Antidot superlattices: Classical chaos and quantum transport

Abstract

Antidot superlattices represent a model system to study electron transport through a periodic potential. Starting from a high-mobility two-dimensional electron gas a periodic array of potential pillars exceeding the Fermi energy in height can be fabricated by various technological means. Usually the electron mean free path is much larger than the lattice period while the Fermi wavelength is typically an order of magnitude smaller than characteristic features of the artificial superlattice. In this so-called classical ballistic transport regime pronounced maxima occur in the magnetoresistance being related to regular electron trajectories around groups of antidots. Theories based on the classical chaotic motion of the electrons in the antidot potential landscape are able to explain the experimental observations quantitatively. In a rectangular geometry the transport properties depend on the direction of the current flow with respect to the lattice orientation. If the electrons flow through the closely spaced antidots electron orbits around one, two or more antidots that are now symmetry allowed lead to maxima in the magnetoresistance. In contrast if the current flows through the wide open channels between the rows of antidots the magnetoresistance is only influenced by electron orbits whose cyclotron diameter is comparable in size to the large period of the lattice. Basic symmetry relations (e.g. Onsager's relation) can be tested with these experiments. Since the antidot systems are so well understood in the classical limit the experiments can be used to play with various lattice symmetries. The basic results of these observations persist into the quantum mechanical regime. Finite antidot lattices are fabricated where an array of e.g. 9 x 9 antidots is surrounded by a square geometry. For very low temperatures, T < 100 mK, electron-electron scattering is reduced and the phase coherence length of the electrons may exceed the size of the total systems. The classical commensurability oscillations are now superimposed by strong reproducible fluctuations that die out for large magnetic fields, at which the cyclotron diameter becomes smaller than the lattice period. A Fourier analysis reveals B-periodic features in the magnetic field regime where the electrons classically encircle groups of antidots. We find that the area, that can be calculated from the sequential addition of a flux quantum, corresponds to the area of the classical cyclotron orbit around a group of antidots. We argue that the electrons travel phase coherently along classical trajectories. In finite rectangular lattices this argument is supported by the fact that such B-periodic oscillations only occur in the current direction where transport is influenced by the respective classical orbit. We conclude that antidot superlattices represent a versatile system to study experimentally the crossover from classical electron trajectories to quantum mechanical wave functions.