Andreev bound states for cake shape superconducting-normal systems

TL;DR

This paper analyzes the energy spectrum of cake-shaped normal-superconducting systems, deriving analytical expressions and comparing them with numerical results, revealing a gapped spectrum with edge singularities and effects of mismatch and barriers.

Contribution

It provides analytical formulas for the density of states in cake-shaped NS systems considering interface mismatch and barriers, validated against numerical calculations.

Findings

Spectrum has an energy gap with singular density of states at the edge.

Analytical and numerical results show excellent agreement.

Mismatch and barriers influence the size of the energy gap.

Abstract

The energy spectrum of cake shape normal - superconducting systems is calculated by solving the Bogoliubov-de Gennes equation. We take into account the mismatch in the effective masses and Fermi energies of the normal and superconducting regions as well as the potential barrier at the interface. In the case of a perfect interface and without mismatch, the energy levels are treated by semi-classics. Analytical expressions for the density of states and its integral, the step function, are derived and compared with that obtained from exact numerics. We find a very good agreement between the two calculations. It is shown that the spectrum possesses an energy gap and the density of states is singular at the edge of the gap. The effect of the mismatch and the potential barrier on the gap is also investigated.