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.

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1 Comment »

  1. […] that in this post, we established a critical wavenumber for a perturbation to grow via Mullins-Sekerka instability. […]

    Pingback by Mullins Sekerka Instability and Liquid Cores « Kris's Research Notes — April 26, 2012 @ 10:55 pm


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