# Kris's Research Notes

## March 12, 2012

### 2+1 GaAs Simulations

Filed under: GaAs Simulations — Kris Reyes @ 4:21 pm

Our KMC code can be easily extended to 2+1 dimensions by altering the underlying lattice to include a z-component. In this note, I describe how this change was made and present interesting simulation results from a brief set of runs with the 2+1 system. While the parameters used have not been tuned to experimental evidence, we see that the simulations are able to capture a wide range of phenomena that depend on both energy and experimental parameters. This can serve as a starting point for future work.

## The Zinceblende Lattice

Atoms are placed in space with integer coordinates. In the lattice, an atom has four nearest neighbors and 12 next-nearest neighbors. Consider an atom with integer coordinates (x, y, z). The nearest neighbor bonds of this atom are specified below:

In this way, the plane determine by the two bonds above an atom is perpendicular to the plane determine by the bottom two bonds. Next-nearest neighbor bonds form an fcc lattice:

In the z-direction, atomic planes repeat with periodicity 4, corresponding to a structure isomorphic to $\mathbb Z / 4\mathbb Z \times \mathbb Z / 2\mathbb Z$. That is, the offset $(dx, dy)$ of a plane at height $z$ is given by

$\displaystyle dx = z \mod 4$,

$\displaystyle dy = z \mod 2$.

N.B. The lattice does not occupy the entire integer lattice $\mathbb Z^3$. Rather, the coordinates (x,y,z) for points on the zincblende lattice have the form:

$\displaystyle x = 4i + (2\mod j) + (k\mod 4)$,

$\displaystyle y = 2j + (k\mod 2)$,

$\displaystyle z = k$,

where $i, j, k \in \mathbb Z$ and $a \mod b$ here means simply the remainder of the division $\frac{a}{b}$.

Throughout this note, we consider simulations on a 64×64 atom domain.

## GaAs Substrate Growth

As before, our initial set of runs simulate substrate growth. Ga flux is fixed at $F_{Ga} = 0.1$ ML/sec, while temperature is varied between 625 and 1000 K. The bond strengths used are summarized below:

$\gamma_{GA} = 0.5$ eV,

$\gamma_{GG} = 0.4$ eV, $\gamma^\prime_{GG} = 0.1$ eV,

$\gamma_{AA} = 0.1$ eV, $\gamma^\prime_{AA} = 0.1$ eV,

$\mu_{AS} = 0.6$ eV.

Moreover, all atom-atom exchanges were turned off so that surface diffusion was the only mechanism for atom movement.
Both Ga and As were deposited with fluxes $F_{Ga}, F_{As}$, respectively, at the selected temperature until five monolayers of material were deposited. The resulting profile was imaged top-down to produce a 2D plot both in color and in grayscale. Also, note that the fundemental lattice parallelogram is a cube of side length 4. The numbers on the axes for the plots in this note report length, so in order to get the number of lattice cells, one should divide the length by 4.

### Effect of $F_{As}$

Fix $T = 666K$ and vary $F_{As} = 0.05, 0.1, 0.2, ... 1.0 eV$. Here are the results:

 $F_{As} = 0.05$ ML/sec $F_{As} = 0.1$ ML/sec $F_{As} = 0.2$ ML/sec $F_{As} = 0.4$ ML/sec $F_{As} = 0.6$ ML/sec $F_{As} = 0.8$ ML/sec $F_{As} = 1.0$ ML/sec

There are many interesting features in the results above. First, we see that for low As flux $F_{As} = 0.05$ ML/sec, the surface is Ga terminated. In fact, a faceted Ga droplet has formed. Below it, there are islands that appear to prefer growth in the $[\bar{1}10]$ direction (assuming I understand how matlab indexes an image).

Increasing the As flux to $F_{As} = 0.1$ ML/sec, the surface Ga concentration has fallen to 44% and the islands don’t seem to have a prefered direction of growth. As the As flux is increased, islands preferentially grow in the $[110]$ direction.

Moreover, in the high As flux cases, the growth mode switches from island formation to highly direction rows several atoms thick. Here is a portion of one such row, showing only the top two layers of atoms:

We observe that the rows are As terminated, and there appears to be a preference for As atoms to occur in pairs. Could this be an attempt by the simulation to dimerize As atoms on the surface while staying on lattice?

### Effect of $T$

Here, we fix $F_{As} = 0.4$ ML/sec, and vary $T$ between 625 and 1000K:

 $T = 625$ K $T = 666$ K $T = 714$ K $T = 769$ K $T = 833$ K $T = 909$ K $T = 1000$ K

We see similar behavior as before. In the low temperature case, there is preferential growth in the $[110]$ direction as As atoms form rows 2-4 atoms wide. In the mid-temperature regime, growth is less anisotropic, resulting in larger and wider island formation (e.g.$T = 769$ K). The resulting substrate at T=833 case is the transition between an As-terminated and Ga-terminated system. The morphology here seems confused — growth occurs in both the $[110]$ and $[\bar{1}10]$ case, resulting in a crisscrossed pattern. Past this, for higher temperatures, growth is toward the $[\bar{1}10]$ direction and the system is Ga terminated.

