Quantum conductance fluctuations and classical short-path dynamics

Abstract

We present numerical results for ballistic-electron quantum transport through weakly open integrable circle and chaotic stadium billiards. The geometry of the pair of conducting leads is chosen in accordance with recent experiments for semiconductor microstructures I Marcus et al , Phys. R. ev. Lett. 69, 506 (1992)].The conductance as a function of the Fermi wave number displays characteristic noisy fluctuations for both the integrable and the chaotic systems. We show that structures in the conductance autocorrelation function as a function of the Fermi wave number are related to short-length classical orbits. This correspondence permits incorporation of effects of phase decoherence due to incoherent scattering into the quantum calculation.During the last decade, remarkable advances in semiconductor fabrication technology have made it possible to produce quantum devices of submicrometer size. ' The size of such mesoscopic devices is large on the atomic scale but comparable to or smaller than the phase-coherence length l~d ue to inelastic electron-electron (and electron-phonon) scattering at low temperatures ((0.1 K). Both experimentally and theoretically, there has been considerable interest in the electronic and magnetic properties of such systems and as a result, a number of physical phenomena have been observed and predicted including universal conductance fiuctuations, weak localization, and Aharonov-Bohm conductance oscillations. One of the most interesting aspects is chaotic boundary scattering in systems formed in highmobility GaAs/Al"Ga, As heterostructures, where the system size is less than the elastic mean free path of electrons. In such systems, the two-dimensional motion of noninteracting electrons confined by a wall of arbitrary geometry is ballistic. In the ballistic regime, the shape of walls directly determines the motion of electrons colliding with them, and the classical "billiard" model successfully explains the experimental results of those systems. Nonlinear dynamics of the electrons play an important role in quantum transport through the microstructures (see, e. g., Ref. 6, and references therein).One of the prototypes of conservative chaotic systems is the "stadium billiard. " The study of its quantum mechanics has played an important role in the identification of quantum manifestations of classical chaos. ' Studies of opened billiards pertaining to transport through mesoscopic devices have only recently begun. ' Major progress has been made in relating quantum conductance fluctuations with classical (non)integrable dynamics' ' employing Miller's semiclassical formulation of the S matrix. Generic fingerprints of regular and chaotic dynamics were identified that are related to long classical orbits and are universal (i.e., system independent). Several quantum calculations (e.g., Refs. 6, 13, and 17) were performed for a symmetric geometry for which entrance and exit leads are on opposite sides of the device facing each other. Very recently, Marcus et al. ' reported experimental results of...

Modular recursive Green’s function method for ballistic quantum transport

“…Studying the power spectrum derived from the quantum transmission amplitudes ͉ t nm (L)͉ 2 , we can gain information about the underlying classical dynamics of the scattering process. [13][14][15]38 High resolution of the L spectra requires data for a large interval of wave numbers ͓k min ,k max ͔ such that the resulting width of structures in L is not Fourier transform limited but reveals information about the intrinsic peak width due to the spread of L within the bundles of trajectories. In all the data presented in this paper, the transform limited length width is ⌬L FT ϭ /(k max Ϫk min )Ϸ0.054 for the circle (Ϸ0.11 for the stadium͒.…”

Gauge-invariant theory for semiclassical magnetotransport through ballistic microstructures

“…One important difference to previously investigated length spectra [16][17][18] ͑power spectra of t mn as a function of k) is that the association of pronounced peaks with individual bundles of trajectories is not clear cut. One reason for this is that the directed area, unlike the length, is not positive definite.…”

“…The power spectrum at fixed k F can be identified with the spectrum of areas enclosed by classical trajectories. Similar to the case of the length spectrum for the field free transmission, [16][17][18]28 the choice of the integration interval of the Fourier integral ͓0,B max ͔ has nontrivial consequences beyond the limitation of the resolution. Since the curvature of the paths and, hence, the enclosed areas change as functions of the magnetic field, the resulting area spectrum is to be understood as an average over area distributions pertaining to different fields.…”

“…͑14͔͒, some of which have already been discussed in the literature. [10][11][12][13][14][15][16][17][18][19] We recall that Eq. ͑14͒ employs the semiclassical Green's function, or Fourier-Laplace transform of the Van Vleck propagator, 22,9 for the motion in the interior of the billiard structure.…”

“…Several semiclassical approaches have been applied to the open billiard problem. [10][11][12][13][14][15][16][17][18][19] While significant progress has been made, there remain considerable obstacles to a consistent semiclassical description of ballistic quantum transport in the presence of a magnetic field, some of which are the subject of this paper.…”