# Kris's Research Notes

## November 19, 2010

### Droplet Size Experiments

Filed under: GaAs Simulations — Kris Reyes @ 4:18 am

In this post we discuss $Ga$ droplet formation as a function of model parameters $\gamma_{droplet} = \gamma(G0, G0), \gamma_{surface} = \gamma(G2, A4)$ and physical parameters $T, r_{\downarrow Ga}$ and $d$, the amount of Gallium deposited (monolayers).

We measure droplet size using the auto-correlation  of the height profile of the droplets. Let’s briefly recall the properties of the autocorrelation function. By definition, the auto-correlation function $S(t)$ of a signal $h(x)$ is given by the convolution

$\displaystyle S(t) = \int_{-\infty}^\infty h(x)h(x-t) dx$.

Note that $S(-t) = S(t)$ and if $h(x)$ is periodic with period $\rho$ then so is $S(t)$. Further

$\displaystyle S\left(\frac{\rho}{2}-t\right) = \int_{-\infty}^\infty h(x) h\left(x - \frac{\rho}{2}+t\right) dx.$

Under the translation $u = x + \frac{\rho}{2}+t$, we see that

$\displaystyle S\left(\frac{\rho}{2}-t\right) = \int_{-\infty}^\infty h\left(u-\left(\frac{\rho}{2}+t\right)\right)h(u-\rho) du,$

which is just $S\left(\frac{\rho}{2}+t\right)$. That is, $S(t)$ is symmetric about $t = \frac{\rho}{2}$. Therefore it suffices to specify $S(t)$ in the interval $\left[0, \frac{\rho}{2}\right]$.

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

In the experiments, we fix $\gamma_{surface} = 1.0 eV$, $\gamma_{droplet} = 0.30 eV$ and all other $Ga-Ga$ bonds to $0.25 eV$. We also fix $T = 600 K$. We deposit Gallium on a flat substrate with deposition rate varying in

$r _{\downarrow Ga} in \left\{ 0.2, 0.3, \hdots 0.9 \right\}$ monolayers/second.

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

$d in \left\{ 2, 3, 4, 5\right\}$ monolayers.

After depositing we allow the droplets to anneal for 30 seconds.At the end, we measure the height profile $h(x)$ and calculate the corresponding autocorrelation $S(t)$.

For a fixed pair of experiment parameters $(r_{\downarrow Ga}, d)$ we repeat the a trial 16 times. and consider the average autocorrelation obtained from each of the 16 autocorrelation functions $S(t)$ we obtain. We may do this because the autocorelation of $h(x)$ is invariant under shifts $h(x + c)$, 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 $8 \times 4 = 32$ trials.

 $r_{\downarrow Ga}$ $d$ 2 3 4 5 0.2 50.3231 41.528 52.2903 55.3552 0.3 36.43 44.0216 50.5379 49.3419 0.4 46.145 47.9206 47.6792 51.4427 0.5 56.9722 42.7745 52.4556 49.7989 0.6 37.7097 41.8768 44.747 50.7267 0.7 48.7011 48.3853 44.752 47.9207 0.8 36.5 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: