# Kris's Research Notes

## April 26, 2012

### A Mullins-Sekerka Instability for Crystallization

Filed under: GaAs Simulations — Kris Reyes @ 9:05 am

In this note, we show how the deposition of As into liquid Ga exhibits a Mullins-Sekerka instability.

## Quasi-static Planar Growth

We consider the deposition of As at a flux of $F_{As}$ onto a rectangular domain of liquid Ga in contact with a GaAs slab:

The domain has height $L$ and periodic boundaries on the side. We make two assumptions:

1. Quasi-static deposition: the $As$ atoms diffuse quickly relative to deposition rate so that the system is always in equilibrium.
2. Moving boundary: We assume the interface moves so that the (average) distance between the interface and top boundary is always $L$.

This leads to the diffusion equation:

$\displaystyle D\nabla^2 c = c_t \approx 0$

$\displaystyle \ell D c_y \big|_{y=L} = F_{As}$

$\displaystyle c\big|_{y=0} = c_0$

$\displaystyle \ell^2 D (\nabla c \cdot \mathbf{n}) = v_n$

where $\ell$ is an atomic length, $D$ is the diffusion coefficient of $As$ in liquid $Ga$, and $c_0$ is the concentration of $As$ on the interface. The last equation above gives the evolution of the interface in terms of its normal velocity $v_n$ and the outward pointing normal vector $\mathbf{n}$.

This system admits a quasi-static, planar solution

$c_{st}(y) = \nu y + c_0$

where $\nu = \frac{F_{As}}{\ell D}.$ From the last equation in the above system, $v_n = \alpha \nu$ (i.e. constant), where $\alpha = \ell^2 D$. That is, the interface growth is uniform with respect to $x$, and hence we have planar growth.

## Linear Perturbation Analysis

We analyze the linear stability of this system by introducing a perturbation to the interface (and hence to the underlying concentration field):

Specifically, we consider a sinusoidal perturbation $h(x)$ to the flat interface of the form

$\displaystyle h(x) = h_0 \sin(kx) e^{\omega t}$

We consider the resulting concentration as a perturbation from the stationary solution above, writing

$\displaystyle c(x,y,t) = c_{st}(y) + u(x,y,t),$

where we assume the form

$\displaystyle u(x,y,t) = u_{k}(y) \sin(kx)e^{\omega t}$,

for some function $u_{k}(y)$ to be determined later.

### Boundary Conditions

The boundary conditions on the interface $y = h(x)$ are subsequently altered when a perturbation is introduced. By the Gibbs-Thomson relationship, the concentration on the interface is given by

$\displaystyle c(x, h(x)) = c_0 \exp\left[\frac{\ell \gamma \kappa(x)}{k_b T}\right],$

where $\gamma$ is the liquid-solid interfacial energy and $\kappa(x)$ is the curvature of the interface. Linearizing both the exponential expression and $\kappa$ we obtain:

$\displaystyle c(x, h(x),t) \approx c_0 + \sigma \left| h_{xx}\right|$,

where $\sigma = c_0 \frac{\ell \gamma}{k_b T}$. The left-hand side admits linearizations as well, by Taylor expansion about $y = 0$:

$\displaystyle c(x, h(x),t) \approx \nu h(x) + c_0 + u(x,0,t) + \mathcal O\left(h u_y\right)$.

Comparing the two expressions for $c(x, h(x), t)$, we obtain the linearized boundary condition:

$\displaystyle u(x, 0, t) = \sigma \left| h_{xx} \right| - \nu h$.

The normal velocity boundary condition may be similarly linearized. On one hand, we have (by level set methods):

$\displaystyle v_n = h_t + \alpha\nu$,

which decomposes the normal velocity of the interface in terms of the velocity of the perturbation and the planar velocity $\alpha \nu$. Th left-hand side $\alpha(\nabla c \cdot \mathbf n)$ may be linearized by observing

$\displaystyle \mathbf n = \left(\mathcal O(h), 1 + \mathcal O(h^2)\right)$

so that

$\displaystyle \nabla c \cdot \mathbf n = \nu + \mathcal O(u_x h) + u_y + \mathcal O(h^2 u_y^2)$

Combining these two expressions yields the boundary condition:

$\displaystyle \alpha u_y(x, h(x), t) = h_t.$

### Critical Wave Number

Plugging in the expression for $c$ in terms of $c_{st}$ and $u$ as well as the above boundary conditions, we obtain the following system:

$\displaystyle \nabla^2 u = 0$,

$\displaystyle u_y(x, L, t) = 0$,

$\displaystyle u(x, 0, t) = \sigma\left|h_{xx}\right| - \nu h$

$\displaystyle u_y(x, h(x), t) = \frac{1}{\alpha} h_t$.

Solving this system yields a relationship between $\omega$ and $k$:

$\displaystyle \omega = \alpha k(\nu -\sigma k^2)\left(\frac{e^{2kL}-1}{e^{2kL} + 1}\right)$,

Assuming $e^{2kL} \gg 1$, we observe $\omega > 0$ exactly when $k < \sqrt\frac{\nu}{\sigma}$. Recall the expression for the perturbation

$\displaystyle h(x, t) = h_0\sin(kx)e^{\omega t},$

so that if $\omega > 0$, the perturbation grows, while $\omega < 0$ implies decay. Hence, we arrive at a critical wave number

$\displaystyle k_c = \sqrt{\frac{\nu}{\sigma}}$

so that $k < k_c$ yields an unstable perturbation:

Recalling $\nu = \frac{F_{As}}{\ell D}, \sigma = c_0\frac{\ell\gamma}{k_b T}$ and writing $D$ in Arrehnius form:

$\displaystyle D = R_0 \exp\left[ -\frac{E_{As}}{k_b T} \right],$

and $c_0$ as in previous posts:

$\displaystyle c_0 = \exp\left[\frac{E_A - E_D}{k_B T}\right]$,

we get an expression for $k_C$ in terms of the energy barriers $E_{As} E_A E_D$ as well as experimental parameters $F_{As}$ and $T$:

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

We see that increasing $T$ and $F_{As}$ leads to larger $k_c$, implying more modes are unstable.

For example, consider a slab of liquid Ga on top of a GaAs substrate with a sinusoidal perturbation of the solid/liquid interface with wave number $k = \frac{2\pi}{128}$. Fix $T = 623$ K and vary $F_{As} = 0.5, 1.0, 2.0$ monolayers/second. Here are the final atom configurations after 60 seconds of depositing As:

 0.5 monolayers/second 1.0 monolayers/second 2.0 monolayers/second

Planar growth is observed when $F_{As} = 0.5$ and $1.0$ monolayers/second, but an instability occurs at the high As flux of $F_{As} = 2.0$.