Ledge growth of facetted interfaces

Figure 27: The growth of a facet by ledge mechanism. The ledge grows laterally at a velocity of $u$, for a distance $\ell$; the height of the ledge is $h$. Due to the growth of the ledge, the facet thickens (grows normal to itself) at a velocity of $v$.

[scale=0.4]Figures/LedgeGrowthSchematic.pdf

The facetted interfaces usually grow by ledge mechanism. Consider the schematic of a facet growing by the ledge mechanism (Fig. 27). The facet grows a distance $\ell$ by the movement of the step of height $h$. If the step moves at a velocity of $u$, the velocity of the interface normal to itself is given by $v$, and,

$$v = \frac{uh}{\ell}$$ (13)

In this case also, as in Eq. 62, we can obtain the lengthening rate $u$ (by replacing $r$ by $h$, and not incorporating the Gibbs-Thomson correction)

$$u = \frac{D}{X_e^{\beta} - X_e} \frac{\Delta X_0}{kh}.$$ (14)

Thus, the ledge growth rate is obtained as

$$v = \frac{D}{X_e^{\beta} - X_e} \frac{\Delta X_0}{k \ell}$$ (15)