Chaos and Fractals in Nanostructure Billiards

Nanotechnology is used to construct semiconductor devices that induce chaos (an exponential sensitivity to initial conditions) in the flow of electrons over nano-scale distances. Due to the spectacular advances made in semiconductor growth and fabrication techniques, it is possible to study ballistic electrons - where the host material is so pure and the channel size so small that electrons travel along classical trajectories determined by the shape of the device channel (Fig. 1 bottom) rather than material induced scattering events (Fig. 1 top).

Figure 1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Figure 2

Figure 2 shows how the channel walls are formed using electrostatic gates deposited on the surface of the semiconductor. By applying a negative gate bias, depletion regions are formed in the sheet of electrons located below the surface at the interface between the layers of GaAs and AlGaAs. By shaping the gate patterns to form an enclosed region, the device becomes analogous to a billiard table (Fig. 3).

Figure 3 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Figure 4

The current project investigates billiard shapes designed to induce chaos in the classical electron trajectories. The Sinai billiard, shown in Fig. 4, is of particular interest. The 'empty' square has been predicted to support stable (i.e., non-chaotic) trajectories. By inserting a circle at the centre of the square, the billiard is transformed into the 'Sinai' geometry, named after the Russian chaologist who, back in 1972, predicted that this billiard would generate chaotic trajectories. Figure 5 shows the state-of-the-art multilevel gate architecture we use to investigate Sinai's proposal - the fundamental transition to chaotic behaviour in a controllable, physical environment. Whereas transitions to chaos have previously been observed in systems such as a pendulum and a dripping tap, here we induce the transition in the flow of fundamental particles - electrons. In addition to addressing fundamental aspects of chaology, the results are of interest to the electronics industry, where the ability to exploit the extreme sensitivity of chaotic behaviour is important. This work serves as a demonstration of the precision with which semiconductor technology can tune electronic properties of small devices.

Figure 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Figure 6

At milli-Kelvin temperatures, the quantum wave properties of electrons becomes important, allowing the study of 'quantum chaos' - the quantum behaviour of classically chaotic systems. As highlighted in Fig. 6, quantum behaviour can often be both surprising and remarkable! We found that the transition to the Sinai billiard was accompanied by the emergence of a form of fractal behavior known as exact self-similarity (ESS) in the billiard's magnetoconductance. This is shown in the left-hand image of Fig. 7. Exact self-similarity - the exact repetition of a pattern at different magnifications - is rare in physical systems. Whereas it is common in mathematical systems (see Fig. 7(left)), physical systems such as coastlines (see Fig. 7(right)) are described by another form of fractal behavior - statistical self-similarity (SSS), where the patterns at different magnifications are described simply by the same statistics. In contrast to the Sinai billiard, the 'empty' billiard's magnetoconductance obeys SSS. Thus the device of Fig. 4 represents a unique physical system where both forms of fractal behaviour - ESS and SSS - can be induced and the transition between the two forms studied.

Figure 7

Figure 8 shows a billiard which, by tuning the relative biases applied to two central gates, is being used to identify the precise geometry required to generate ESS. If this can be established, the result opens up many fascinating possibilities.

Figure 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Figure 9

One example, shown in Fig. 9, is designed to study how fractal systems 'add'. If two systems - one generating ESS (left) and the other SSS (right) - are added in series, is the combined system fractal? And if so, does the current direction determine whether the magnetoresistance shows ESS or SSS? This 'artificial' capacity to design fractal systems is not possible in nature. Another fundamental question being asked is 'how does fractal behaviour disappear?' We are answering this question by studying the dependence of the fractal dimension - the parameter frequently used to quantify fractal behavior. Perhaps surprisingly, we are finding that as we adjust the billiard parameters and suppress the quantum chaos, the range of magnifications over which we observe the fractal behaviour does not diminish. Instead, the fractal dimension gradually reduces until a the non-fractal value (unity) is reached. This work is being carried out in a collaboration with Arizona State University and RIKEN laboratories (Japan).

