Kris's Research Notes

February 15, 2012

Bulk Diffusion and Liquid Cores

Filed under: GaAs Simulations — Kris Reyes @ 6:27 pm

Recall, we had discovered that antisite defects in GaAs diffused through the crystal quite quickly. In this note, we discuss a bulk diffusion barrier that addresses this problem.

Consider the exchange of an As atom and Ga atom at the liquid/solid interface (the transition $X \rightarrow Y$ in the figure below) and the resulting diffusion of the antisite defect through the bulk of the solid ($Y \rightarrow Z$):

By bond counting, the transition $X \rightarrow Y$ has an energy barrier of 1.7 eV, whereas the transition $Y\rightarrow Z$ has a barrier of $\gamma_{HH} = 0.7$ eV. This means that an antisite defect introduced into the bulk phase diffuses rapidly throughout the solid. We had previously seen this in the crystallization of liquid cores — liquid Ga in the cores were able to diffuse away from the core, displacing As in the process. This displaced As would end up in the liquid core, and eventually crystallizing it.

It is undesirable to have such fast bulk diffusion rates. Indeed, it often reasonable to eliminate bulk diffusion entirely from the model — it occurs at such a long time scale to be irrelevant for the processes we simulate. In order to have a separate bulk diffusion rate, we do not use bond counting to calculate the associated energy barrier. Instead, we must recognize anti-site defects by examining an atom’s local neighborhood, and give such defects the bulk diffusion barrier. This is similar to how we assign liquid diffusion rates to As atoms in Ga droplets.

An atom at or near an anti-site defect is classified as such by counting the deviation of its local neighborhood from a perfect crystal local neighborhood. If the neighborhood has exactly one defect (relative to a perfect crystal neighborhood), we label the atom at or near an antisite defect:

An exchange between two atoms is given a bulk diffusion barrier $\sigma_{\text{Bulk}}$ if and only if both atoms have such a neighborhood — i.e. if and only if both atoms are at or near an antisite defect. Note that this satisfies detailed balance. When two bulk atoms exchange, the resulting state will necessarily have two atoms that are at or near a anti-site defect. Hence the barrier for the reverse event is the same bulk diffusion barrier. The diffusion of antisite defects will not change the total energy, and hence detailed balance is satisfied (that is $\frac{\pi_X}{\pi_Y} = 1 = \frac{r_{Y\rightarrow X}}{r_{X \rightarrow Y}}$).

Here is a movie of a Ga liquid core inside GaAs. Here we have set $\sigma_{\text{Bulk}} = 10$ eV, effectively killing any bulk diffusion. The temperature is 623K, and the total simulation time is one second. The liquid core stays coherent inside the crystal:

Indeed, we expect this to be the case, as there is an interfacial energy between liquid and solid phases. To confirm this, we distribute some volume of Ga into a GaAs crystal to create several antisite defects. We set $\sigma_{\text{Bulk}} = 1.20$ eV — just some reasonable number to allow for some bulk diffusion. We then anneal the system and measure the total energy as a function of time.

First, here is the movie for the first 0.001 seconds:

We see the liquid Ga coarsens. Here is the total energy vs. time:

As the system coarsens, the energy decreases. Here is the system over the first 1 second:

Lastly, here is corresponding energy of the system over the first 1 second: