Fractal Conductance Fluctuations Research Articles

Fractal conductance fluctuations of classical origin

In mesoscopic systems, conductance fluctuations are a sensitive probe of electron dynamics and chaotic phenomena. We show that the conductance of a purely classical chaotic system, with either fully chaotic or mixed phase space, generically exhibits fractal conductance fluctuations unrelated to quantum interference. This might explain the unexpected dependence of the fractal dimension of the conductance curves on the (quantum) phase breaking length observed in experiments on semiconductor quantum dots.

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

Motivated by recent experiments and theoretical works that contradict the original explanation for fractal conductance fluctuations (FCF) in electron billiards, based on the mixed structure of the classical phase space, we propose an alternative approach to investigate FCF using Random Matrix Theory (RMT). By means of a semiclassical estimate for value of the magnetic correlation field BC we conclude that most of the experiments on FCF were performed for magnetic fields around or greater than BC. This strongly suggests that the appropriate explanation for the observed FCF should rely on the absence of long-range correlations and not on the structure of the classical phase space. This idea is supported by a numerical study of parametric variations within the framework of RMT, which validates our surmise that the observed FCF actually reflect a diffusive scenario for electronic transport.

Unified model of fractal conductance fluctuations for diffusive and ballistic semiconductor devices

We present an experimental comparison of magnetoconductance fluctuations measured in the ballistic, quasiballistic, and diffusive scattering regimes of semiconductor devices. In contradiction to expectations, we show that the spectral content of the magnetoconductance fluctuations exhibits an identical fractal behavior for these scattering regimes and that this behavior is remarkably insensitive to device boundary properties. We propose a unified model of fractal conductance fluctuations in the ballistic, quasiballistic, and diffusive transport regimes, in which the generic fractal behavior is generated by a subtle interplay between boundary and material-induced chaotic scattering events.

Simulation of electron transport through a quantum dot with soft walls

We numerically investigate classical and quantum transport through a soft-wall cavity with mixed dynamics. Remarkable differences to hard-wall quantum dots are found which are, in part, related to the influence of the hierarchical structure of classical phase space on features of quantum scattering through the device. We find narrow isolated transmission resonances which display asymmetric Fano line shapes. The dependence of the resonance parameters on the lead mode numbers and on the properties of scattering eigenstates are analyzed. Their interpretation is aided by a remarkably close classical-quantum correspondence. We also searched for fractal conductance fluctuations. For the range of wave numbers $k_F$ accessible by our simulation we can rule out their existence.

Three key questions on fractal conductance fluctuations: Dynamics, quantization, and coherence

Recent investigations of fractal conductance fluctuations (FCF) in electron billiards reveal crucial discrepancies between experimental behavior and the semiclassical Landauer-Buttiker (SLB) theory that predicted their existence. In particular, the roles played by the billiard's geometry, potential profile and the resulting electron trajectory distribution are not well understood. We present measurements on two custom-made devices - a 'disrupted' billiard device and a 'bilayer' billiard device - designed to probe directly these three characteristics. Our results demonstrate that intricate processes beyond those proposed in the SLB theory are required to explain FCF.

Imaging fractal conductance fluctuations and scarred wave functions in a quantum billiard.

We present scanning-probe images and magnetic-field plots which reveal fractal conductance fluctuations in a quantum billiard. The quantum billiard is drawn and tuned using erasable electrostatic lithography, where the scanning probe draws patterns of surface charge in the same environment used for measurements. A periodicity in magnetic field, which is observed in both the images and plots, suggests the presence of classical orbits. Subsequent high-pass filtered high-resolution images resemble the predicted probability density of scarred wave functions, which describe the classical orbits.

The influence of confining wall profile on quantum interference effects in etched Ga 0.25In 0.75As/InP billiards

We present measurements of the potential profile of etched GaInAs/InP billiards and show that their energy gradients are an order of magnitude steeper than those of surface-gated GaAs/AlGaAs billiards. Previously observed in GaAs/AlGaAs billiards, fractal conductance fluctuations are predicted to be critically sensitive to the billiard profile. Here we show that, despite the increase in energy gradient, the fractal conductance fluctuations persist in the harder GaInAs/InP billiards.

