Magnetoconductance fluctuations and weak localization in quantum dots: Reliability of the semiclassical approach

Abstract

We utilize a semiclassical ͑SC͒ approach to calculate the conductance and weak-localization ͑WL͒ corrections in a triangular billiard of a given shape in the presence of nonzero magnetic field. The semiclassical conductance is given as a sum of all classical trajectories between the leads, each of them carrying the quantum-mechanical phase. The present SC approach is numerically exact ͑i.e., free from any approximations͒, explicitly includes diffractive effects in the leads, and is valid for arbitrary ͑low͒ mode numbers in the leads. We find however that the symmetry of the SC conductance/reflectance with respect to the direction of magnetic field or direction of the current is not satisfied, as well as that the WL corrections for the conductance and reflectance are inconsistent with each other; the SC approach does not satisfy the current conservation requirements and does not reproduce the corresponding exact quantum-mechanical results. The reason for that is traced to the topological difference in the sets of classical trajectories between the leads for different current or magnetic field directions which determine the conductance in the SC approximation. Our findings raise a question as to what extent one can rely on numerous predictions for statistical properties of the conductance oscillations of ballistic cavities including the WL line shapes and fractal conductance which were essentially based on the standard SC approach.

Diffraction and boundary conditions in semiclassical open billiards

“…These trajectories are not included in this SC model. 18,20 The included billiard geometries have been specially chosen to avoid that ''ghost'' trajectories coincide in length with normal trajectories which otherwise makes comparison with QM computations difficult.…”

“…One should, however, be careful in relying on these results. 18 The SC approach can also provide an interpretation of specific frequencies in the conductance oscillations, by relating them to specific classical trajectories in a billiard. In calculating conductance or transmission amplitudes of a system, a SC approximation of the systems Green's function is used.…”

Diffractive paths for weak localization in quantum billiards

“…For regular billiards, classical paths alone are insufficient to reproduce quantum transport. 19 To uncover which mechanism for WL is at work for this class of systems, we focus, in this paper, on a prototype structure for a ballistic regular cavity: the circular quantum billiard with perpendicular leads ͓Fig. 1͑a͔͒.…”

Stochastic Carrier Dynamics in Semiconductor Superlattices

“…In semiconductor physics, chaotic electron transport has been explored using a variety of two-dimensional billiard and antidot barrier structures [1][2][3][4][5][6][7][8][9][10][11][12], in superlattices (SLs) [13][14][15], and in resonant tunnelling diodes containing a wide quantum well with a tilted magnetic field [1,[16][17][18]. Despite the diversity of these experiments, they all involve systems in which the transition to chaos occurs by the gradual and progressive destruction of stable orbits in response to an increasing perturbation.…”