Kris's Research Notes

October 6, 2010

Crystal Growth Again

Filed under: GaAs Simulations — Kris Reyes @ 1:44 pm

I repeated the experiment in this post, but I set \mu_{As} = 0.25. This means it is easier for As atoms to desorb.

T \ r_{\downarrow As} 2 4 6 8 10 12 14 16 18 20
600 movie movie movie movie movie movie movie movie movie movie
700 movie movie movie movie movie movie movie movie movie movie
800 movie movie movie movie movie movie movie movie movie movie
900 movie movie movie movie movie movie movie movie movie movie
1000 movie movie movie movie movie movie movie movie movie movie

We may also plot surface Ga concentration as a function of T and r_{\downarrow As}.

This shows the percent surface Ga concentration for the \mu_{As} = 0.25 case on top of the \mu_{As} = 0.35 case.

Advertisements

October 4, 2010

Comparing Marginal Height Profiles – Part 2

Filed under: Stat. Mech for SOS Model — Kris Reyes @ 3:58 pm

This is a follow up to this post.  Recall, in our KMC simulations, we had originally assigned atom hopping rates

r_{i, i\pm1} =\begin{cases} \frac{1}{2}R_i & i = 2, \hdots , N-1 \\ R_i & i = 1, N. \end{cases}

where R_i = \Omega^\prime e^{-\beta \gamma n_i}, and n_i is the number of lateral bonds of the top atom at site i (and we count the atom-wall bond at site N as half a bond). Here \Omega^\prime = 5\times 10^7 and \gamma = 0.25 eV. The different cases here in r_{i, i\pm1} arise due to how the simulations sample a hop event — the code first samples a site according to rate R_i, then hops the atom to the left or right with equal probability (hence the 1/2) if i \neq 1, N and directly to the right or left if i = 1 or N, respectively (and hence no 1/2 factor).  This is incongruous to our analysis of the average height profile and its evolution.

We can rectify this in two ways. First, we can change the simulation so that r_{N, N-1} = \frac{1}{2} R_N and r_{1, 2} = \frac{1}{2} R_1.  Second, we can change the evolution of \bar h(t) by replacing expressions \frac{1}{2} R_1 with R_1 and similarly \frac{1}{2} R_N with R_N. We opt to make the first change. Then we have the usual expressions for the evolution of \bar h_k:

\displaystyle \frac{d \bar h_k}{dt} = \begin{cases}\frac{\Omega}{2}\left(-e^{\beta\mu_1} + e^{\beta\mu_2}\right) & k = 1 \\ \frac{\Omega}{2}\left(e^{\beta\mu_{i-1}} - 2e^{\beta\mu_i} e^{\beta\mu_{i+1}}\right) & k = 2, \hdots, N-1 \\ \frac{\Omega}{2}\left(e^{\beta\mu_{N-1}} - e^{\beta\mu_N}\right) & k =N \\ \end{cases},

where \Omega = \Omega^\prime e^{-\beta\gamma}.

This allows us to approximate the evolution of the average height profile. In this run, we have 4000 KMC simulations of an initially sinusoidal height profile with period 64 and amplitude 64. We measure the height profile every 0.01 second. This allows to calculate the empirical average height profile h_{emp}(t) and the empirical distributions for the height differences. We compare this to the expected height profile \bar h, which we compute using the evolution equation above and initial conditions \bar h(0.01) = \bar h_{emp}(0.01). We also compare empirical vs. predicted marginal distributions.