Fractal conductance fluctuations in a quantum-dot molecule

Exactly self-similar, and statistically self-similar, features have been studied in the magneto-conductance of an open quantum-dot molecule. The fractal dimensions determined from the two different features are found to differ significantly from each other. While we do not understand the detailed origins of this difference at present, our results suggest that different physical processes are responsible for the two different types of fractal structure.

Dependence of fractal conductance fluctuations on soft-wall profile in a double-layer semiconductor billiard

We present a semiconductor system featuring two billiards located one on top of the other. We use this system to study the dependence of fractal conductance fluctuations on soft-wall potential profile and show the fluctuations to be surprisingly robust to changes in profile.

Discrete energy level spectrum dependence of fractal conductance fluctuations in semiconductor billiards

We present experimental results demonstrating that fractal conductance fluctuations are remarkably robust to deviations from the semiclassical conduction regime. We find that the fractal dimension depends solely on an empirical parameter that quantifies the resolution of the billiard's discrete energy level spectrum. Non-fractal fluctuations are obtained only in the extreme classical and quantum-mechanical conduction regimes.

The dependence of fractal conductance fluctuations on semiconductor billiard parameters

We present a comprehensive study of the dependence of fractal conductance fluctuations (FCF) on semiconductor billiard parameters. We find that the fractal dimension depends solely on an empirical parameter that quantifies the resolution of the discrete energy level spectrum of the billiard. We have also investigated the effect of profile softness using billiards formed in a double two-dimensional electron gas heterostructure and find that FCF are robust to the changes in profile softness achieved.

The dependence of fractal conductance fluctuations on soft-wall profile in a double-2DEG billiard

We have developed and investigated a double-2DEG billiard to study the dependence of fractal conductance fluctuations on the soft-wall potential profile in semiconductor billiards. We use numerical modeling to establish how the profile differs between the billiards. We present preliminary results showing that the changes in profile modify the fractal dimension rather than suppress the fractal behavior, and that the fractal dimension depends inversely on the softness of the potential profile.

Numerical simulation of fractal conductance fluctuations in soft-wall quantum cavities

Magnetoconductance fluctuations in quantum cavities are numerically calculated for soft-wall confinement potentials. Employing a tight-binding lattice consisting of $200\ifmmode\times\else\texttimes\fi{}200$ sites, the conductance fluctuations are confirmed to be fractal for one order of magnitude in magnetic-field scale. We find no emergence of self-affine conductance fluctuations with fine scales in a Sinai billiard, suggesting that the hierarchical phase space structure in the underlying classical dynamics alone cannot explain recent experimental observations.

Comment on “Fractal Conductance Fluctuations in a Soft-Wall Stadium and a Sinai Billiard”

A Comment on the Letter by A. S. Sachrajda et al., Phys. Rev. Lett. 80, 1948 (1998). The authors of the Letter offer a Reply.

Observation of Fractal Conductance Fluctuations over Three Orders of Magnitude

Fractal magneto-transport properties of mesoscopic semiconductor billiards are highly topical. In these studies, the magnetic field range over which fractal behaviour can be observed is crucial. Previous observations have been limited to approximately one order of magnitude. We present fractal conductance fluctuations observed over three orders of magnitude and discuss the physical conditions required to extend this range.

Fractal Conductance Fluctuations in Gold Nanowires.

A detailed analysis of magneto-conductance fluctuations of quasiballistic gold-nanowires of various lengths is presented. We find that the variance $<(\Delta G)^2> = < (G(B)-G(B+\Delta B))^2>$ when analyzed for $\Delta B$ much smaller than the correlation field $B_c$ varies according to $<(\Delta G)^2>\propto \Delta B^{\gamma}$ with $\gamma < 2$ indicating that the graph of $G$ vs. $B$ is fractal. We attribute this behavior to the existence of long-lived states arising from chaotic trajectories trapped close to regular classical orbits. We find that $\gamma$ decreases with increasing length of the wires.

Fractal conductance fluctuations in generic chaotic cavities.

It is shown that conductance fluctuations due to phase coherent ballistic transport through a chaotic cavity generically are fractals. The graph of conductance vs. externally changed parameter, e.g. magnetic field, is a fractal with dimension D=2-b/2 between 1 and 2. It is governed by the exponent b (<2) of the power law distribution P(t) ~ t^{-b} for a classically chaotic trajectory to stay in the cavity up to time t, which is typical for chaotic systems with a mixed (chaotic and regular) phase space. The phenomenon should be observable in semiconductor nanostructures and microwave billiards.