Proximity-induced subgaps in andreev billiards

Area of Science:

• Condensed matter physics
• Quantum chaos
• Mesoscopic physics

Background:

• Andreev billiards are mesoscopic systems exhibiting quantum interference effects.
• The density of states in such systems is crucial for understanding their electronic properties.
• Previous studies often relied on random matrix theory, which may not fully capture all physical aspects.

Purpose of the Study:

• To investigate the density of states in Andreev billiards.
• To identify the conditions leading to an energy gap.
• To develop a new semiclassical approximation for the density of states and energy gap.

Main Methods:

• Exact quantum mechanical calculations for Andreev billiards.
• Semiclassical analysis incorporating energy-dependent phase shifts for Andreev reflections.
• Derivation of a new semiclassical Bohr-Sommerfeld approximation.

Main Results:

• A finite upper cutoff in the path length distribution P(s) leads to an energy gap on the scale of the Thouless energy.
• Exact quantum calculations show good agreement with semiclassical predictions when phase shifts are included.
• A simple formula for the energy gap was derived from the new approximation.
• The calculated energy gap, in units of Thouless energy, can exceed predictions from random matrix theory for chaotic billiards.

Conclusions:

• The presence of an energy gap in Andreev billiards is linked to path length distribution cutoffs.
• The developed semiclassical Bohr-Sommerfeld approximation offers accurate predictions for the density of states and energy gap.
• This work provides new insights into quantum transport phenomena in mesoscopic systems and challenges existing theoretical predictions.