# Kris's Research Notes

## October 27, 2010

### Model Parameters

Filed under: GaAs Simulations — Kris Reyes @ 2:16 pm

In this post, we examine the effect of the model parameters. Many parameters depend on the definition of an atom’s local neighborhood, and throughout this note, we define the local neighborhood of an atom to be the vector of neighbor counts $(n_V, n_{Ga}, n_{As})$, where e.g. $n_{Ga}$ is the number of Gallium nearest neighbors. However, we often simplify things by considering only the neighbor count for the species the atom is not. That is, we often shall only be concerned with, for example, $n_{Ga}$ for $As$ atoms, and vice versa. Yet there are instance where we do use the other counts, as we will detail below. Regardless, the atom species along with this counts vector determine the atom’s extended species and we shall write $As(n_V, n_{Ga}, n_{As}), Ga(n_V, n_{Ga}, n_{As})$ to indicate the extended species. We will also write $As(n_{Ga}), Ga(n_{As})$ when we only care about those counts — meaning the other counts may be anything.

Model Parameters

1. $\mu_{As}$ — The $As$ desorption potential. The desorption rate of an $As$ atom $A$ is

$r_{\uparrow As} = \Omega \exp[-\beta(E(A) + \mu_{As})]$,

where $E(A)$ is the local energy of atom $A$. I normally fix this to $0.35 eV$, though this was a somewhat arbitrary choice.

2. $\alpha(A, B)$ — The “diffusion coefficients”. These numbers determine the swapping rate between atoms $A$ and $B$:

$r_{swap}(A,B) = \Omega \exp\left[-\beta \alpha(A,B)(E(A) + E(B))\right].$

There is one for each pair of extended types, but we make several simplifications in the code, as we detail below. The larger $\alpha(A,B)$ are, the more difficult it is to swap the atoms. These numbers are symmetric in the arguments and when we fix $\alpha(A, B)$ below, we often will omit the fact that this fixes $\alpha(B,A)$.

3.$\gamma(A,B)$ — Pairwise bonding energies. These determine the local energy of atom $A$:

$E(A) = \sum_{i} \gamma(A, B_i),$

where the summation is over the nearest neighbors of $A$. There is one for each pair of  extended types, but as with diffusion coefficients, we shall make some simplifications. The larger these are, the more static $A$ is. These numbers are also symmetric in the arguments and as in the case for diffusion coefficients, when we fix $\gamma(A,B)$ below, we also implicitly fix $\gamma(B,A)$.

Simplifications

Given there are $4 \times \binom{6}{4} = 60$ extended species and hence $\binom{60}{2} = 1770$ total diffusion coefficients and the same number of pairwise bonding energies, we must make some simplifications to reduce the number of parameters of our model.

Diffusion Coefficients

We set $\alpha(A,B) = \infty$ except for three cases: diffusion on the surface, diffusion in a $Ga$ droplet, and diffusion on the droplet/crystal interface.

1. Diffusion on the surface. If either $A$ or $B$ is vacuum but not both, then $\alpha(A, B) = 1$. Then (assuming $A$ is the non-vacuum), the swapping rate for $A$ with vacuum is

$r_{swap}(A, B) = \Omega \exp[ -\beta \alpha(A,B)(E(A) + E(B))] = \Omega \exp[ -\beta E(A)].$

2. Diffusion in droplet. We characterize an $Ga$ atom as a droplet $Ga$ atom if it is of extended type $Ga(0)$ or possibly $Ga(1)$. We characterize an $As$ atom in a Gallium droplet as the extended type $As(4)$ — an Arsenic atom completely surrounded by Gallium. We would like an $As$ atom to diffuse rapidly through the droplet and so we set $\alpha(Ga(1), As(4)) = \epsilon$, for a small parameter $\epsilon$. This handles the case of $As$ atom already within the droplet, but not the case of an $As$ atom on the surface. There, the Arsenic atom is not fully coordinated and so is not $As(4)$. For now, we handle this by setting setting the diffusion coefficient between any Arsenic atom and $Ga(1)$ to be small:

$\alpha(Ga(1), As) = \epsilon$.

In the future, we may want to change this. Note this has some implications about the behavior near the droplet/crystal interface that is independent of the next point. Consider a crystal that has nucleated within the droplet and consider an Arsenic atom at the top layer of this crystal:

Here Arsenic is green and Gallium is red. The shaded Arsenic and Gallium atoms are of extended species $As(4), Ga(1)$, respectively. The diffusion coefficient is small for this pair, so the indicated swap is likely. This may continue until the entire nucleated atom has dissolved into the droplet. Note however, that the rate of the indicated swap is different than that of a freely diffusing Arsenic atom since the local energies of the two scenarios are different. Here, we may specify a higher local energy of the swapping Arsenic atom because it has one $Ga(4)$ neighbor, one $Ga(3)$ neighbor, one $Ga(2)$ neighbor and one $Ga(1)$ neighbor as opposed to a freely diffusing Arsenic atom, which has four $Ga(1)$ neighbors. Thus we may somewhat control the speed a nucleating crystal dissolves by specifying sufficiently large $As(4)-Ga(i)$ bond strengths for $i > 1$, but in general: it is harder for an crystal to exist within the droplet than outside of it.

