On the spectral gap in Andreev graphs

TL;DR

This paper investigates how Andreev scattering at the edges of quantum star graphs influences their spectral properties, revealing conditions for the emergence of a spectral gap near the Fermi energy.

Contribution

It introduces Andreev scattering into quantum star graphs and analyzes its impact on the spectral gap, considering scattering amplitudes and vertex properties.

Findings

Spectral gap can form near the Fermi energy due to Andreev scattering.

The gap size depends on scattering amplitudes and vertex scattering matrix.

Bond length fluctuations have minimal effect on the spectral gap.

Abstract

We introduce Andreev scattering (electron-hole conversion at an interface of a normal conductor to a superconductor) at the outer vertices of a quantum star graph and examine its effect on the spectrum. More specifically we show that the density of states in Andreev graphs is suppressed near the Fermi energy where a spectral gap may occur. The size and existence of such a gap depends on one side on the Andreev scattering amplitudes and, on the other side,on the properties of the electron-electron scattering matrix at the central vertex. We also show that the bond length fluctuations have a minor effect on the spectrum near the Fermi energy.