependence of the local density of states on increasing asymmetry $a=G_1/G_2$. The lower part of each plot shows the well-known minigap, the upper part shows the secondary gap. In each plot $E_{\text{Th}}=\mathrm{\Delta }$. For a symmetric setup (left plot) the result is equivalent to Fig. 1. With increasing asymmetry $a=10$ (central plot) and $a=100$ (right plot) both gaps are stable, however, additional gaps appear around $\phi =\pi$.