### Other Observations

We have seen several examples of liquid Ga formation above. The droplets are heavily faceted, implying that the bonds strengths used were perhaps too high if we wished to accurately simulate liquid behavior. Moreover, the preferred growth direction of $[\bar{1}10]$ in Ga rich environments imply an asymmetrical shape of the resulting Ga droplets. In the above trials, we encountered exaggerated instances of this. For example, when $\mu_{As} = 0.2$ eV, $T = 769$ K and $F_{As} = 0.4$, the substrate becomes Ga terminated early on. Here is the resulting configuration:

Here we see an elongated liquid Ga droplet — a quantum dash.

Other things observed in the above trials:

• Oriented island formation:
 $\mu_{As} = 0.8$ ev$T = 909$ K$F_{As} = 0.05$ ML/sec $\mu_{As} = 0.8$ ev$T = 909$ K$F_{As} = 0.10$ ML/sec $\mu_{As} = 0.8$ ev$T = 909$ K$F_{As} = 0.20$ ML/sec $\mu_{As} = 0.8$ ev$T = 909$ K$F_{As} = 0.80$ ML/sec $\mu_{As} = 0.8$ ev$T = 909$ K$F_{As} = 1.0$ ML/sec
• Possible snail trails?:
 $\mu_{As} = 0.4$ ev$T = 833$ K$F_{As} = 0.05$ ML/sec $\mu_{As} = 0.4$ ev$T = 833$ K$F_{As} = 0.10$ ML/sec
• Droplet formation on a trench-edge:
 $\mu_{As} = 0.8$ ev$T = 769$ K$F_{As} = 0.05$ ML/sec

## Droplet Formation and Crystallization

In the second set of trials, we attempt to grow liquid Ga droplets on an initially As-terminated substrate. Ga atoms were deposited at a rate of 0.1 ML/sec until 4 monolayers were deposited. The system was then allowed to anneal for 60 seconds without any deposition of material. Then As flux was turned on for 10 seconds at a rate of $F_{As}$ ML/sec, which we varied between 0.1 ML/sec and 1.0 ML/sec, at a temperature $T$, which we varied between 573 and 623 K. The temperature was then increased to 623 K for another 10 seconds, maintaining the same As flux. The same energies above were used, fixing $\mu_{As} = 0.6$ eV.

### $T = 573$K

 $F_{As} = 0.1$ ML/secN.B. At a low flux, there was not enough material to fully crystallize the droplet. $F_{As} = 0.2$ ML/sec $F_{As} = 0.4$ ML/sec $F_{As} = 0.6$ ML/sec $F_{As} = 0.8$ ML/sec $F_{As} = 1.0$ ML/sec

We observe that for most cases, the droplet fully crystallized (I haven’t checked for the presence of a liquid core, however). In the low flux case $F_{As} = 0.10$ ML/sec, not enough material was deposited to fully crystallize the droplet. However, this is an interesting case as it shows the presence of a ring during intermediate stages of droplet crystallization.

### $T = 623$K

 $F_{As} = 0.1$ ML/sec $F_{As} = 0.2$ ML/sec $F_{As} = 0.4$ ML/sec $F_{As} = 0.6$ ML/sec $F_{As} = 0.8$ ML/sec $F_{As} = 1.0$ ML/sec

There are several interesting results to observe. First, for the low As flux cases ($F_{As} \leq 0.2$ ML/sec), flat substrate formed instead of a quantum dot. When $F_{As} = 0.4$ ML/sec, a nanoring formed. Here is a higher-contrast picture of that case:

Here is a movie of the ring’s formation:

Another interesting phenomena is that although the liquid droplet seems to have an prefered growth direction, the crystallized droplets (especially in the high As flux cases) tend to be more symmetric and hemispherical.

## Summary

We simulated GaAs substrate growth and GaAs quantum dot formation on a 2+1 zincblende lattice. Among the different phenomena we were able to capture were:

• Oriented growth and surface termination as a function of As flux and temperature. Perhaps this is a quasi surface reconstruction phenomena in which surface atoms are required to reside on the same lattice as bulk atoms. A marked phase change in orientation occurs at the same growth conditions in which surface termination changes.
• Droplet formation. Liquid droplets are able to form, but are faceted for our choice of bonding energies
• Quantum Dot formation. Upon the presence of an As flux, the liquid droplets were crystallized. We saw some dependence on growth conditions, where morphology varied between flat substrate, nanoring formation and compact dots. We have not investigated the presence of liquid cores.
• Liquid droplet formation trenches. This is preliminary and we have not investigated this further.
• Possible existence of snail trails. As above, this is preliminary.

Unfortunately, due to time constraints, I don’t anticipate being able to investigate the 2+1 case for a while. However, this should give us some idea of what to tweak in case we wish to pursue this later.