SEMICLASSICAL THEORY OF TRANSPORT IN ANTIDOT LATTICES

Motivated by a recent experiment by Weiss et al. [Phys. Rev. Lett. 70, 4118 (1993)], we present a detailed study of quantum transport in lar ge antidot arrays whose classical dynamics is chaotic. We calculate th e longitudinal and Hall conductivities semiclassically starting from t he Kubo formula. The leading contribution reproduces the classical con ductivity. In addition, we find oscillatory quantum corrections to the classical conductivity which are given in terms of the periodic orbit s of the system. These periodic-orbit contributions provide a consiste nt explanation of the quantum oscillations in the magnetoconductivity observed by Weiss et al. We find that the phase of the oscillations wi th Fermi energy and magnetic field is given by the classical action of the periodic orbit. The amplitude is determined by the stability and the velocity correlations of the orbit. The amplitude also decreases e xponentially with temperature on the scale of the inverse orbit traver sal time h/T-y. The Zeeman splitting leads to beating of the amplitude with magnetic field. We also present an analogous semiclassical deriv ation of Shubnikov-de Haas oscillations where the corresponding classi cal motion is integrable. We show that the quantum oscillations in ant idot lattices and the Shubnikov-de Haas oscillations are closely relat ed. Observation of both effects requires that the elastic and inelasti c scattering lengths be larger than the lengths of the relevant period ic orbits. The amplitude of the quantum oscillations in antidot lattic es is of a higher power in Planck's constant h and hence smaller than that of Shubnikov-de Haas oscillations. In this sense, the quantum osc illations in the conductivity are a sensitive probe of chaos.