# 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.

## 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: