PHY 598 (Venables) Sect E2

Notes for PHY 598 Sect E2 (Venables)

© Arizona Board of Regents for Arizona State University and John A. Venables

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Lecture notes by John A. Venables. Lectures given 28 March 96 and updated 12 Dec 96. I then gave a series of lectures at EPFL in the Fall of '97, and EPFL Lecture #4 is on closely related topics.

E2. Atomistic Models and Rate Equations

  • Rate Equations, Controlling Energies, and Simulations
  • We have considered simple rate equations for adatom concentrations in section 1.3, and in problem 2, adding a diffusion gradient in problem 3. Now we need to add non-linear terms to describe clustering and nucleation of 2D or 3D islands. These equations are governed primarily by energies, which appear in exponentials, and also by frequency and entropic preexponential factors.

    This is the main advantage of such ‘mean field’ models. They are known not to describe fluctuations very well, so various quantities, such as size distributions of clusters, are not described accurately. In current research, using fast computational techniques such as ‘Kinetic Monte Carlo’ (KMC), the early stages can be simulated on moderate size lattices. These KMC ‘experiments’ using the same assumptions can then be used to check whether mean field treatments work for a particular quantity. The CSSS seminar on April 15th is by one of the leaders in this area, Dr Steve Bales (Sandia, Livermore). This will be a good chance to compare these different approaches.

    The emergence of computer simulation as a third way between experiment and theory is clearly a growth area of our time. To make progress in this area, one has to start with the simplest models, and stick with them until they are really understood. You need to beware generating more heat than light, and in particular of generating special cases which may or may not be of real interest. Simulations can however be very illuminating, and may suggest inputs for simple models that one hadn’t thought of. Animations are immediately appealing, and if Spielberg can do it, why shouldn’t we? The problem lies only in the subsequent claims for correspondence with reality; then a measure of self-discipline is needed, both from the lecturer/writer and the listener/reader.

  • Elements of Rate Equation Models
  • But with the above provisos, here goes with the atomistic models. We consider rate equations for the various sized clusters and then try to simplify them. If only isolated adatoms are mobile on the surface, we have

    nucleus size. In its simplest form, this means that a) we can consider all clusters of size > i to be ‘stable’, in that another adatom usually arrives before the clusters (on average) decay; the reverse is true for clusters of below critical size; b) these subcritical clusters are in local equilibrium with the adatom population.

    The capture numbers are related to the size, stability and spatial distribution of islands, and solving this problem has caused a lot of words to be spilled; it isn’t over yet. The simplest mean field model, which I and others worked on long ago, and which several people are working on now, considers a typical cluster of size k immersed in the average density of islands of all sizes. Then one can set up a diffusion equation for the adatom concentration in the vicinity of the k-cluster (size specific), or x-cluster (the average size cluster), which has a Bessel Function solution.

  • Regimes of Condensation
  • Bales and coworkers have been studying this model using a combination of rate equations and KMC techniques. So far the comparison for i = 1 has been published (G.S. Bales and D.C. Chrzan, Phys. Rev B50 (1994) 6057); other treatments are in the pipeline. My 1987 PRB paper concentrated on including vibrations in a self-consisent way, within the mean field framework outlined above. In the last few years there have been many related treatments by several groups, mostly in response to the new UHV STM-based experimental results. We describe some of these results in section E3.

    Continue to section E3

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