Figure 10

Coupled with these investigations, we are maximising the fractal effect (presently observed over 3 orders of magnitude in magnetic field) by refining the semiconductor environment. In a collaboration with the University of NSW in Australia, billiards are being investigated in a semiconductor system where electrons travel over a remarkable 100 microns (ie 100 times larger than the billiard) before suffering a material-induced scattering event. In a collaboration with Cambridge University (UK), the relationship between fractal behaviour and the degree of 'softness' in the billiard's electrostatic potential profile is being studied. This is demonstrated in Fig. 10. The system consists of two parallel electron sheets located at different depths. Formed by the same surface-gates, the billiard defined in the 'deeper' electron sheet is shaped by a 'softer' profile than for the 'shallow' case. This system will also be used to study the effect of electron interaction effects on the fractal phenomenon.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxx Figure 11 xxxxxxxxxxxxx xxxxxxxxx Figure 12 xxxxxxxxxxxxxxxxxxxx Figure 13

A collaboration with Nottingham University (UK) models the classical and quantum behavior of the billiards. The results indicate that the softness of the potential profile - shown in the simulation in Fig. 11 - is crucial. The softness generates a 'mixed trajectory system' composed of both chaotic and stable trajectories. Figure 12 shows a Poincare plot of the classical trajectories, revealing a remarkably rich behavior. Figure 13 shows a classical trajectory superimposed on the quantum wave function in the bottom-left corner of the Sinai Billiard. This comparison reveals a remarkable correspondence between classical and quantum behaviour - a phenomenon called 'scarring'. The central plunger is thought to act as a 'trajectory selector', controlling how the two families (chaotic and stable) of trajectories interact.

Figure 14

Figure 14 shows the resulting ESS for the model Sinai billiard. These theoretical studies are being extended to, and compared with, wave chaos in light. Recent work indicates that an analogous chaotic effect can occur in optical billiards (shaped glass cavities). This phenomenon is being pursued both in terms of fundamental research and potential applications.

Chaos and Fractals in Optical Billiards

Until recently, experiments on optical wave chaos have focussed on microwave cavities. However, following a recent theoretical paper that highlighted the potential of optical systems for the study and technical exploitation of wave chaos, the above model for soft-walled electron billiards has been extended to consider hard-walled optical billiards. These calculations show that an analogous mixed trajectory system and fractal behavior can be generated by specific billiard geometries. Whereas the electron quantum interference was varied using magnetic fields, in the optical system, the interference is varied by adjusting the incident light's wavelength. The ratio of billiard size to wavelength required for the predicted fractal effect is broad. Initial studies will focus on the direct analogies with fractals in electron billiards. In particular, a 'trajectory filter' procedure can also be used in the optical billiards to generate exact self-similarity. The exact self-similarity is generated by specific stable orbits that hit the billiard walls at selected locations. Thus if the optical billiard's walls are left unsilvered except at these specified locations, the 'unwanted' trajectories that generate statistical self-similarity will escape. Figure 15 shows Poincare sections obtained in simulations of optical billiards performed by our collaborators at the Nottingham University. These Poincare sections illustrate the evolution from a non-chaotic to a chaotic phase-space as the ends of a rectangular optical cavity are tilted with respect to the sides (ie. the cavity is transformed into a parallelogram). Experiments to examine such a system will commence in the coming few months.

Figure 15

Studies will be later extended to include a range of complex interference effects associated with the fractal ray dynamics, including the investigation of potential applications in the optics industry.

Selected References (1997-2006)

 

