Excitation spectra for Andreev billiards of Box and Disk geometries

Abstract

We study Andreev billiards of box and disk geometries by matching the wave functions at the interface of the normal and the superconducting region using the exact solutions of the Bogoliubov-de Gennes equation. The mismatch in the Fermi wavenumbers and the effective masses of the normal system and the superconductor, as well as the tunnel barrier at the interface are taken into account. A Weyl formula (for the smooth part of the counting function of the energy levels) is derived. The exact quantum mechanical calculations show equally spaced singularities in the density of states. Based on the Bohr-Sommerfeld quantization rule a semiclassical theory is proposed to understand these singularities. For disk geometries two kinds of states can be distinguished: states either contribute through whispering gallery modes or are Andreev states strongly coupled to the superconductor. Controlled by two relevant material parameters, three kinds of energy spectra exist in disk geometry. The first is dominated by Andreev reflections, the second, by normal reflections in an annular disk geometry. In the third case the coherence length is much larger than the radius of the superconducting region, and the spectrum is identical to that of a full disk geometry.

arXiv comment23 pages, 15 figures

References in corpus (12)

• Random-Matrix Theory of Quantum Transport
• Random Matrix Theory of a Chaotic Andreev Quantum Dot
• Induced superconductivity distinguishes chaotic from integrable billiards
• Andreev Reflection in Strong Magnetic Fields
• Angle dependence of Andreev scattering at semiconductor-superconductor interfaces
• Chaos in Andreev Billiards
• Density of States and Energy Gap in Andreev Billiards
• The Role of Diffraction in the Quantization of Dispersing Billiards
• Nonlinearity in NS transport: scattering matrix approach
• Superconductor-proximity effect in chaotic and integrable billiards
• Paramagnetic reentrant effect in high purity mesoscopic AgNb proximity structures
• Excitation spectrum of Andreev billiards with a mixed phase space

Cited by in corpus (10)

• Andreev billiards
• Multiple scattering theory for superconducting heterostructures
• Ring--Shaped Andreev Billiards in Quantizing Magnetic Fields
• Magnetic focusing in normal-superconductor hybrid systems: a semiclassical analysis
• Bound states in Andreev billiards with soft walls
• A logarithmic contribution to the density of states of rectangular Andreev billiards
• Andreev bound states for superconducting-ferromagnetic box
• Non-retracing orbits in Andreev billiards
• Andreev bound states for cake shape superconducting-normal systems
• Decreasing excitation gap in Andreev billiards by disorder scattering