Models of step bunching: Turning repulsion into attraction

TL;DR

This paper introduces two models of vicinal surface motion, demonstrating how repulsive and attractive interactions influence step bunching, with results aligning with experimental observations and revealing new scaling behaviors.

Contribution

The paper develops two novel models of step dynamics, showing how attraction can emerge from repulsion and analyzing their scaling properties.

Findings

Step bunching occurs with a time scaling exponent of ~0.18.

In LW model, the number of steps in a bunch depends only on parameter b.

Surface slope in bunches remains constant regardless of bunch size.

Abstract

We report numerical results for two models of vicinal motion. The first, LW, aims at crystal evaporation when the detachment from steps is slow [Liu and Weeks, PRB 57, 23 (1998) 14891]. The source of destabilization is electromigration force acting on the adatoms. The destabilizing part of equation(s) of the step velocity is linear in the widths of the adjacent terraces with larger contribution of the terrace behind. This asymmetry is controlled by a single parameter b. The stabilization part accounts for the tendency to equidistant spacing dictated by the interstep repulsions. We construct the second model, LW2, from LW in the same manner as was constructed Minimal Model 2 (MM2) from another minimal model [B.Ranguelov et al., Nanoscience and Nanotechnology 6, (2006) 31] - keeping the 'repulsions term' from LW and introducing a similar one with opposite sign as 'attractions term'. For LW we obtain for first time that in the pre-factor of the time scaling of the number of steps in the bunch N enters only the parameter b. In LW2 we find the same type of step bunching as in MM2 - the surface slope in the bunch is constant and not function of N. Further, we obtain the time scaling of N with exponent ~0.18, found also in experiments, and the minimal distance in the bunch as function of the model parameters.