Anomalous Diffusion and Quantum Interference Effect in Nano-scale Periodic Lorentz Gas

Recent advances in submicrometer technology have made it possible to confine the two-dimensional electron gas into high-mobility semiconductor heterostructures. Such structure with a lattice of electrondepleted circular obstacles are called quantum antidot lattices, or quantum Lorentz gas systems. By using the semiclassical scattering theory, we show that quantum interference in finite-size open Lorentz gas systems is expected to reflect the difference between normal and anomalous diffusions, i.e., Lévy flights. Recent advances in submicrometer technology have made it possible to confine the two-dimensional electron gas into high-mobility semi-conductor heterostructures. Such structure with a lattice of electron-depleted circular obstacles are called antidot lattices [1] and can be regarded as nano-scale periodic Lorentz gas. In the hexagonal lattice Lorentz gas with R/L < √ 3/4 (R and L are the radius of the circle and the width of the unit-cell, respectively), there exist arbitrarily long paths along which classical particles can move freely without touching the hard discs (antidots). Thus, the diffusion in this system becomes anomalous and can be modeled by Lévy flights [2]. In the case of sufficiently large radius compared to the lattice constant, i.e., R/L > √ 3/4, on the other hand, collisionless long trajectories can no longer exist and the diffusion becomes normal. Therefore, we can expect that the quantum interference between electron paths in these systems is expected to reflect the difference between normal and anomalous diffusions. In this paper, we shall investigate the anomalous diffusion of quantum particles in finite-size Lorentz gas attached to the lead wires by use of the semi-classical theory.