Chris Pownall, Postdoctoral Research Fellow Lehrstuhl 5 für theoretische Physik Physik Department Technische Universität München  James-Franck-Strasse D85747 Garching bei München Germany Telephone: +49-89-289-12356 Room number: 3223 E-mail: cdp@ph.tum.de

Career:

PhD Thesis:

The nucleation, growth and coalescence of islands on stepped substrates is investigated by Monte Carlo simulations and analytical theories. Substrate steps provide a preferential site for the nucleation of island, making many of the important processes one-dimensional in nature, and of potentially major importance in the development of low-dimensional structures as a means of growing highly ordered chains of `quantum dots' or continuous quantum wires.

A model is developed in which island nucleation is entirely restricted to the step edge. Islands grow in compact morphologies by monomer capture, and eventually coalesce with one another until a single continuous cluster of islands covers the entire step. A series of analytical theories is developed to describe the dynamics of the whole evolution. The initial nucleation and aggregation regimes are modelled using the traditional approach of rate equations, rooted in mean field theory, but incorporating corrections to account for correlations in the nucleation and capture processes. This approach is found to break down close to the point at which the island density saturates and a new approach is developed based upon geometric and probabilistic arguments to describe the saturation behaviour, including characteristic dynamic scaling which is found to persist through the coalescence regime as well. A further new theory, incorporating arguments based on the geometry of Capture Zones, is presented which reproduced the dynamics of the coalescence regime.

The latter part of the thesis considers the spatial properties of the system, in particular the spacing of the islands along the step. An expression is derived which describes the distribution of gap sizes, and this is solved using a recently-developed relaxation method. An important result is the discovery that larger critical island sizes tend to yield more evenly spaced arrays of islands. The extent of this effect is analysed by solving for critical island sizes up to i=20 and through the development of a new semi-analytical simulation combining the results from rate equations with a Monte-Carlo based growth mechanism. The results from these simulations provide further evidence of extremely narrowly distributed gap and island sizes at high critical island size. These results have positive implications for the precision manufacture of nanoscale electronic components using self-organised growth techniques.

Current work:

Self-Organized Criticality (SOC) is the term coined by Bak used to describe systems which, regardless of the starting conditions, will evolve naturally into a critical state which has no characteristic length or time scales. The SOC Forest Fire Model (SOC FFM) developed by Drossel and Schwabl is one of the most well-known examples of such a system. After the initial transient period the system will settle into a state where energy which is put into the system (in the form of bolts of lightning) will be dissipated by rare events which occur on all length scales (that is forest fires which can burn down anything from one or a few trees to patches of forest as large as the system itself.)

Present work concerns a model where a bias is introduced whereby trees grow preferentially next to existing trees. This leads to the growth of distinct, compact forest clusters. This is compared to the standard forest fire model where unconventional scaling behaviour is caused by two different types of fire, one which strikes small, fractal forest clusters, and the other which strikes large, mature, compact forest clusters. In my model all forest clusters are all compact, but new complications are introduced by the variable growth rates of the clusters.

A further variation on the model is to introduce quenched disorder, in the form a proportion of random selected sites which are `immune' to fire. We find that making up to about 30% of sites immune does not seem to change the scaling behaviour from that of the standard SOC FFM, indicating the the universality of the SOC behaviour.
 

Fragmentation models

Fragmentation is a ubiquitous, yet poorly understood phenomenon due to its complexity and the inherent difficulty involved in observing the details of such processes as they occur. The primary output from most experiments in this area is the distribution of fragment sizes. These can often be modelled by standard distributions, most commonly the lognormal and powerlaws. The presence of the latter has lead some researchers to suggest that certain fragmentation systems may be exhibit Self-Organised Criticality (see above.) In certain studies a transition in the form of the distribution from the lognormal to the powerlaw form was found as the energy put into the system was increased, suggesting some change in the mode of fragmentation. However, the situation is complicated by the fact that as the variance of lognormal increases, its right-hand tail tends to look like a powerlaw. Thus experimental data, which often only covers the largest two or three decades of fragment sizes, may not be sufficient to determine which of the two distributions most accurately models the complete distribution.

I have been investigating the fragmentation process theoretically through a series of simple statistical models to try to identify what are the general principles which determine the type of distribution obtained from a particular fragmentation process. The process can be encapsulated within simple rules such as:

• Choice of next fragment(s) to break
• How each fragment breaks (eg. how many pieces, in equal or different sizes)
• Conditions under which fragments become stable (ie don't break any further)

10% coverage	20% coverage
50% coverage	60% coverage

We can model this process quantitatively using a technique based on `Capture Zones', that is the amount of bare substrate from which an island tends to collect monomers. Capture Zones are identified by a network of Voronoi polygons with centres based on the nucleation sites of the islands. As the islands tend to nucleate within a relatively short space of time, the capture zone network remains stable for most of the growth process. During the initial DLA phase, the shape of the islands will tend to mimic that of their capture zones, reflecting the rate at which monomers are likely to arrive from different directions. As the island grows further, and its extremities approach the edge of the capture zone, DLA type growth ceases and direct capture takes over. As the growth rate of the islands is directly proportional to the capture zone area, we can predict the relative rates of the DLA and direct capture growth modes throughout the evolution of the system, and thus model the change in island morphology as the system evolves. Whereas the radius of the island increases rapidly under DLA growth, in the direct hit growth phase it increases very slowly and the island resist coalescence until much higher coverages than is seen with models where the growth is entirely compact.

The pictures on the right show snapshots from part of a simulation used in the studies at a range of substrate coverages. Colours are used to distinguish different clusters (these are randomly allocated, so some neighbouring but still separate islands may happen to have the same colour). It can be seen that most nucleation takes place before 10% coverage. The initial tree-like structures are evident in the first two pictures, whereas the compact but `feathery' structure can be seen in the later two shots. Notice how coalescence does not begin until at approximately 50% coverage, and how the gaps between island are of approximately equal size, resembling a network of cracks. .

Publications List:

• C. D. Pownall and P. A. Mulheran, Dynamic Scaling in simulations of island growth and coalescence on substrate steps, Phys. Rev. B, 57, R9463 (1998).
• C. D. Pownall and P. A. Mulheran, Simulation and theory of island nucleation, growth and coalescence on stepped substrates, Phys. Rev. B, 60, 9037 (1999).
• C. D. Pownall and P. A. Mulheran, Simulations of growing ordered arrays of islands on substrate steps, to be published in Phil. Mag. B.