Semiclassical theory of transport in antidot lattices

We present a detailed study of quantum transport in large antidot arrays whose classical dynamics is chaotic. We calculate the longitudinal and Hall conduc- tivities semiclassically starting from the Kubo formula. The leading contribu- tion reproduces the classical conductivity. In addition, we find oscillatory quantum corrections to the classical conductivity which are given in terms of the periodic orbits of the system. These periodic-orbit contributions provide a consistent explanation of the quantum oscillations in the magnetoconductivity observed by Weiss et al.\ [Phys.\ Rev.\ Lett.\ {\bf70}, 4118 (1993)]. We find that the phase of the oscillations 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 exponentially with temperature on the scale of the inverse orbit traversal time $\hbar/T_\gamma$. We also present a semiclassical derivation of Shubnikov-de Haas oscillations where the corresponding classical motion is integrable. We show that the quantum oscillations in antidot lattices and the Shubnikov-de Haas oscillations are closely related. However, the amplitude of the quantum oscillations in antidot lattices is of a higher power in Planck's constant $\hbar$ and hence smaller than that of Shubnikov-de Haas oscillations. In this sense, the quantum oscillations in the conductivity are a sensitive probe of chaos.