The thermopower of a multiprobe ballistic conductor in the form of caterpillarlike Sinai billiard is experimentally investigated. The magnetic-field dependence of both longitudinal thermopower and Nernst-Ettingshausen effect exhibits commensurability oscillations, which are more pronounced than the corresponding oscillations in the magnetoresistance. Results of computer calculations based on the generalized Landauer-Büttiker approach are in agreement with experiment. The observed features in the thermopower originate from drastic difference between the transmission coefficients of quasielectrons ͑above the Fermi level͒ and quasiholes ͑below the Fermi level͒ in the vicinity of geometrical resonances.

We report on B-periodic oscillations in the magnetoresistance of a circular magnetic billiard. These arise from singly periodic orbits that return to their point of departure after n bounces on the billiard. A semiclassical calculation of the magnetic field increment required to depin annulus n and n + 1 is in good agreement with the measured period of the resistance oscillations.The magnetoresistance of two-dimensional electron billiards depends critically on the shape of sample boundaries and gives unique insight into the development of classical chaos. 1 Precision lithography and dry etching of high mobility two-dimensional electron gases (2DEGs) have made possible the realization of the well-defined boundaries that are required for these studies. Recent work has focused on antidot lattices, 1-3 elliptical antidots, 4,5 and magnetic stadia. 6 These geometries acquire nonintegrable electron dynamics once a magnetic field is applied. Their phase space shows stable islands, consisting of periodic orbits, surrounded by a chaotic sea. Magnetoresistance peaks were explained by the pinning of periodic orbits to hard wall scatterers. 1 Chaotic trajectories, by contrast, were assumed to run away hence contributing to the conductivity. This pinned/runaway picture was placed on a firm theoretical ground by recent numerical simulations that showed chaotic orbits to be easily swept away by a small electric field. 4,7 Periodic orbits were found to have a long dwell time and to affect the magnetoresistance through both the pinning/depinning mechanism 1,4 and their correlations to the chaotic sea. 7 Integrable systems, such as circular billiards, are also very interesting because the regular/chaotic duality is replaced by the single/ multiperiodicity of the orbits. 8 Single periodic orbits return to the point of departure after just one revolution around the billiard. They only differ by the number of bounces, n, it takes to encircle the billiard. By contrast, multiperiodic orbits return to their point of departure after two or more revolutions and as such have much longer periods. Real billiards have finite mean free path and are subject to small electric fields during measurements. One may therefore expect that the predominant features of the magnetoresistance will arise from the pinning of the shortest orbits that are the most robust against these perturbations. 9In this paper we demonstrate the discrete annular structure of single period magnetic edge states 10-13 that drift in a loop of magnetic gradient. This semiclassical annular structure is a universal feature of circular potentials, whether magnetic or electrostatic, when placed in a magnetic field. Its observation demonstrates the wavelength of the meandering magnetic edge states as a physical length scale. A thin ferromagnetic disk was fabricated at the surface of a shallow GaAs/ AlGaAs heterostructure to modulate the twodimensional electron gas with corral-shaped magnetic field profile. The magnetoresistance measured across the corral showed quasiperiodic oscilla...

We have measured the phonon-drag thermopower of a periodically modulated two-dimensional electron gas, and report on a complex series of oscillations developing in the presence of a perpendicular magnetic field. At low electron density these oscillations are in phase with the commensurability resistance oscillations, however they become increasingly antisymmetrical with respect to B at higher electron density. We are able to explain this magnetic field dependence by proposing that the periodic potential lowers the symmetry of electron-phonon interaction. We calculate the thermopower by solving the Boltzmann equation given an electron-phonon scattering term with two-fold symmetry. Our fit of the experimental curves shows that the Brillouin zone folding enhances the electron-phonon scattering rate by a factor of two along the periodic potential and that a small misalignment (6 • ) of the heat gradient with the direction of the periodic potential is sufficient to explain the antisymmetrical oscillations. Our experiment demonstrates phonon drag as a very sensitive tool to probe the electronic and vibronic anisotropy in mesoscopic systems.

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