Mean Free Path and Energy Fluctuations in Quantum Chaotic Billiards

TL;DR

This paper investigates how surface roughness affects the mean free path and energy level fluctuations in quantum chaotic billiards, revealing a minimum mean free path and RMT-like correlations at small scales.

Contribution

It introduces a model incorporating microscopic surface roughness and analyzes its impact on mean free path and energy fluctuations in quantum billiards.

Findings

Mean free path decreases with surface roughness W, reaching a minimum around L/2.

Energy fluctuations follow RMT predictions at small scales, deviating at larger scales depending on mean free path.

Surface roughness causes a non-monotonic behavior of energy level fluctuations relative to RMT.

Abstract

The elastic mean free path of carriers in a recently introduced model of quantum chaotic billiards in two and three dimensions is calculated. The model incorporates surface roughness at a microscopic scale by randomly choosing the atomic levels at the surface sites between -W/2 and W/2. Surface roughness yields a mean free path l that decreases as L/W^2 as W increases, L being the linear size of the system. But this diminution ceases when the surface layer begins to decouple from the bulk for large enough values of W, leaving more or less unperturbed states on the bulk. Consequently, the mean free path shows a minimum of about L/2 for W of the order of the band width. Energy fluctuations reflect the behavior of the mean free path. At small energy scales, strong level correlations manifest themselves by small values of the number of levels variance Sigma^2(E) that are close to Random Matrix Theory (RMT) in all cases. At larger energy scales, fluctuations are below the logarithmic behavior of RMT for l > L, and above RMT value when l < L.