Mean Free Path and Energy Fluctuations in Quantum Chaotic Billiards

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 $\ensuremath{-}W/2$ and $W/2$. Surface roughness yields a mean free path $l$ that decreases as ${L/W}^{2}$ as $W$ increases, $L$ being the linear system size. 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 bandwidth. Energy fluctuations reflect the behavior of the mean free path. At small energy scales, strong level correlations manifest themselves by small values of ${\ensuremath{\Sigma}}^{2}(E)$ that are close to random matrix theory (RMT) in all cases. At larger energy scales, fluctuations are below the logarithmic behavior of RMT for $lgL$, and above RMT value when $llL$.