For example, here are two movies of a $GaAs$ crystal hemisphere dissolving within a $Ga$ droplet. In the first movie, we set $As(4)-Ga(i) = 0.5 eV$ for $i = 1, 2, 3$ and $As(4)-Ga(4) = 1.0 eV$. In the second movie, we lower $As(4)-Ga(1)$ to $0.10 eV$. According to our above discussion, the droplet should dissolve quicker in the second case.

Note that in the first movie, the crystal in the droplet has not completely dissolved after 20 seconds. The second movie is quiet surprising. The crystal does indeed dissolve much more quickly, but there is crystal nucleation at the edges of, but still within the crystal. How do these crystals form and survive in the droplet? I can’t explain it at this time.

As noted before, a crystal that nucleates within the droplet will eventually dissolve because of the presence of a $Ga(1)-As$ bond, which occur at the “corners” of crystal within the droplet. In particular, the underlying substrate will not dissolve via this mechanism because all such substrate $As$ atoms will be connected to $Ga(i)$ atoms, with $i> 1$:

More generally, if an entire layer of $As$ atoms can nucleate within the droplet (that is, as wide as the droplet is at that height), it will not dissolve via this mechanism. This would be possible if e.g. the $As$ flux is sufficiently large.

3. Diffusion at the Droplet/Crystal interface. As noted above, the underlying $GaAs$ substrate remains intact underneath the Gallium droplet. However, we may want the droplet to etch into the crystal. That is, we would like the $As$ atoms at the droplet/crystal interface to detach from the crystal and diffuse into the droplet. In order to do this, it must swap with one of its $Ga(2)$ neighbors. We may be tempted to specify $\alpha( As(4), Ga(2) ) = \epsilon$, but this would also imply that an Arsenic atom may swap with a Gallium atom at the surface of a Gallium terminated crystal:
Consider the two examples above. In both cases, the shaded $Ga$ atom is of extended species $Ga(2)$, but one is on the droplet/crystal interface whereas the other is a surface $Ga$ atom. This is easily addressed if we consider the Gallium neighbor counts for the Gallium atom. That is, we set $\alpha(As(4), Ga(2,2,0)) = \epsilon$.

Pairwise Bonding Energies

For now, bonding energies are specified in terms of $Ga(i)-Ga(j), As(i)-As(j),$ and $Ga(i)-As(j)$ pairs, instead of pairs between full extended species (i.e. neighbor counts vectors). In the future, we may want to change this.

1. Gallium-Gallium Bonds. We set $\gamma(Ga(i),Ga(j))= 0.25 eV$ unless $i = j = 0$. We fix $\gamma(Ga(0),Ga(0))$ to $0.28 eV$. As discussed previously, this bond strength determines Gallium droplet formation.
2. Arsenic-Arsenic Bonds. We set bond strengths$As(i)-As(j) = 0.10 eV$. This number was arbitrary, but we want it to be somewhat small.
3. Gallium-Arsenic Bonds.  These are the interesting bond strengths.
• The bond strength $\gamma(As(4), Ga(4))$ determines the strength of the bonds in the bulk of the crystal. We set this to a large number, $1.0 eV$.
• The $Ga(1)-As(i)$ bonds are involved in $As$ atoms diffusing on or in the Gallium droplet. For example, if  the $Ga(1)-As(4)$  is sufficiently large, with respect to the corresponding diffusion coefficient $\alpha(Ga(1), As(4))$, we can effectively slow or prevent Arsenic atoms from diffusing in the droplet.

For example, if we fix $\alpha(Ga(1), As(4)) = 0.2$ and $r_{\downarrow As} = 1$ monolayer/second and consider the cases $\gamma(Ga(1), As(4)) = 1.0 eV, 0.5eV$ and $0.1eV$ and fix $\alpha(Ga(1), As(4)) = 0.2$, we get the following movies:

Note that for the $\gamma(Ga(1), As(4)) = 1.0 eV$ case, the droplet forms a crystal shell. The bond strength is so large that the deposited $As$ atoms cannot diffuse quickly through the shell before crystals start to nucleate.  In the case $\gamma(Ga(1), As(4)) = 0.5 eV$, Arsenic atoms can diffuse more readily through the droplet, but crystals still nucleate within the droplet. In the final case, crystals do not nucleate in the droplets as the $As$ atoms are extremely mobile. Note further that there are now vacancies within the droplet. Thus, there is a correlation between $As$ mobility and the appearance of vacancies within the droplet.

