Role of step-flow advection during electromigration-induced step bunching

Abstract

We propose a one-dimensional model based on the Burton-Cabrera-Frank equations to describe the electromigration-induced step bunching instability on vicinal surfaces. The step drift resulting from atomic evaporation and/or deposition is explicitly included in our model. A linear stability analysis reveals several stability inversions as the evaporation rate varies, while a deposition flux is shown to have a stabilizing effect.

Electromigration in Macroscopic Relaxation of Stepped Surfaces

“…An assumed condition is |v| k where k is a typical kinetic rate of atom attachment-detachment at a step; see Remark 3.2. This condition implies that the adatom drift is slow relative to atom attachmentdetachment at step edges and is met in physical situations; see, e.g., [4,13].…”

“…Previous studies of the microscopic dynamics for electromigration have shown that steps can bunch or meander [4,12,16,21,36,42]. The macroscopic PDE here does capture the tendency of steps to bunch, as discussed in section 6.1.…”

“…At the level of steps, the electric field amounts to the addition of a convective (linear in the density gradient) term in the diffusion equation for the adatom density; see (3.2). Regarding (1.1), |Z * | is larger than unity for metals but can be much smaller than unity for semiconductors [4]. This Z * has been estimated in experiments via comparisons of data for decay rates with predictions of the microscopic model; see, e.g., [13].…”

“…An externally applied (vector-valued) electric field, E, tends to force adatoms to move in its direction, causing a drift velocity v. This v is related to E by [4,13] (…”

“…where τ is the characteristic desorption time; cf. (1) in [4]. We use the quasi-steady approximation, ∂ t C i ≈ 0, by assuming that step motion is much slower than diffusion.…”