• "Can Ohmic Spikes Define Quantum Systems?" R.P. Taylor et al., Japanese Journal of Applied Physics 36, 3964, (1997).
• "Self-similar Magnetoresistance in a Semiconductor Sinai Billiard" R.P. Taylor et al., Physical Review Letters 78, 1952, (1997). This work is also the subject of the review "Fractal Resistance in a Transistor" by T.M. Fromhold, Nature 386, 124, (1997).
• "Correlation Analysis of Self-similarity in Semiconductor Billiards" R.P. Taylor et al., Physical Review B 56, R12733, (1997).
• "Unification of Exact and Statistical Self-similarity in Semiconductor Billiards" R.P. Taylor et al., Proceedings of the 24th International Conference on the Physics of Semiconductors (1998), Published in World Scientific , Chapter 7, A14 (1999).
• "Physical Realisation of Weierstrass Scaling Using a Quantum Interferometer" A.P. Micolich et al., Proceedings of the 11th International Conference on Superlattices, Microstructures, and Microdevices (1998), Published in Superlattices and Microstructures 25, 207 (1999).
• "Temperature Dependence of the Fractal Dimension of Magneto-conductance Fluctuations in a Mesoscopic Semiconductor Billiard" A.P. Micolich et al., Proceedings of the 11th International Conference on Superlattices, Microstructures, and Microdevices (1998), Published in Superlattices and Microstructures 25, 157 (1999).
• "Scale Factor Mapping of Self-similarity in Semiconductor Billiards" R.P. Taylor et al., Proceedings of the 1998 Conference on Optoelectronic and Microelectronic Materials and Devices, Published in IEEE Journal ISBN 0-7803-4513-4, 475 (1999).
• "Temperature Dependent Fractal Electron Transmission in Mesoscopic Billiards" A.P. Micolich et al., Proceedings of the 1998 Conference on Optoelectronic and Microelectronic Materials and Devices, IEEE Journal ISBN 0-7803-4513-4, 471 (1999).
• "Physical Realisation of Weierstrass Scaling in a Soft-wall Antidot Billiard" A.P. Micolich et al., Proceedings of the 1998 Conference on Optoelectronic and Microelectronic Materials and Devices, IEEE Journal ISBN 0-7803-4513-4, 468 (1999).
• "Geometry-induced Fractal Behaviour in a Semiconductor Billiard" A.P. Micolich et al., J. Phys. Condens. Matter 10, 1339 (1998).
• "Exact and Statistical Self-similarity in Magnetoconductance Fluctuations: a Unified Picture" R.P. Taylor et al., Physical Review B 58, 11107 (1998).
• "Observation of Fractal Conductance Fluctuations over Three Orders of Magnitude" R.P. Taylor et al., Proceedings of the 8th Gordon Workshop on Condensed Matter Physics, Australian Journal of Physics 52, 887 (1999).
• "Chaotic Ray Dynamics and Fast Optical Switching in Micro-cavities with a Graded Refractive Index" P.B. Wilkinson et al., Proceedings of the 11th International Conference on Hot Carriers in Semiconductors (1999), Physica B 272, 484 (1999).
• "Voltage and Temperature Limits for the Operation of a Quantum Dot Ratchet" H. Linke et al., Proceedings of the 11th International Conference on Hot Carriers in Semiconductors (1999), Physica B 272, 61 (1999).
• "Comment on Fractal Conductance Fluctuations in a Soft-wall Stadium and a Sinai Billiard" R.P.Taylor et al., Physical Review Letters 83, 1074 (1999).
• "Experimental Tunnelling Ratchet" H. Linke et al., Science 286, 2314 (1999).
• "A Physical Explanation for the Origin of Self-similar Magnetoconductance Fluctuations in Semiconductor Billiards" C.R. Tench et al., Proceedings of the 9th International Conference on Modulated Semiconductor Systems (1999), Physica E 7, 726 (2000).
• "Temperature and Size Dependence of Fractal MCF in Semiconductor Billiards" A.P. Micolich et al., Proceedings of the 3rd International Conference on Low Dimensional Structures and Devices (1999), Microelectronics Engineering 51-52, 241 (2000).
• "An Investigation of Weierstrass Self-similarity in a Semiconductor Billiard" A.P. Micolich et al., Europhys. Lett. 49, 417, (2000).
• "Electron Tunneling Ratchets" H. Linke et al., Proceedings of the 25th International Conference on the Physics of Semiconductors (2000), World Scientific CD-ROM (2001).
• "Semiconductor Billiards - a Controlled Environment to Study Fractals" R.P. Taylor et al., Physica Scripta T90, 41 (2001).
• "Evolution of Fractal Patterns in a Classical-Quantum Transition" A.P. Micolich et al., Physical Review Letters 87, 036802 (2001).
• "Electromagnetic Wave Chaos in Gradient Refractive Index Optical Cavities" P.B. Wilkinson et al., Physical Review Letters 86, 5466 (2001).
• "Chaos in Quantum Ratchets" H. Linke et al., Physica Scripta T90, 54 (2001).
• "Effects of Geometrical Ray Chaos on the Electromagnetic Eigenmodes of a Gradient Index Optical Cavity" P.B. Wilkinson et al., Physical Review E 64, 026203 (2001).
• "A Compact Fourth-Order Finite Difference Method for Solving Differential Equations" P.B. Wilkinson et al., Physical Review E 64, 047701, (2001).
