Evolution of Fractal Patterns during a Classical-Quantum Transition

Abstract

We investigate how fractals evolve into nonfractal behavior as the generation process is gradually suppressed. Fractals observed in the conductance of semiconductor billiards are of particular interest because the generation process is semiclassical and can be suppressed by transitions towards either fully classical or fully quantum-mechanical conduction. Investigating a range of billiards, we identify a "universal" behavior in the changeover from fractal to nonfractal conductance, which is described by a smooth evolution rather than deterioration in the fractal scaling properties.

Fractal conductance fluctuations in electron billiards: a random matrix theory approach

“…We now can use the semiclassical result (15) to estimate the value of the magnetic correlation field B C in recent experiments on FCF [2][3][4][5]. In these experiments, one has typically N c ∼ 2, n e ∼ 10 2 , A ∼ 1µm 2 and CL ∼ 10 −1 so that the correlation field is of the order of B C ∼ 10 −3 T. Some of these experiments [2][3][4] were performed with magnetic field values around or greater than B C ∼ 10 −3 T, which we believe to make the interpretation of the observed conductance fluctuations as genuine FCF ambiguous.…”

“…Triggered by the pioneer theoretical prediction by Ketzmerick [1], fractal conductance fluctuations (FCF) have been experimentally observed in semiconductor billiards by several groups [2][3][4][5]. These billiards are sub-micron sized electron cavities bounded by shaped walls and connected to twodimensional electron gas reservoirs by quantum point contact leads.…”

Effects of Dissipation and Noise on Chaotic Transport in Superlattices

“…In semiconductor physics, chaotic electron transport has been explored using a variety of two-dimensional billiard structures [1][2][3][4][5][6][7][8][9][10], antidot arrays [1,2,[11][12][13] and resonant tunneling diodes containing a wide quantum well enclosed by two tunnel barriers [1,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Despite the diversity of the systems studied in these previous works, 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.…”

Simulation of electron transport through a quantum dot with soft walls

“…First experimental data 14,15,16,17,18 appear to support this notion. However, in the corresponding numerical studies, no fractal structure in the conductance fluctuations could be found, 5,6,7,19 which, in part, has led to a number of theoretical works that propose alternative and sometimes even contradictory explanations for FCF.…”

“…14,15,16,17,18 These experiments were in the regime where only a rather limited number of transmitting modes m, n = 2 − 6 is open. 15 Our results clearly show that the presence of soft walls in the experiment can be ruled out as the source for FCF at moderate k F . Note that this observation is in close correspondence to recent findings which suggest that "more complicated processes than those predicted in the semiclassical models are responsible for the observed behavior of FCF".…”