Superconductor-proximity effect in hybrid structures: Fractality versus Chaos

TL;DR

This paper investigates how the superconductor proximity effect manifests in systems with fractal spectra, revealing the absence of a spectral gap and deriving the distribution of the smallest excitation energy, supported by numerical validation.

Contribution

It provides an analytical expression for the smallest excitation eigenvalue distribution in superconductor-normal hybrid structures with fractal spectra, linking it to the fractal dimension.

Findings

No spectral gap in fractal spectrum systems, even with chaotic classical dynamics.

Small-scale eigenvalue distribution decays algebraically with exponent related to fractal dimension.

Numerical results confirm the analytical predictions.

Abstract

We study the proximity effect of a superconductor to a normal system with fractal spectrum. We find that there is no gap in the excitation spectrum, even in the case where the underlying classical dynamics of the normal system is chaotic. An analytical expression for the distribution of the smallest excitation eigenvalue $E_g$ of the hybrid structure is obtained.On small scales it decays algebraically as ${\cal P}(E_g)\sim E_g ^{-D_0}$, where $D_0$ is the fractal dimension of the spectrum of the normal system. Our theoretical predictions are verified by numerical calculations performed for various models.