In this post we discuss droplet formation as a function of model parameters and physical parameters and , the amount of Gallium deposited (monolayers).

We measure droplet size using the auto-correlationÂ of the height profile of the droplets. Let’s briefly recall the properties of the autocorrelation function. By definition, the auto-correlation function of a signal is given by the convolution

.

Note that and if is periodic with period then so is . Further

Under the translation , we see that

which is just . That is, is symmetric about . Therefore it suffices to specify in the interval .

Consider the autocorrelation of . When $t = 0$ the two convolving waves are exactly in phase, and we get a large positive number as . When however, the two convolving waves are exactly out of phase, and we get a large negative number as . This suggests that in general the first local minimum of corresponds to droplet width, and we use this below.

In the experiments, we fix $\gamma_{surface} = 1.0 eV$, and all other bonds to . We also fix . We deposit Gallium on a flat substrate with deposition rate varying in

monolayers/second.

We deposit until a fixed amount of Gallium has been reached, and we vary that amount in

monolayers.

After depositing we allow the droplets to anneal for 30 seconds.At the end, we measure the height profile and calculate the corresponding autocorrelation .

For a fixed pair of experiment parameters we repeat the a trial 16 times. and consider the average autocorrelation obtained from each of the 16 autocorrelation functions we obtain. We may do this because the autocorelation of is invariant under shifts , and we assume that the droplets will form equal to one another in the 16 trials modulo such a shift.

Here are the location of the first local minima of the average autocorrelation functions for each of the trials.

2 | 3 | 4 | 5 | |

0.2 | 50.3231 | 41.5280 | 52.2903 | 55.3552 |

0.3 | 36.4300 | 44.0216 | 50.5379 | 49.3419 |

0.4 | 46.1450 | 47.9206 | 47.6792 | 51.4427 |

0.5 | 56.9722 | 42.7745 | 52.4556 | 49.7989 |

0.6 | 37.7097 | 41.8768 | 44.7470 | 50.7267 |

0.7 | 48.7011 | 48.3853 | 44.7520 | 47.9207 |

0.8 | 36.5000 | 41.4402 | 51.3904 | 51.5909 |

0.9 | 41.4574 | 47.6389 | 46.8415 | 48.2489 |

Here are the plots of the autocorelation (ordered as the transpose of above table):

We can plot droplet size as a function of depth.We may also plot as a function of deposition rate:

[…] This is a follow-up to this post. […]

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