• The $Ga(1)-As(i)$ bond strengths determine the behavior of $As$ atoms on the surface of the crystal. Consider an Arsenic atom on the surface of the droplet:It may either swap with a droplet $Ga$ atom, or diffuse on the surface. To simplify things, let’s assume all $Ga(1)-Ga$ bonds are equal say $0.25 eV$. Let’s also denote $\alpha = \alpha(G(1), A(1))$ and $\gamma_i = \gamma(G(1), A(i))$ to simplify notation. Then, the corresponding energies for these transitions are $\alpha \left(0.75 + 2\gamma_1\right)$ and $\gamma_1$, respectively. If we wish to encourage diffusion through the droplet, we require

$\alpha\left(0.75 + 2\gamma_1\right) \leq \gamma_1.$

That is, if $\gamma_1$ is fixed, we require

$\displaystyle \alpha \leq \frac{\gamma_1}{\left(0.75 + 2\gamma_1 \right) }$,

and if $\alpha$ is fixed we require

$\displaystyle \gamma_1 \geq \frac{0.75 \alpha}{1-2\alpha}$

More generally: we require:

$\displaystyle \alpha \leq \frac{i \gamma_i}{\left(0.75 + (i+1)\gamma_i \right) }$,

or

$\displaystyle \gamma_i \geq \frac{0.75\alpha}{i - \alpha(i+1)}$,

for $i = 1, 2, 3$.

For example, if we fix $\alpha = 0.2$, then the bounds above are:

$\displaystyle \gamma_1 \geq 0.25$,

$\displaystyle \gamma_2 \geq 0.107$,

$\displaystyle \gamma_3 \geq 0.07$,

Therefore, if we set $\gamma_1 = \gamma_2 = \gamma_3 = \gamma_4 = 0.05$, we would expect the Arsenic atoms to diffuse on the Gallium droplet, rather than on it. To see where the $As$ atoms land, we set $As(4)-G(i)$ bond strengths to $1.0 eV$, a large number, for $0 < i < 4$. Here is what happens if we deposit Arsenic at the rate of 1 monolayer per second over 5 seconds:

We observe some nucleation at the bottom of the droplet which indicates that there are some $As$ atoms diffusing within the droplet:

If we pick $\gamma_1 = 0.25, \gamma_2 = 0.107, \gamma_3 = 0.07, \gamma_4 = 0.05$, we expect an $As$ atom to diffuse in or on the droplet with an equal probability.

Here we observe much more crystal nucleation at the bottom of the droplet than in the previous case.

If we pick $\gamma_1 = 0.35, \gamma_2 = 0.207, \gamma_3 = 0.17, \gamma_4 = 0.05$ we expect an $As$ atom to diffuse within the droplet more often than diffusing on the droplet:

The amount of nucleation is about the same as the previous case. One interesting thing to observe, however, is that there are no vacancies in the droplet, which was not the case in the previous two cases. I don’t know why this is the case.

Note however, increasing the bond strengths $\gamma_i$ does not imply the $As$ is extremely mobile at the surface. It just means that it is more likely to swap with a droplet Gallium atom than diffuse on the surface. In fact, increasing the bond strength will decrease the probability of either event. To observe this, consider the exaggerated case $\gamma_1 = 0.95, \gamma_2 = 0.807, \gamma_3 = 0.77, \gamma_4 = 0.05$.

Here we see that surface $As$ atoms are very static on the surface. The simulation neither wants to swap them with droplet Gallium atoms nor diffuse them on the droplet surface. As a result, $GaAs$ crystals start to nucleate at the surface of the droplet.

Summary Of Gallium-Arsenic Pairs Associated with Various Phenomena

 $A(1)$ $A(2)$ $A(3)$ $A(4)$ $G(1)$ $As$ atom on Droplet Surface [Droplet Diffusion] $As$ atom on Droplet Surface [Droplet Diffusion] $As$ atom on Droplet Surface [Droplet Diffusion] $As$ atom in Droplet Bulk[Droplet Diffusion]  Bond near Droplet/Crystal interface [Dissolving Crystal] Vacancies in the droplet. $G(2)$ – – – Surface $Ga$  $Ga$ atom at droplet/crystal interace [Etching] Bond near Droplet/Crystal interface [Dissolving Crystal] $G(3)$ – – – Bond near Droplet/Crystal interface [Dissolving Crystal] $G(4)$ – – – Crystal Bulk