We plot the empirical data along with the predicted data below. Each row corresponds to t \in \left\{0.01, 0.04, 0.09, 0.14, 0.19, 0.24\right\}. The first plot in a row is the plot of \bar h and \bar h_{emp}. Subsequent plots in a row are expected and observed plots of the marginal distribution for h_i(t) - h_{i-1}(t) for i \in \left\{1, 16, 32, 48, 64\right\}. In these graphs, blue plots are data observed from the simulations while red plots are the expected average height profile and marginal distributions. We note two things. First, the average height profile predicts the simulation data very well. Second, except for the case where i=1, we get a good match in the marginal distributions as well. When t = 1, 4 or 9, the marginal distribution for h_1 is not good. Note these correspond to the case when h_1 is much larger than h_0 =0. (more…)

October 1, 2010

Some Movies of Crystal Growth

Filed under: GaAs Simulations — Kris Reyes @ 4:02 pm

In this run, I changed the prefactor \Omega in the swapping rates of two atoms A, B:

r_{swap}(A, B) = \Omega e^{-\beta \Delta E_{A,B}},

where \Delta E_{A,B} is the (neighborhood-specific) some measure of the change in energy when swapping atoms A and B. Before it was set at 10^{13}, and I lowered it to 10^9. This has the effect to scaling back the swapping rates and desorption while holding deposition rates constant. This has essentially the same effects of scaling up the deposition rates, but allows us to work with deposition rates that are not as crazy as e.g. 10000 monolayers per second. It also is the one which we haven’t really calculated the exact values for, whereas we had calculated (or know) to some degree the other parameters — except for desorption potential.

I changed the width to 1024 atoms. It was 256 atoms previously.

I also changed bonds between “extended types” slightly. Recall an extended type for an atom encodes in some way the local neighborhood and species of that atom. We had defined 6 types of atoms, which depend on the number of different-species neighbors an atom had:

  • Ga^{(0)} has zero As neighbors;
  • Ga^{(1)} has one, two or three As neighbors;
  • Ga^{(2)} has four As neighbors;

and similarly for As^{(0)}, As^{(1)}, As^{(2)}. The pairwise bond strengths (in eV) are now:

Ga^{(0)} Ga^{(1)} Ga^{(2)} As^{(0)} As^{(1)} As^{(2)}
Ga^{(0)} 0.3 0.25
Ga^{(1)} 0.25 0.25 0.25 0.95
Ga^{(2)} 0.95 1.0
As^{(0)} 0.05 0.05
As^{(1)} 0.25 0.95 0.05 0.05
As^{(2)} 0.95 1.0

The desorption potential for As was set at \mu_{As} = 0.35 eV and \mu_{Ga} = \infty. The deposition rate of Ga was set at r_{\downarrow Ga} = 2 monolayers/sec. I varied the deposition rate for As

r_{\downarrow As} \in \left\{2, 4, 6, \hdots, 20\right\} monolayers/sec.

I also varied the temperature

T \in \left\{ 600, 700, 800, 900, 1000\right\} K.

I ran each trial for a simulation time of 10 seconds. The following table has are links to the movie files. (Note: The horizontal/vertical aspect ratio is almost 1:1 and so because of the width of the crystal, the images are quite long in the horizontal direction. Also note that if you open these in Firefox, you may have to “unzoom” the image by clicking on it, since Firefox tries to fit the entire image into your browser window.)

T \ r_{\downarrow As} 2 4 6 8 10 12 14 16 18 20
600 movie movie movie movie movie movie movie movie movie movie
700 movie movie movie movie movie movie movie movie movie movie
800 movie movie movie movie movie movie movie movie movie movie
900 movie movie movie movie movie movie movie movie movie movie
1000 movie movie movie movie movie movie movie movie movie movie

For each temperature, and Arsenic deposition rate, we can calculate the average (over time) surface Ga concentration, where we count an atom as a surface atom if in contact with the vacuum. For a fixed temperature, we may plot this average as a function of Arsenic deposition rate, and compare the different plots as we vary temperature.This is what the following graph shows. The temperatures T = 600, 700, 800, 900, 1000 correspond to the red, green blue, black and magenta plots, respectively.

We may further increase r_{\downarrow As} for the T= 1000 case if we wanted to see how the plot behaves. Here I plotted the average surface Ga concentration, adding r_{\downarrow As} \in \left\{30, 40, 50\right\}. Here is the plot:

« Newer Posts