Time evolution of initially localized states in two-dimensional quantum dot arrays in a magnetic field

The evolution of non-stationary localized states jCðt 1⁄4 0Þi is investigated in two-dimensional tight binding systems of N potential wells with and without a homogeneous field perpendicular to the plane. Most results are presented in analytical form, what is almost imperative if the patterns are as complex as for rings in a magnetic field, where the qualitatively different features arise depending on rational or irrational numbers. The systems considered comprise finite linear chains ðN 1⁄4 2; 3Þ, finite rings (N 1⁄4 3–6), infinite chains, finite rings (N 1⁄4 3–6) in a magnetic field, and rings with leads attached to each ring site. The position of the particle at time t is described by the projection of the wave function PmðtÞ 1⁄4 jhmjCðtÞij onto the localized basis function at site m. For finite chains and rings with N 1⁄4 3; 4; 6 the time evolution is periodic, whereas it is non-periodic for N 1⁄4 5 and N greater then 6. Rings in a magnetic field show a rich spectrum of different features depending on N and the number of flux quanta through the ring, including periodic oscillation and rotation of the charge as well as non-periodic charge fluctuations. r 2004 Elsevier B.V. All rights reserved. PACS: 73.23. b; 73.40.Gk; 73.50. h