# Local density of states in clean two-dimensional superconductor--normal metal--superconductor heterostructures

## TL;DR

This paper provides a comprehensive theoretical analysis of the local density of states in clean 2D superconductor-normal metal-superconductor heterostructures, revealing how Andreev bound states and magnetic flux influence the LDOS and supercurrent.

## Contribution

It introduces a detailed quasiclassical framework for analyzing phase-dependent LDOS in 2D SNS junctions, including effects of geometry, reflectivity, and magnetic fields, aligning with recent experimental findings.

## Key findings

- Andreev bound states shape the phase-dependent LDOS.
- Magnetic flux causes complex interference effects.
- Analytical relation between LDOS and supercurrent is established.

## Abstract

Motivated by recent advances in the fabrication of Josephson junctions in which the weak link is made of a low-dimensional non-superconducting material, we present here a systematic theoretical study of the local density of states (LDOS) in a clean 2D normal metal (N) coupled to two s-wave superconductors (S). To be precise, we employ the quasiclassical theory of superconductivity in the clean limit, based on Eilenberger's equations, to investigate the phase-dependent LDOS as function of factors such as the length or the width of the junction, a finite reflectivity, and a weak magnetic field. We show how the the spectrum of Andeeev bound states that appear inside the gap shape the phase-dependent LDOS in short and long junctions. We discuss the circumstances when a gap appears in the LDOS and when the continuum displays a significant phase-dependence. The presence of a magnetic flux leads to a complex interference behavior, which is also reflected in the supercurrent-phase relation. Our results agree qualitatively with recent experiments on graphene SNS junctions. Finally, we show how the LDOS is connected to the supercurrent that can flow in these superconducting heterostructures and present an analytical relation between these two basic quantities.