• "Semiconductor Billiards: a Controlled Environment to Study Fractals" R.P. Taylor et al., Contribution to the Nobel Foundation book Y2K Quantum Chaos, World Scientific ISBN 981 02 4711-7 and The Royal Swedish Academy of Sciences, 41 (2001) (INVITED).
• "The Dependence of Fractal Conductance Fluctuations on Soft-wall Profile in a Double-2DEG Billiard" A.P. Micolich et al., Proceedings of the 14th International Conference on the Physics of Two Dimensional Systems (2001), Physica E 12, 841 (2002).
• "Discrete Energy Level Spectrum Dependence of Fractal Conductance Fluctuations in Semiconductor Billiards" A.P. Micolich et al., Proceedings of the 10th International Conference on Modulated Semiconductor Structures (2001), Physica E 13, 683-686 (2002).
• "The Dependence of Fractal Conductance Fluctuations on Semiconductor Billiard Parameters" A.P. Micolich et al., Proceedings of the 12th International Conference on Nonequilibrium Carrier Dynamics in Semiconductors (HCIS-12), Physica B 314, 477 (2002).
• "Quantum Ratchets Act as Quantum Heat Pumps" H. Linke et al., Proceeding of the 12th International Conference on Nonequilibrium Carrier Dynamics in Semiconductors (HCIS-12), Physica B 314, 464 (2002).
• "Quantum Ratchets and Quantum Heat Pumps" H. Linke et al., Applied Physics A 75, 237 (2002).
• "The Dependence of Fractal Conductance Fluctuations on Soft-wall Profile in a Double-layer Semiconductor Billiards" A.P. Micolich et al., Applied Physics Letters 80, 4381 (2002).
• "Dependence of Fractal Conductance Fluctuations on Semiconductor Billiard Parameters" A.P. Micolich et al., Proceedings of the Australian Institute of Physics Congress (2002), 15th Biennial Congress, Australian Institute of Physics, 367 (2002).
• "Fractal Conductance Fluctuations in Single and Double Layer Billiards" R.P. Taylor et al., 26th International Conference on the Physics of Semiconductors (2002).
• "Reversible Quantum Brownian Heat Engines for Electrons" T.E. Humphrey et al., Physical Review Letters 89, 116801 (2002).
• "A Review of Fractal Conductance Fluctuations in Ballistic Semiconductor Devices" R.P. Taylor et al., Chapter appearing in Electron Transport in Quantum Dots, ed. Jon Bird (Kluwer Academic/Plenum Publishers, New York, 2002).
• "Generic Fractal Behaviour of Ballistic Devices" R.P. Taylor et al., Proceedings of the 2002 Conference on Optoelectronic and Microelectronic Materials and Devices, 394, ed. M. Gal, Spinning Head Publications ISBN 0-7803-7571-8 (2003).
• "Fractal Transport Behaviour in Coupled-Dot Systems" Y. Ochiai et al., Proceedings of "Localization 2002" Tokyo (2003).
• "Geometry-Independence of Fractal Ballistic Processes" R.P. Taylor et al., Physica E 19, 225-229 (2003).
• "Surviving Conduction Symmetries in Non-linear Response" C.A. Marlow et al., Superlattices and Microstructures 34, 173 (2003).
• "The Influence of Confining Wall Profile on Quantum Interference Effects in Etched GaInAs/InP Billiards" T.P. Martin et al., Superlattices and Microstructures 34, 179 (2003).
• "Three Key Questions on Fractal Conductance Fluctuations: Dynamics, Quantization and Coherence" A.P. Micolich et al., Physical Review B 70, 085302 (2004) and The Virtual Journal of Nanoscale Science and Technology (APS), 10 (2004) (www.vjnano.org).
• "Symmetry of Two-Terminal Nonlinear Electric Conduction" A. Lofgren et al., Physical Review Letters 92, 046803 (2004).
• "Preserved Symmetries of Non-linear Electronic Conduction" C.A. Marlow et al., To be published in the Proceedings of the 27th International Conference on Physics and Semiconductors (2004).
• "Fractal Study of Coupling Transitions in Ballistic Quantum Dot Arrays" T.P. Martin et al., To be published in the Proceedings of the 27th International Conference on Physics and Semiconductors (2004).
• "A Novel Quantum Interference Probe of the Energy Spectrum of Coupled Nanodevices" T.P. Martin et al., Proceedings of the 2nd International Conference on Advanced Materials and Nanotechnology (2005), To be published in Current Applied Physics (Elsevier).
• "Series Summation of Fractal Fluctuations in Electron Billiards Arrays" T.P. Martin et al., To be published in the Proceedings of EP2DS (2005).
• "Non-linear Effects on Quantum Interference in Electron Billiards" C.A. Marlow et al., To be published in the Proceedings of the 14th International Conference on Hot Carriers in Semiconductors, Springer (2005).
• "A Unified Model of Electron Quantum Interference For Ballistic and Diffusive Semiconductor Devices" C.A. Marlow et al., Submitted to Physical Review B (2005).
• "Self-Propelled Film-Boiling Liquids" H. Linke et al., Submitted to Physical Review Letters (2005).
• "Experimental Investigation of the Breakdown of the Onsager-Casimir Relations" C.A. Marlow et al., Submitted to Physics Review Letters (2005).
• "Symmetry of Magnetoconductance Fluctuations of Quantum Dots in the Nonlinear Response Regime" C.A. Marlow et al., Submitted to Physical Review B (2006).
• "Mapping the Wall Profile of Etched GaInAs/InP Billiards" T.P. Martin et al., To be submitted, Journal of Applied Physics (2006).