BSc, PhD Nottingham
University
Richard P. Taylor
C.A.D. Manchester School of ArtHead of Condensed Matter
Physics
Queen Elizabeth II Fellow
Condensed Matter Physics
In April 2000, I will be become an Associate Professor in the Physics Department at the University of Oregon, USA. Facilities in the lowtemperature laboratory will be based around a Kelvinox AST (advanced sorption pump technology) dilution fridge. Working with a postdoc (position to be advertised in Physics Today) and PhD students, the fridge will be used to study both the fundamental and applied physics of mesoscopic semiconductor systems. Current collaborations with Nottingham (UK), Cambridge (UK), UNSW (Australia) and Arizona State Universities will continue, and these will be integrated with projects within the Materials Science Institute at the University of Oregon. My current research is described below.
Due to the spectacular advances made in semiconductor growth and fabrication technologies, it is now possible to study billiards  electronic devices where the host material is so pure that electrons travel along classical trajectories determined by the shape of the device cavity rather than by materialinduced scattering events. In Fig. 1, surface gates are used to electrostatically define a submicron billiard within the two dimensional electron gas (2DEG) located at the interface of an AlGaAs/GaAs heterostructure.
The current project investigates billiard shapes designed to induce chaos (an exponential sensitivity to initial conditions) in the classical electron trajectories. The Sinai billiard, shown in Fig. 2, is of particular interest. The 'empty' square has been predicted to support stable (ie nonchaotic) 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. Fig. 3 shows the stateoftheart 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.
At milliKelvin 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. 4, quantum behaviour can often be both surprising and remarkable! We found that the transition to the Sinai billiard was accompanied by the emergence of exact selfsimilarity (ESS) in the billiard's magnetoresistance. This is shown in the middle image of Fig. 5. Exact selfsimilarity  the exact repetition of a pattern at different magnifications  is rare in physical systems. Whereas it is common in mathematical systems (see Fig. 5(a) ), physical systems such as coastlines (see Fig. 5(b) ) are described by another form of fractal behaviour  statistical selfsimilarity (SSS), where the patterns at different magnifications are described simply by the same statistics. In contrast to the Sinai billiard, the 'empty' billiard's magnetoresistance obeys SSS. Thus the device of Fig. 3 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. Possible applications include multilogic and quantum computing.
Fig. 6 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.
One example, shown in Fig. 7, 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 temperature dependence of the fractal dimension  the parameter frequently used to quantify fractal behaviour. Perhaps surprisingly, we are finding that as we raise the temperature (and therefore suppress the quantum character of the electrons) the range of magnifications over which we observe the fractal behaviour does not diminish. Instead, the fractal dimension gradually reduces until a the nonfractal value (unity) is reached. This work is being carried out in a collaboration with Arizona State University and RIKEN laboratories (Japan).
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 Semiconductor Nanofabrication Facility (SNF) in Sydney, billiards are being constructed in a semiconductor system where electrons travel over a remarkable 100 microns (ie 100 times larger than the billiard) before suffering a materialinduced 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. 8. The system consists of two 2DEG layers located at different depths. Formed by the same surface gates, the billiard defined in the 'deeper' 2DEG 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.

A collaboration with Nottingham University (UK) models the classical and quantum behaviour of the billiards. The results indicate that the softness of the potential profile  shown in the simulation in Fig. 9  is crucial. The softness generates a 'mixed trajectory system' composed of both chaotic and stable trajectories. The central plunger is thought to act as a 'trajectory selector', controlling how the two families of trajectories interact. Fig. 10 shows a classical trajectory superimposed on the quantum wave function in the bottomleft corner of the Sinai Billiard. This comparison reveals a remarkable correspondence between classical and quantum behaviour  a phenomenon called 'scarring'. Fig. 11 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.
Pattern analysis techniques developed to detect fractals in data plots of the semiconductor devices are being refined to analyse a wide range of different patterns and this has triggered a number of successful interdisciplinary collaborations both internationally and within Australia. For example, a project with the Prince of Wales Hospital, Sydney, analyses tissue patterns in the temporal region of the human brain to investigate behaviour such as speech limitations. The use of fractal analysis as a medical diagnostic tool to distinguish healthy tissue is also being considered for commercial development. The same technique has been refined to bring a scientific objectivity to the aesthetic evaluation of abstract patterns in modern art. In particular, the infamous paintings of the Abstract Expressionist Jackson Pollock have been shown to be fractal. This result was the subject of a 30 minute TV special (ABC TV's 'The Art of Science') broadcast in May 1998. One of Pollock's fractal paintings is shown in Fig. 12.
For Further Information see Fractal Expressionism
School of Physics
The University of New South Wales
SYDNEY 2052
Australia
rpt@phys.unsw.edu.au
61 2 9385 4572
61 2 9385 6060