Secondary "Smile"-gap in the density of states of a diffusive Josephson junction for a wide range of contact types

TL;DR

This paper investigates the secondary 'smile'-shaped energy gap in the density of states of diffusive Josephson junctions, extending previous theories to various contact types and establishing conditions for its existence.

Contribution

It generalizes the theory of secondary gaps in Josephson junctions beyond ballistic contacts and derives a criterion linking the gap to transmission eigenvalues.

Findings

Secondary gap exists beyond ballistic contacts.

Existence of the secondary gap depends on small transmission eigenvalues.

Density of states shows singular behavior at the superconducting gap edge.

Abstract

The superconducting proximity effect leads to strong modifications of the local density of states in diffusive or chaotic cavity Josephson junctions, which displays a phase-dependent energy gap around the Fermi energy. The so-called minigap of the order of the Thouless energy $E_{\mathrm{Th}}$ is related to the inverse dwell time in the diffusive region in the limit $E_{\mathrm{Th}}\ll\Delta$, where $\Delta$ is the superconducting energy gap. In the opposite limit of a large Thouless energy $E_{\mathrm{Th}}\gg\Delta$, a small new feature has recently attracted attention, namely, the appearance of a further secondary gap, which is around two orders of magnitude smaller compared to the usual superconducting gap. It appears in a chaotic cavity just below the superconducting gap edge $\Delta$ and vanishes for some value of the phase difference between the superconductors. We extend previous theory restricted to a normal cavity connected to two superconductors through ballistic contacts to a wider range of contact types. We show that the existence of the secondary gap is not limited to ballistic contacts, but is a more general property of such systems. Furthermore, we derive a criterion which directly relates the existence of a secondary gap to the presence of small transmission eigenvalues of the contacts. For generic continuous distributions of transmission eigenvalues of the contacts, no secondary gap exists, although we observe a singular behavior of the density of states at $\Delta$. Finally, we provide a simple one-dimensional scattering model which is able to explain the characteristic "smile" shape of the secondary gap.