Kris's Research Notes

November 3, 2010

GaAs Crystal Growth: Phase Diagram

Filed under: GaAs Simulations — Kris Reyes @ 10:53 pm

Recall we wished to examine how a GaAs crystal is terminated as a function of deposition rates r_{\downarrow Ga}, r_{\downarrow As} (measured in monolayers per second) and temperature T (measured in degrees Kelvin). Specifically, we wish to determine the proportion of  Gallium surface atoms (defined to be Gallium atoms exposed to vacuum) to surface atoms in general. In the following trials, we vary temperature

T \in \left\{ 400, 417, 435, 455, 476, 500, 526, 555, 588, 625, 666, 714, 769, 833, 909, 1000 \right\}

and deposition ratio \rho = r_{\downarrow As}/r_{\downarrow Ga}

\rho \in \left\{ 0.1, 0.2, 0.4 ,0.6, 0.8, 1, 2, 4, 6, 8, 10, 20 , 40, 60 , 80, 100 \right\}.

The energy \gamma(Ga(4), As(2) is the relevant bond strength for an Arsenic atom on the surface of the crystal, and I vary this as well. The surface Gallium bond strength \gamma(Ga(2), As(4)) was fixed at 0.7 eV, which was left over from the droplet experiments (in a later post, I will change this to a more symmetric case).

Below I have plotted the contours of the Percent Surface Gallium as a function of \rho and \frac{1}{T}, which I did for three cases. Note, I ran each simulation for a maximum of three hours and several trials did not finish completely. I shall discuss this below. (Note the horizontal label is incorrect. It should read r_{\downarrow As}/r_{\downarrow Ga}.)

r_{\downarrow Ga} = 1 monolayer/second. \gamma(Ga(4), As(2)) = 0.95 eV (base case);

.

1/1000 1/909 1/833 1/769 1/714 1/666 1/625 1/588 1/555 1/526 1/500 1/476 1/455 1/435 1/417 1/400
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r_{\downarrow Ga} = 5 monolayer/second. \gamma(Ga(4), As(2)) = 0.95 eV (the effect of raising Gallium deposition);

1/1000 1/909 1/833 1/769 1/714 1/666 1/625 1/588 1/555 1/526 1/500 1/476 1/455 1/435 1/417 1/400
100 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
80 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
60 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
40 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
20 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
10 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
8 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
6 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
4 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
2 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
1 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
0.8 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
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r_{\downarrow Ga} = 1 monolayer/second. \gamma(Ga(4), As(2)) = 0.85 eV (the effect of lowering surface Arsenic bond strength).

1/1000 1/909 1/833 1/769 1/714 1/666 1/625 1/588 1/555 1/526 1/500 1/476 1/455 1/435 1/417 1/400
100 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
80 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
60 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
40 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
20 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
10 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
8 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
6 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
4 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
2 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
1 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
0.8 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
0.6 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
0.4 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
0.2 movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie movie
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Discussion

Recall what we observe in experiments. For low temperatures, the deposition ratio at which the phase change occurs is constant and at higher temperatures, this ratio increases exponentially with respect to decreasing inverse temperature (so that in our plots, since the deposition ratio is plotted on a log scale, we should see a linear increase in the critical deposition ratio as the inverse temperature approaches 0). It is hard to tell whether we see this behavior. We do see (if we examine e.g. the green 0.5 contour) the critical deposition ratio is constant for lower temperatures. With increasing temperatures, however, the growth in the critical deposition ratio does not seem to be linear.

Also note that the lower contours (say % surface Gallium = 0.1 or 0.2 — the darker blue contours) exhibit a much different behavior than the other contours. This can be explained easily. For higher temperature, the crystal grows fairly flat and the layer growth exhibits the “zipper” effect — a new layer grows as a few contiguous nucleation sites expands and coalesce  until the entire layer is filled. This happens because the atoms on the surface can diffuse more rapidly with increasing temperatures. Here is an example typical nucleation site in such a regime:

This will expand from the left and right ends until the entire layer is filled. For the lower temperature case, the growth does not occur like this. Instead, many more nucleation sites occur and instead of the flat surface above and so we tend to see a much rougher surface. This explains the increase in surface Gallium concentration. In the above case there are only two exposed Gallium per nucleation site because of the flatness of growth. In the low temperature case, there may be more than two exposed Gallium because of the roughness of the crystal. More over, there are many such sites on the surface, and hence more exposed Gallium. For example, consider the following typical section of the crystal growth for this low-temperature regime.

It is interesting that only the low % Surface Gallium contours exhibit this property. Perhaps if we adjust the bond strengths, we could see a more uniform behavior of the contours.

One more thing to note is that several trials did not complete fully and so for many of the points in the above plots, the sample mean %Surface concentration (the “Z” value) is computed with less samples than other points. I was worried about this, so I calculated the sample variance of my samples as well, but I have not analyzed this.

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2 Comments »

  1. […] GaAs Crystal Growth: Phase Diagram pt. 2 Filed under: GaAs Simulations — Kris Reyes @ 7:03 pm This is a follow-up of this post. […]

    Pingback by GaAs Crystal Growth: Phase Diagram pt. 2 « Kris's Research Notes — November 4, 2010 @ 7:03 pm

  2. […] am currently in the process of fitting the surface termination phase diagram (see this post) to experiments. While doing this, I have learned more about the model. Specifically, I more […]

    Pingback by Model Parameters — Part 2. « Kris's Research Notes — February 4, 2011 @ 9:30 pm


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