Facet evolution on supported nanostructures: Effect of finite height

The surface of a nanostructure relaxing on a substrate consists of a finite number of interacting steps and often involves the expansion of facets. Prior theoretical studies of facet evolution have focused on models with an infinite number of steps, which neglect edge effects caused by the presence of the substrate. By considering diffusion of adsorbed atoms (adatoms) on terraces and attachment-detachment of atoms at steps, we show that these edge or finite height effects play an important role in the structure’s macroscopic evolution. We assume diffusion-limited kinetics for adatoms and a homoepitaxial substrate. Specifically, using data from step simulations and a continuum theory, we demonstrate a switch in the time behavior of geometric quantities associated with facets: the facet edge position in a straight-step system and the facet radius of an axisymmetric structure. Our analysis and numerical simulations focus on two corresponding model systems where steps repel each other through entropic and elastic dipolar interactions. The first model is a vicinal surface consisting of a finite number of straight steps; for an initially uniform step train, the slope of the surface evolves symmetrically about the centerline, i.e., the middle step when the number of steps is odd. The second model is an axisymmetric structure consisting of a finite number of circular steps; in this case, we include curvature effects which cause steps to collapse under the effect of line tension. In the first case, we show that the position of the facet edge, measured from the centerline, switches from $O\left(t^{1/4}\right)$ behavior to $O\left(t^{1/5}\right)$ (where $t$ is time). In the second case, the facet radius switches from $O\left(t^{1/4}\right)$ to $O\left(t\right)$. For the axisymmetric case, we also predict analytically through a continuum shock wave theory how the individual collapse times are modified by the effects of finite height under the assumption that step interactions are weak compared to the step line tension.

Figure 1

The two model systems studied in this paper. (a) A finite number of straight steps separating two semi-infinite (flat) facets, possibly representing a straight-step bunch. Facet edges, $w_1\left(t\right)$ and $w_2\left(t\right)$, are measured from the centerline. (b) A finite number of circular steps, representing an axisymmetric nanostructure. A macroscopic facet, with radius $r\left(t\right)$, develops at the top of the structure as innermost steps collapse.

Figure 2

Simulation data for the (discrete) straight-step density $F_i$ as a function of the (scaled) step position $x_i$. The profiles were obtained by solving Eqs. (7, 9, 10, 11, 12) numerically with the number of steps $N=31$ and the dimensionless constant [defined in Eq. (8)] $\gamma =1$.

Figure 3

(a) Circular-step (radial case) simulation data for the scaled deviations $E_n$ defined in Eq. (22) with the step-interaction parameter $g=0.01$. Step collapse times for an infinite structure, $t_n\left(\infty ,g\right)$, were approximated with $t_n\left(N=700,g\right)$. (b) The similarity collapse data from (a) by Eq. (23). Inset shows a local maximum of $E_n/N^4$ for $n/N\approx 0.45$.

Figure 4

Absolute minimum (in $n$) of scaled deviations $E_n/N^4$ from Fig. 3b. This minimum follows the scaling law $\left|{\text{min}}_n\left(E_n/N^4\right)\right|=O\left(g^{1/2}\right)$ for sufficiently small step-step interaction parameter $g$.

Figure 5

Simulation data for the straight-step density obtained by integrating Eqs. (7, 9, 10, 11, 12) using a semi-infinite number of steps and the dimensionless constant $\gamma =1$ [defined in Eq. (8)]. Inset: data collapse consistent with Eq. (38).

Figure 6

(a) Simulation data for straight-step densities obtained by solving Eqs. (7, 9, 10, 11, 12) with the dimensionless constant $\gamma =1$ [see Eq. (8)] and the number of steps $N=61$. (b) Collapse of the step simulation data (symbols) and numerical solution (solid curve) of Eq. (42) with Eq. (41). The data collapse verifies the similarity solution of Eq. (41), predicting a $O\left(t^{-1/5}\right)$ scaling for the entire profile and a $O\left(t^{1/5}\right)$ scaling for the facet edge position (cf. Fig. 5).

Figure 7

(a) For an axisymmetric nanostructure, the evolving facet is represented by a propagating shock wave whose position in the $s$ (Lagrangian) coordinate is described by either $s_1\left(\tau \right)$, $s_2\left(\tau \right)$, or $s_3\left(\tau \right)$, depending on whether finite height effects are significant. When $s<s_{1,2,3}\left(\tau \right)$, the step radius is $\rho =0$. (b) Characteristics (thin lines) in $\left(s,\tau \right)$ space. The position of the facet edge in Lagrangian coordinates is given by $s_1\left(\tau \right)$, $s_2\left(\tau \right)$, and $s_3\left(\tau \right)$ (thick lines). Note the presence of a rarefaction fan originating at $\left(s,\tau \right)=\left(1,0\right)$, representing the effect of the motion of the base step. Characteristics also emanate from $s=1$ where the boundary conditions are unknown.

Figure 8

Collapse time simulation data and related quantities, together with analytic predictions, for an axisymmetric nanostructure. (a) The facet radius as a function of time switches from $O\left(t^{1/4}\right)$ to $O\left(t\right)$, confirming prediction (49), shown as a solid line. At the collapse times $t_n$, the facet radius was approximated by the radius of the innermost step $r_{n+1}\left(t_n\right)$. The number of initial steps is $N=120$ and the step-interaction parameter is $g={10}^{-8}$. (b) Collapse times from simulation data compared with theoretical prediction (50) when the initial number of steps $N=120$ and the step-step interaction parameter $g={10}^{-8}$. (c) Simulation data for scaled deviations, $E_n/N^4$, from the collapse times of an infinite structure are compared to analytic prediction (51) for different values of the step-interaction parameter $g$. In the data, the collapse times for an infinite structure, $t_n\left(\infty ,g\right)$, were approximated by $t_n\left(N=500,g\right)$.