Signature of microscale kinetics in mesoscale description of epitaxial growth

We describe the effect of kinetic interactions of adsorbed atoms in a mesoscale model of epitaxial growth without elasticity. Our goal is to understand how atomic correlations due to kinetics leave their signature in mechanisms governing the motion of crystal line defects (steps) at the nanoscale. We focus on the key atomistic processes related to external material deposition, desorption, and asymmetric energy barriers on a stepped surface. By starting with a kinetic, restricted solid-on-solid model in 1+1 dimensions, we derive laws that govern the motion of a single step when deposition is nearly balanced out by desorption. These mesoscale laws reveal how kinetic processes, e.g., bond breaking at the step edge, influence step motion via the correlated motion of atoms.

Figure 1

Schematic for main assumptions of the KRSOS model. Bottom panel: Movable atoms are shown in dark gray. The kinetic rates are $D$, for atom hopping away from the step; $D{\varphi }_{\pm }$ (dashed atoms and arrows) for atom attachment to the step from the upper ($-$) or lower ($+$) terrace; $Dk{\varphi }_{\pm }$ for atom detachment or attachment at step; $F$ for deposition from above; and ${\tau }^{-1}$ for desorption. The factors ${\varphi }_{\pm }=exp[-E_{\pm }/\left(k_{B}T\right)]$ account for Ehrlich-Schwoebel barriers, $E_{\pm }$ [5, 6]. The step position is $s(\mathit{\alpha },m)$. Top panel: 1D energy-barrier landscape.

Figure 2

Illustration of KRSOS configurations in which adatoms interact kinetically with the step [(a),(b)] and other adatoms (c). In panel (a), an adatom resting on top of the step atom prevents detachment (yielding a zero detachment rate), but attachment and terrace hopping are still possible. Similarly, no detachment is permitted in panel (b); however, attachment is also forbidden since it would cause the step to advance by more than one lattice site. Panel (c) illustrates that only topmost adatoms in multiply occupied lattice sites may hop to adjacent sites. Mobile atoms are indicated in dark gray.

Figure 3

Snapshots of KMC simulations (symbols) for Lagrangian density $c_{\widehat{ȷ}}$ and corresponding high-occupancy corrections ${\widehat{R}}_{\widehat{ȷ}}$. (a) KMC results for $c_{\widehat{ȷ}}$ compared to prediction (solid line) for equilibrium density, $c^{\text{eq}}$. (b) KMC results for ${\widehat{R}}_{\widehat{ȷ}}$ [the Lagrangian counterpart to $R_j\left(t\right)$ appearing in (10)] compared to estimate $k{c}^{\text{eq}}$ for corrections (solid line) [cf. (11)]. Error bars depict the standard deviation of ten ensembles of ${10}^5$ KMC simulations. Parameters: $N=50,k=0.1,D={10}^{10}{\mathrm{s}}^{-1},{\varphi }_{\pm }=1,F=2\times {10}^7$ atoms/s, and ${\tau }^{-1}={10}^7{\mathrm{s}}^{-1}$. These values are typical for KMC simulations in 1D [9].