# Kris's Research Notes

## August 23, 2012

### Reaction Events

Filed under: KMC, Nanowire — Kris Reyes @ 9:53 pm

In this note, we describe a new reaction event which is incorporated in our model in order to correctly simulate VLS nanowire growth.

A reaction event is when an atom changes it species $A$ to another species $B$, leading to a transition $X \rightarrow Y$ in configuration space:

The activation energy for the reaction is of a form similar to all other activation energies in our model:

$\displaystyle E_{a}(X, Y) = \Delta E(X,Y) + \rho(A, B)$,

where $\Delta E(X,Y)$ is the difference in energy between the current state $X$ and some intermediate state $X\wedge Y$ occurring during the reaction and $\rho(A,B)$ is an additive reaction barrier that depends on the species $A, B$. As per our model, the intermediate state is one in which the reacting atom is replaced with an intermediate species $AB$, representing a combination of $A$ and $B$ species. The energy of the intermediate state is obtained by bond counting, where bond strengths are determined by averaging:

$\displaystyle \gamma(AB, C) = \frac{1}{2}\left(\gamma(A, C) + \gamma(B, C)\right).$

The barrier $\rho(A,B)$ may be assigned according to the neighborhood about the reacting atom. If the reacting atom is in a liquid neighborhood, we use liquid barriers $\rho_{liquid}(A,B).$ This is used in our VLS simulations by assigning relatively low reaction barriers within the liquid droplet, which models the catalytic effect of the liquid.
Similar barriers $\rho_{bulk}$ may be assigned for bulk neighborhoods.

Assuming the reaction does not affect the categorization of neighborhood (i.e. normal, liquid or bulk), the above activation energies lead to rates that satisfy detailed balance if and only if the barriers are symmetric with respect to the reacting and reacted species: $\rho(A, B) = \rho(B, A)$. Note that because of the energy dependent term $\Delta E$, this does not imply that the rates of reaction $A \rightarrow B$ and its reverse $B \rightarrow A$ are the same. If the barrier is not symmetric, then the reaction is irreversible in the sense that the reaction rates are not time reversible (i.e. do not satisfy detailed balance).

## VLS Nanowire Growth

We simulate VLS nanowire growth by considering a liquid droplet of species B on a substrate of material C on a hexagonal lattice. Material A is deposited uniformly on the surface. Atoms of material A can react and become atoms of material C, and the barrier for this reaction in the liquid $\rho_{liquid}(A,C)$ is different from the barrier of the reaction away from the liquid $\rho(A,C)$. No other reactions are considered. This is done by setting all other reaction barriers nominally high. In particular, the reverse reaction $C \rightarrow A$ is not allowed.

We consider the effect of $\rho(A,C)$ and $\rho_{liquid}(A,C)$ on the resulting nanowire. The following bond strengths (in eV) were used:

 $\gamma$ A B C A 0.10 0.10 0.10 B 0.10 0.40 0.35 C 0.10 0.35 0.50

The desorption barrier for A was set at $\mu(A) = 0.50$ eV. All other species were not allowed to desorb.
We see that the unreacted species $A$ is weakly bonded with all other species and can desorb readily. Exchange barriers are $\epsilon_{D} = 0.7$ eV and $\epsilon_{D} = 1.0$ eV for exchanges in and away from liquid neighborhoods, respectively. Bulk events were not allowed. We vary $\rho(A,C)$ as

$\displaystyle \rho(A,C) \in \left\{ 0.5, 1.0, 1.5, \infty \right\}$ eV.

In the small barrier regime, the vapor species (A) is likely to react to form the solid species (C) uniformly in the domain, and hence we do not expect nanowire growth. In the large barrier regime, atoms of material A can only possibly become solid atoms in the droplet (though this depends on $\rho_{liquid}(A,C)$.

The liquid barrier for reaction is varied as

$\displaystyle \rho_{liquid}(A,C) \in \left\{ 0.5, 0.75, 1.0, 1.25, 1.5\right\}$ eV.

In the low barrier regime, the reaction $A \rightarrow C$ occurs quickly, so we expect the wire to grow fast. In the high barrier regime, the reaction does not occur in a reasonable time scale, so we do not expect to see growth of the solid material underneath the droplet.

Material A was deposited for 6000 seconds at a rate of 0.1 ML/sec at a temperature or 300C. Here are the results, organized from left to right with increasing $\rho_{liquid}$ and from top to bottom with increasing $\rho$. The liquid droplet of species B is colored green, while the original solid material of species C is colored purple. The solid material which was initially deposited as species A and reacted to form species C is blue.

Qualitatively, the simulations behave as expected. In the low $\rho$ case, we see a planar front of solid material, suggesting that the deposited A material turned to a solid material C uniformly throughout the domain. This persists up to a barrier value of $\rho = 1.0$ eV. For higher values, reactions are not favored anywhere away from the droplet. We note that for values of $\rho \geq 1.5$ eV, we observe nanowire growth. In the case of low $\rho_{liquid}$, more solid material C is present. This is consistent with a small value for $\rho_{liquid}$, implying that the deposited material A was readily converted to material C before escaping the system. We observe that as $\rho_{liquid}$ is increased, the amount of solid material present decreases, so that in the extreme case of large $\rho_{liquid}$ and $\rho$, there is no growth of the solid material.