Phase–coherence Effects in Antidot Lattices: A Semiclassical Approach to Bulk Conductivity

We derive semiclassical expressions for the Kubo conductivity tensor. Within our approach the oscillatory parts of the diagonal and Hall conductivity are given as sums over contributions from classical periodic orbits in close relation to Gutzwiller’s trace formula for the density of states. Taking into account the effects of weak disorder and temperature we reproduce recently observed anomalous phase coherence oscillations in the conductivity of large antidot arrays. 03.65Sq, 73.50.Jt Typeset using REVTEX Unité de Recherche des Universités Paris XI et Paris VI associée au CNRS 1 Interference phenomena in quantum electron transport through small microstructures are usually interpreted within two complementary frameworks: The Landauer–Büttiker formalism commonly describes transmission through single phase coherent devices [1]. The current is expressed in terms of (a sum over) conductance coefficients between channels in different leads attached. On the other hand linear response theory (Kubo formalism) has proved useful to treat bulk transport properties of samples with a size exceeding the phase breaking length. Besides coherence effects related to the presence of disorder (e.g., universal conductance fluctuations, weak localization) the development of high mobility devices has opened experimental access to the ballistic limit where the elastic mean free path is considerably larger than the system size and the conductance reflects the geometry or potential landscape of the microstructures. This has especially oriented interest to questions how classically regular or chaotic electron dynamics manifests itself on the level of quantum transport [2–4]. In this spirit a semiclassical approach to conductance within the Landauer–Büttiker framework has already been successfully performed expressing the conductance coefficients in terms of interfering electron paths [4]. Recent experiments on magnetotransport in antidot structures unveiled the lack of a corresponding semiclassical approach to Kubo bulk conductivity [3,5,6]. Antidot superlattices consist of arrays of periodically arranged holes ”drilled” through a two dimensional electron gas (2DEG). Since the lattice constants a (of typically 200–300 nm) are significantly larger than the Fermi wavelength λF ∼ 50 nm the dynamics of the electrons moving in between the repulsive antidots can be considered to be of semiclassical nature. The combined potential of the superlattice and a perpendicular magnetic field gives rise to a variety of peculiar effects: The diagonal magnetoresistivity ρxx exhibits pronounced peaks due to trapping of electrons encompassing a particular number of antidots at magnetic field strengths related to specifc values of the normalized cyclotron radius Rc/a [7]. Superimposed upon these resistivity peaks (reflecting the classical chaotic electron dynamics of the antidot geometry [8]) additional quantum resistivity oscillations of anomalous periodicity had been observed at sufficiently low temperature indicating phase coherence phenomena [3,5]. They cannot 2 be attributed to interfering electron waves traversing the whole device (as in the case of single junctions) since the antidot arrays are too large to maintain phase coherence. However, assuming that ρxx reflects density of state oscillations we were able to describe the periodicity of the modulations observed in terms of quantized periodic orbits in the antidot array [3]. Nevertheless, a complete direct semiclassical approach to the conductivity tensor was still missing [9] in spite of important work by Wilkinson providing a semiclassical evaluation of the diagonal conductivity in a somewhat different context [10]. Such an approach is also desirable since the antidot measurements have not yet been completely reproduced by quantum mechanical calculations which turn out to be rather involved [11]. In this letter we derive semiclassical Kubo–type transport formulas by stationary phase evaluation of the disorder averaged two–particle Green functions. We obtain diagonal and Hall conductivity oscillations in terms of sums over periodic orbit contributions with the classical actions of the orbits determining the periodicities. Using a model antidot potential and working at finite temperature we are able to reproduce the amplitudes and periodicities of measured quantum oscillations in antidot superlattices. Within the Kubo formalism the static conductivity tensor at temperature T is given by