Island-dynamics model for mound formation: Effect of a step-edge barrier

We formulate and implement a generalized island-dynamics model of epitaxial growth based on the level-set technique to include the effect of an additional energy barrier for the attachment and detachment of atoms at step edges. For this purpose, we invoke a mixed, Robin-type, boundary condition for the flux of adsorbed atoms (adatoms) at each step edge. In addition, we provide an analytic expression for the requisite equilibrium adatom concentration at the island boundary. The only inputs are atomistic kinetic rates. We present a numerical scheme for solving the adatom diffusion equation with such a mixed boundary condition. Our simulation results demonstrate that mounds form when the step-edge barrier is included, and that these mounds steepen as the step-edge barrier increases.

Figure 1

Schematic representation of the step-edge barrier in the case with isotropic diffusion (when diffusion rates are scalars). The top panel shows the potential energy surface, and the bottom panel illustrates the corresponding diffusion rates near the step edge (side view).

Figure 2

Schematic of a grid including a computational cell, $C_{i,j}$, cut by an interface curve, $\Gamma$. ${\Omega }^{\pm }$ denotes the region of the upper $(+)$ or lower $(-)$ terrace, and $\Gamma$ is the island boundary.

Figure 3

Island morphology after the deposition of 32 atomic layers for different values of $D^{\prime }/D$. (a) $D^{\prime }/D=0.95$ (almost no step-edge barrier); (b) $D^{\prime }/D=0.3$; (c) $D^{\prime }/D=0.1$; and (d) $D^{\prime }/D=0.01$ (large step-edge barrier). The steepening of mounds for decreasing values of $D^{\prime }/D$ (increasing step-edge barrier) is evident.

Figure 4

Time evolution of the surface roughness, $w={\left[\langle {(h_i-\langle h\rangle )}^2\rangle \right]}^{1/2}$, for different values of $D^{\prime }/D$. (a) $D^{\prime }/D=0.95$ (small step-edge barrier); (b) $D^{\prime }/D=0.3$; (c) $D^{\prime }/D=0.1$; and (d) $D^{\prime }/D=0.01$ (large step-edge barrier). As the step-edge barrier increases, $w$ increases considerably and oscillations of $w$ as a function of time die out. The dashed straight lines are guides to the eye to measure a possible scaling exponent $\beta$.

Figure 5

Schematic of microscopic processes near a straight step edge according to the terrace-step-kink model of [33]. The relevant kinetic transition rates are shown by arrows. ${\Omega }_{\pm }$ denotes the upper ($+$) or lower ($-$) terrace.