In this note, we examine the effect of Ga-Ga bonds on droplet statistics and describe changes to the code that speed up adatom diffusion.

## Droplet Statistics

During the June 23 meeting, we discussed the results of a large run that looped over several values for the five energy parameters , which affect G-G bonds, G-A bonds, atom-atom exchanges, and diffusion through the droplet and As desorption, respectively. The results of the run can be found here. Observe that the size of the droplets smaller than what is expected in experiments. For example, here is a crystallized droplet formed by depositing Ga at a rate of 0.1 monolayers/second for 35 seconds then exposed to an As flux of 1 monolayer/second at a temperature of 473K with parameter values eV, eV, eV, eV and eV:

The domain is 512 atoms wide, and the droplet is at most 120 atoms wide. However, in experiments (see Z. Gong et al., Appl. Phys. Lett. 87, 093116 (2005) ) droplets/nanorings of diameter 150 nm (about 600 atoms) form at 423K, much larger than the above example.

To address this, we consider the effect of and on the size of the droplets. The idea is that on a Ga-terminated surface, by lowering these energies, diffusing Ga adatoms are more likely to find each other over a larger domain. We deposit 3.5 monolayers of Ga on a domain of width 4096 atoms at a temperature of 423K. Using the values in Gong et al. (namely the average diameter is 360 atoms and average height 24 atoms) is and assuming the droplets are elliptical in shape, we expect each droplet to have a mass of about 6786 atoms, half of which are Ga atoms. Since we deposit 14336 Ga atoms total, we want between 4 and 5 droplets to form in our domain, resulting in a droplet density between and (compare this to the experimental density of ).

We vary eV and eV. For each pair of parameters, Ga is deposited at a rate of 0.1 monolayers/second for 35 seconds with substrate temperature of 423K. Statistics are collected over an ensemble of 16 identical runs. Widths are determined using the height profile auto-correlation function. (All results here)

First is a plot droplet width (which we want to be 360/2 = 180 atoms) vs. . The different lines correspond to different values of :

Second is a plot of number of droplets (which we want around 4 or 5) vs. .

These two plots suggest parameter values of on the low side of values tested. Specifically,

- if eV (red), then eV;
- if eV (blue), then eV;
- if eV (green), then eV.

Here is a histogram of how much Ga (monolayers) was actually deposited in the 35 seconds:

Note the standard deviation of 0.06 monolayers Ga. We can repeat the same experiment, this time fixing the amount of Ga deposited to exactly 3.5 monolayers (results here). Here is droplet width vs in that case:

The behavior is similar to the previous case.

## Fast Adatom Diffusion

In order for the above simulations to run in a reasonable amount of time, we changed the code to allow for fast adatom diffusion. This means that when an adatom hops, it may hop several one direction before any bookkeeping is performed. An adatom hops in one direction until its local environment is different from the one it started out in, or until 10 hops are performed. To correct for this, we simply sample adatom hops 10 times less than before. This has a significant impact on the speed of the simulation, but also affects the diffusivity of adatoms in a non-trivial way.

To illustrate this, fix eV and consider the result of depositing 1 monolayer of Ga at a rate of 0.1 monolayer/second. Here is the result with the above fast-adatom diffusion mechanism turned on. Note that there are nine Ga droplets formed. The simulation performed 194,749,977 updates. Here is the result with the fast diffusion turned off. In this case, thirteen droplets have begun to form. Here, the simulation performed 624,706,417 updates, about three times more than when fast adatom diffusion was turned on. We conclude that the Ga adatoms diffuse more rapidly with fast adatom diffusion turned onn even when the rates are scaled by a factor of 0.1.

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