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 = 0.1eV is the surface-liquid interfacial energy,
  • E_{As} = 0.7eV 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.

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