Kris's Research Notes

April 8, 2011

Exchange Rules

Filed under: GaAs Simulations — Kris Reyes @ 7:32 am

In this note, we describe the exchange events which can occur in our simulations and the relevant parameters for these events. We then examine the effect of these parameters using a system completely filled with Ga liquid and GaAs substrate with zero Ga and As flux.

Current Exchange Rules

We have two types of (atom-atom) exchange events in our simulations. First is an As atom diffusing through a droplet. Here the As atom exchanges with any of its Ga neighbors. Second is exchanges at the Ga droplet/GaAs substrate, where an As atom exchanges with a droplet Ga atom. Exchanges are given the form

R(A,B) = R_0 \exp\left[ -\beta\epsilon(A,B)\left(E_\ell(A) + E_\ell(B) - \gamma(A,B) - E^* + E_d\right)\right],

where R_0 = 10^{13}, E^* = 2\gamma_{U}, E_d = 1.0 and

E_\ell(A) + E_\ell(B) - \gamma(A,B)

is the expression for all the bonding energies about atoms A and B. The parameter controlling the rate of exchange is the exchange coefficient \epsilon(A,B), which has the effect of scaling the energy barrier. This parameter is set to specific rules defined by the two types of exchanges detailed above:

  • As atom diffusing through a droplet

    We recognize this scenario by looking for a Ga atom with one As neighbor, which handles As atoms on the surface of a droplet in addition to within one.

    All of the above exchanges are given exchange coefficient \epsilon_D. Note that when two As atoms in the droplet meet at a Ga atom, the coefficient of exchange between either of the As atoms and the Ga atom is large since that Ga atom has more than one As neighbor. Moreover, the bonds between the As atoms and the Ga atom become strong, since they are \gamma(Ga(2), As(4)) = \gamma_{P} bonds.

    As the Ga droplet becomes more saturated with As, the above effect leads to GaAs nucleation.

  • Exchanges at the Ga droplet/ GaAs substrate interface

    Exchanges between As and Ga atoms at the Ga droplet/GaAs substrate interface occur between As atoms and Ga atoms with exactly two Ga neighbors and two As neighbors:

    The exchange coefficient for this event is denoted \epsilon_I. Note by specifying the Ga atom to have exactly two Ga neighbors and two As neighbors (that is, to be of type Ga(2,2)), we do not allow the following exchanges:

    Namely, exchanges at the surface of a (vacuum-exposed, Ga-terminated) substrate and those at the (Ga/GaAs/Vacuum) triple-point. While it is clear that the first example is probably correct — we don’t want exchanges in the substrate, it is not so obvious that the second one is true. Moreover, there are probably other pathological examples which could occur and affect those processes associated to crystallization (e.g. ring formation). That is, those examples where Ga liquid, GaAs nucleation and Vacuum are all present in a small neighborhood.

The effect of \epsilon_D, \epsilon_I on a Ga/GaAs system

Here we consider a fully occupied system with Ga liquid on top of a GaAs substrate:

In the simulations, we use a system of size 64 \times 64.

The only events that may occur are the two described above. We know the qualitative effect of the two exchange coefficients on this system: \epsilon_I governs how readily the GaAs substrate destabilizes under Ga, while \epsilon_D measures how fast an As atom diffuses through Ga.

First, we fix \epsilon_I = \epsilon_D = 0.3, and we let the system evolve for 3 minutes. We consider temperatures 473, 523, 573, 623 K, which is how the movies are ordered below. Experiments suggest that Ga droplets completely diffuse into the substrate at 573 K after no longer than 10 minutes.

I had initially expected the Ga to dissolve into the substrate at higher temperatures. Instead, we see that the GaAs interface is rather fluid while the bulk GaAs remains intact. This doesn’t seem correct. For reference, here are similar simulations, but run for only 10 seconds:

However, we can lower with the parameters to get the desired behavior, but not at the correct time-scale. Indeed, lowering the exchange coefficients to around 0.10 or 0.20 yields really fast As diffusion through the Ga liquid, but implies the clock moves much slower because of the lowered energy barrier for exchange. Here are the moves as we vary

\epsilon_D, \epsilon_I \in \left\{ 0.1, 0.2, 0.3\right\}

and fix T = 573 K. These simulations all ran for the same time (0.00001 seconds). While the time units are somewhat arbitrary, we can still see the effect of the parameters on our system.

Here, the table is ordered with \epsilon_D increasing by column and \epsilon_I increasing by row. Here we see for high \epsilon_I (i.e. third row), the one event is sufficient to advance the clock past the simulation time. For small \epsilon_I, we get good mixing between Ga and GaAs. Consider the effect of \epsilon_D. Specifically, consider the cases \epsilon_D = 0.1, \epsilon_I = 0.1 (upper left corner) vs. \epsilon_D = 0.3, \epsilon_I = 0.1. For the second case, we observe that the GaAs seems to stay coherent, and that the Ga liquid wishes to simply pass through the substrate. This is evident toward the end, where we start seeing regions of liquid Ga at the bottom of the system. We do not observe this in the first case.

Therefore lowering \epsilon_I, \epsilon_D will encourage more Ga diffusion into the GaAs substrate, and I believe this is correct for the case when T \geq 300 C. Lowering these parameters around 0.1 produces the correct behavior, but at the wrong time scale.

We now vary \epsilon_D, \epsilon_I \in \left\{0.25, 0.26, ..., 0.30\right\}. These runs are over 10 seconds. The table is again organized so that \epsilon_D is increasing by column (so that the left-most column corresponds to \epsilon_D = 0.25 and the right-most column is \epsilon_D = 0.30) and \epsilon_I is similarly increasing by row.


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