Computer model of LIFE results

A simple model has been developed to try and replicate the results of the LIFE experiments and the Fourier analysis of the SEM images. It is based around the gradual growth of lines from starting dots, and follows a series of ‘rules’ that have been derived from examining the SEM images.

1. Lines grow from dots.

2. Lines can propagate along one of three directions at 120° to each other. 3. Lines cannot touch or cross over each other.

4. Lines may change direction to avoid another line.

The execution of the program is summarised in Figure 3.37. If a line is detected as being too close to another line it is given the chance to change direction, as shown in the flow diagram. However, if it tries to change direction more than twice in a single cycle, then that line is stopped. When all the lines are marked as stopped the program is complete. This represents a sufficiently long period of etching time so that the lines are completely formed.

GenerateNrandom starting positions

Extend line by 1 pixel

Check line is not too close to another If lineiswithin x pixels of another

Check position of the line

Stop the line

Figure 3.37:Algorithm for LIFE computer model.

The main variable is N, the number of starting dots. The number of dots in a fixed area is equivalent to the density, which in turn is believed to be related to the intensity of light at a particular point on the crystal surface and distance from the

centre of the site. So by changing the value ofN, the program can simulate different areas of site; namely the densely packed centre and the edge regions with lower density and longer features, or lines.

The LIFE program also produced a Fourier transform of the simulation results to allow comparison with the experimental results. The results from a series of simulations with varying values of N, the number of starting points, are shown in Figure 3.38. For a simple model, the results compare well with those obtained experimentally, and show all the characteristic features. The Fourier transforms also bear a close resemblance to those of the SEM images. As N is increased the density of the structures becomes higher, and the length of the lines decrease, as there is less space available. ForN=5000, there are no large-scale patterns formed and the results appears to be very similar to SEM images taken at the centre of the site.

The Fourier transforms of the images with lowNvalues show the same star-shaped pattern as found experimentally, but with some extra detail. This pattern changes as N is increased; the width of the lobes increases and the length (frequency) of each lobe decreases. AtN=5000 the Fourier transform shows only a little direction dependence, again as would be expected from the experimental evidence.

The model initially had a totally random distribution of starting points, but to try and simulate an entire LIFE site this was adapted to feature a Gaussian distribution of starting points. The results of this approach are shown in Figure 3.39.

The success of the model in matching the results obtained by experiment shows that the rules which were defined to create the model must be realistic and somehow arise from the physical properties of the crystal, under exposure to light and etchant. While the assumption that lines will simply stop and change direction once they reach a certain distance from another line is probably an over-simplification it does produce reasonable results. Instead of this ‘hard sphere’ type repulsion a more accurate model would make the repulsion distance-dependent, and so make the effect more gradual. However the improvement this would make probably does not warrant the extra complication of the model, as the results would only be slightly different. The model does show that the assumption of lines growing from starting dots is reasonable, and that the lines can only grow in one of three directions, although perhaps a little simplistic, again gives good results. Finally, the model also supports the suggestion that more of these growth dots are formed towards the centre of the site where the laser intensity is highest, as the patterns and Fourier transforms from the model are very close to those obtained experimentally.

NSpots Model Image Fourier Transform

100

500

1000

2500

Figure 3.38: Results of the LIFE computer model and corresponding Fourier Transforms

NSpots Model Image Fourier Transform

600

1000

2000

10000

Figure 3.39:Results of the LIFE computer model using a Gaussian distribution of starting points and corresponding Fourier transform images.

Finally, it should be noted that although the model grows the lines by extending the existing lines pixel by pixel, this is not believed to be a literal representation of the processes that are occurring. Instead the lines are thought to grow as the areas around them are removed by the action of the acid while the lines themselves resist etching by acid.