Kris's Research Notes

June 7, 2011

Etching without Extended Species

Filed under: GaAs Simulations — Kris Reyes @ 6:25 pm

In this post, we show how droplet etching is possible with the simplified model. Recall that this process was the issue with the old model — we found that the rates of an As atom detaching from the substrate into the droplet and the reverse process were too disparate, which meant that etching would not occur in a reasonable time scale. With the new model, the difference in rates between these two events is now a reasonable amount, allowing for droplet etching.

Intermediate Species and Energy Barriers

In order to satisfy detailed balance any transition X \rightarrow Y, we define an intermediate state latex X \wedge Y = Y \wedge X, and assign the transition rate

\displaystyle r_{X\rightarrow Y} = R_0 \exp\left[\beta(E(X)-E(X\wedge Y)\right].

For atom-atom exchanges, we achieve this intermediate state by replacing the two exchanging atoms with atoms of an intermediate species. Consider the transition between X \rightarrow Y of an As atom detaching from the GaAs substrate into the Ga droplet:

Above, we also illustrate the intermediate state X\wedge Y with intermediate species (which we shall denote as “H”) in blue. The energies of the three states:

\displaystyle E(X) = 4\gamma_{GG} + 4\gamma_{GA} + 3\gamma^\prime_{GG} + 3 \gamma^\prime_{AA},

\displaystyle E(Y) = 4\gamma_{GG} + 4\gamma_{GA},

\displaystyle E(X\wedge Y) = 8\gamma_{GH} + 3\gamma^\prime_{GH} + 3\gamma^\prime_{AH} + \gamma^\prime_{HH}.

Recall: \gamma_{\cdot} is a nearest-neighbor bond while \gamma^\prime_{\cdot} means a next-nearest neighbor bond. We set \gamma^\prime_{GA} = 0, so we do not include it in the above energies. If the intermediate species is thought of half Ga and half As, and the bonds between the intermediate species are appropriately averaged, i.e.

\displaystyle \gamma_{GH} = \frac{1}{2}( \gamma_{GG} + \gamma_{GA} ),

\displaystyle \gamma_{AH} = \frac{1}{2}( \gamma_{AA} + \gamma_{GA} ),

\displaystyle \gamma^\prime_{GH} = \frac{1}{2}( \gamma^\prime_{GG} + \gamma^\prime_{GA} ) = \frac{1}{2}\gamma^\prime_{GG},

\displaystyle \gamma^\prime_{AH} = \frac{1}{2}( \gamma^\prime_{AA} + \gamma^\prime_{GA} ) = \frac{1}{2}\gamma^\prime_{AA},

the energy E(X\wedge Y) can be rewritten

\displaystyle E(X\wedge Y) = 4\gamma_{GG} + 4\gamma_{GA} + \frac{3}{2}\gamma^\prime_{GG} + \frac{3}{2}\gamma^\prime_{AA} + \gamma^\prime_{HH}.

The difference in energies are given by:

\displaystyle E(X) - E(X\wedge Y) = \frac{3}{2}(\gamma^\prime_{GG} + \gamma^\prime_{AA}) - \gamma^\prime_{HH}.

\displaystyle E(X) - E(Y) = 3(\gamma^\prime_{GG} + \gamma^\prime_{AA})

Recall the difference E(X) - E(Y) measures the disparity of rates as in:

\displaystyle \frac{r_{X\rightarrow Y}}{r_{Y\rightarrow X}} = \exp\left[\beta(E(X)-E(Y))\right].

Therefore, if the nearest-neighbor bonds are sufficiently small, this disparity can be a reasonable number, allowing for droplet etching.

Etching Results

We set \gamma_{GG} = 0.3 eV, \gamma_{AA} = 0.10 eV, \gamma_{GA} = 0.6 eV, \gamma^\prime_{GG} = 0.20 eV, \gamma^\prime_{AA} = 0.2 eV, \gamma^\prime_{HH} = -0.5 eV and \mu_{A} = 0.35 eV. The energies above are then

E(X) = 4.8 eV,

E(X\wedge Y) = 4.7 eV,

E(Y) = 3.6 eV

Our initial profile was a Ga droplet on a GaAs substrate. We allowed the system to anneal for 10 seconds at a temperature of 593K.

Here is the result:

In the movie, there are several detachment/attachment events for an As atom at the droplet/substrate interface. This is due to the difference in barrier E(X) - E(Y) = 1.2 eV. But this difference is small enough so that given a reasonable amount of time a detached As atom can escape from the neighborhood close to this interface. That is, while r_{Y \rightarrow X} is large, given a sufficient amount of time, we may see a transition Y \rightarrow Z. With the current set of parameters, we see that this time is on the order of 10 seconds. Once this event does occur, the substrate is somewhat destabilized, which allows the droplet to etch more rapidly.

N.B. There is some evidence for the formation of hills about the etched droplet (at least on one side for this simulation), but the apparent hill is too diffuse and does not grow like what is seen in experiments. I have tried to address this in several ways:

  • CrystallizationMy initial thought was that hills would form if we attempted to crystallize the etched droplet. Here is the same etched droplet, but with As flux turned on at one monolayer/second:

    We see that in fact, the system flattens out. This may indicate that the atoms are diffusing too fast.

  • Lowering the temperature In experiments, at a temperature of T=593K, droplets are believed to have etched into the substrate after a short period of time and produce small rings on an otherwise flat substrate. I tried to lower the temperature to T=523, where there is presumably little etching. I started with a droplet completely on top of the substrate, unetched and crystallized with an As flux of one monolayer per second:

    While this does indeed form hills on the sides of the droplet, there is nucleation on the surface of the droplet, which may not be desirable. If we change \gamma^\prime_{HH} to -0.45 eV, we prevent nucleation on the surface of the droplet, but again the geometry is not correct:

So while we can successfully etch into the substrate, we cannot form hills.

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