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 onto a rectangular domain of liquid Ga in contact with a GaAs slab:

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

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

This leads to the diffusion equation:

where is an atomic length, is the diffusion coefficient of in liquid , and is the concentration of on the interface. The last equation above gives the evolution of the interface in terms of its normal velocity and the outward pointing normal vector .

This system admits a quasi-static, planar solution

where From the last equation in the above system, (i.e. constant), where . That is, the interface growth is uniform with respect to , 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 to the flat interface of the form

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

where we assume the form

,

for some function to be determined later.

### Boundary Conditions

The boundary conditions on the interface are subsequently altered when a perturbation is introduced. By the Gibbs-Thomson relationship, the concentration on the interface is given by

where is the liquid-solid interfacial energy and is the curvature of the interface. Linearizing both the exponential expression and we obtain:

,

where . The left-hand side admits linearizations as well, by Taylor expansion about :

.

Comparing the two expressions for , we obtain the linearized boundary condition:

.

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

,

which decomposes the normal velocity of the interface in terms of the velocity of the perturbation and the planar velocity . Th left-hand side may be linearized by observing

so that

Combining these two expressions yields the boundary condition:

### Critical Wave Number

Plugging in the expression for in terms of and as well as the above boundary conditions, we obtain the following system:

,

,

.

Solving this system yields a relationship between and :

,

Assuming , we observe exactly when . Recall the expression for the perturbation

so that if , the perturbation grows, while implies decay. Hence, we arrive at a critical wave number

so that yields an unstable perturbation:

Recalling and writing in Arrehnius form:

and as in previous posts:

,

we get an expression for in terms of the energy barriers as well as experimental parameters and :

We see that increasing and leads to larger , 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 . Fix K and vary 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 and monolayers/second, but an instability occurs at the high As flux of .

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

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