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).



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 strengthsAs(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) },


      \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


  1. […] where is determined to encourage diffusion of an atom through a droplet rather than on its surface (see this post). […]

    Pingback by Droplet Experiments — Effect of Droplet Bonds « Kris's Research Notes — January 31, 2011 @ 5:51 pm

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

    Pingback by Model Parameters — Part 2. « Kris's Research Notes — February 4, 2011 @ 9:31 pm

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