# Kris's Research Notes

## April 26, 2012

### Mullins Sekerka Instability and Liquid Cores

Filed under: GaAs Simulations — Kris Reyes @ 10:55 pm

In this note, we model the formation of liquid cores during droplet crystallization as a Mullins Sekerka instability. We then compare the theoretical critical curve for the instability with simulation results.

Recall that in this post, we established a critical wavenumber for a perturbation to grow via Mullins-Sekerka instability. This expression was:

$\displaystyle k_c = \sqrt{\frac{F_{As} k_B T}{R_0 \ell^2 \gamma}} \exp\left[ \frac{1}{2}\frac{E_D - E_A + E_{As}}{k_B T}\right]$,

where:

• $F_{As}$ is the As flux during crystallization,
• $T$ is the temperature,
• $R_0 = 10^{13}$ is the rate prefactor,
• $\ell = 1$ is the atomic spacing,
• $\gamma = 0.1$eV is the surface-liquid interfacial energy,
• $E_{As} = 0.7$eV is the barrier for the diffusion of As in liquid Ga,
• $E_D, E_A$ are the detachment and attachment barriers of an As atom from and onto a flat GaAs substrate, respectively.

In order for this perturbation to arise within a liquid droplet of radius $R$, we necessarily require that $R > C\frac{2\pi}{k_c}$, for some constant of proportionality $C \geq 1$, otherwise modes of any perturbation would not have sufficiently small wavenumber (equiv. sufficiently large wavelength) to induce a instability. Recall we modeled the droplet radius as

$\displaystyle R = r_0 \left(\frac{D_G}{F_G}\right)^\alpha = \frac{r_0R^\alpha_0}{F_G^\alpha}\exp\left[ -\frac{\alpha E_G}{k_B T} \right],$

where is $r_0 = 11.34$ constant of proportionality, $F_G = 0.1$ monolayers/second is the flux at which Ga was deposited, $E_G = 0.9$ eV is the energy barrier for Ga-on-Ga diffusion and $\alpha = 0.182$ is the power-law the radius obeys. Hence the Mullins-Sekerka condition $R \geq C\frac{2\pi}{k_C}$ may be written as:

$\displaystyle \frac{r_0R^\alpha_0}{F_G^\alpha}\exp\left[ \frac{\alpha E_G}{k_B T} \right] \geq C 2\pi \sqrt{\frac{R_0 \ell^2 \gamma}{F_{As} k_B T}} \exp\left[ -\frac{1}{2}\frac{E_D - E_A + E_{As}}{k_B T}\right]$

Solving for $F_{As}$, we obtain

$\displaystyle F_{As} = C^2\frac{R^{1-2\alpha}_0 F_G^{2\alpha}\ell^2\gamma}{r_0^2k_B T} \exp\left[ \frac{E_A - E_D - E_{As} + 2\alpha E_G}{k_B T}\right]$

Compare this to the expression for when surface nucleation occurs (without making any approximations):

$\displaystyle F_{As} = \frac{R_0^{1-\alpha}F_{Ga}^\alpha}{r_0} \exp\left[ \frac{E_A - E_D - E_{As} + \alpha E_{Ga}}{k_B T}\right]\left( \exp\left[\frac{E_{nuc}}{k_B T}\right] - 1 \right)$.

Before, we had set $E_A = 0.33, E_D = 1.29$ eV (values obtained from simulations) and fit $E_{nuc} = 0.36$ again to simulation data. We had argued that the barriers $E_A, E_D$ closely matched the attachment and detachment events illustrated below:

This is incorrect! Instead, we must consider the attachment of an As atom onto a flat substrate and its reverse event:

Here, $E_A = 0.4$ eV and $E_D = 0.9$ eV. We may then fit the nucleation barrier $E_{nuc} = 0.01$ eV, a very small barrier to nucleation. Picking these values yield the following phase diagram:

This is a comparison between simulation data and our theoretical curves above. Red crosses correspond to simulations where liquid cores were observed. Blue crosses correspond to simulations where nucleation is present. The red line is the critical line for the Mullins-Sekerka instability, while the blue line is the theoretical critical curve for nucleation.

We observe an accurate fit. It is important to note: only the energy $E_{nuc}$ was fit to simulation data, and its value is consistent to the idea that the barrier to nucleation is very small. The other parameters were obtained from first principles.