Time Scaling Relations for Step Bunches from Models with Step-Step Attractions (B1-Type Models)

TL;DR

This study investigates time-scaling relations for step bunches caused by step-step attractions in three models, revealing how the scaling exponent depends on the attraction power and demonstrating self-similar surface pattern formation.

Contribution

It introduces a unified approach to analyze time-scaling in B1-type models with step-step attractions, including a new minimal model and a scheme for scaling ODEs in vicinal studies.

Findings

Time-scaling exponent ta = 1/(3+p) for MM2 model

Hypothesized ta = 1/(5+p) for LW2 and TE2 models

Surface patterns are self-similar with a single length scale

Abstract

The step bunching instability is studied in three models of step motion defined in terms of ordinary differential equations (ODE). The source of instability in these models is step-step attraction, it is opposed by step-step repulsion and the developing surface patterns reflect the balance between the two. The first model, TE2, is a generalization of the seminal model of Tersoff et al. (1995). The second one, LW2, is obtained from the model of Liu and Weeks (1998) using the repulsions term to construct the attractions one with retained possibility to change the parameters in the two independently. The third model, MM2, is a minimal one constructed ad hoc and in this article it plays a central role. New scheme for scaling the ODE in vicinal studies is applied towards deciphering the pre-factors in the time-scaling relations. In all these models the patterned surface is self-similar - only one length scale is necessary to describe its evolution (hence B1-type). The bunches form finite angles with the terraces. Integrating numerically the equations for step motion and changing systematically the parameters we obtain the overall dependence of time-scaling exponent \b{eta} on the power of step-step attractions p as \b{eta} = 1/(3+p) for MM2 and hypothesize based on restricted set of data that it is \b{eta} = 1/(5+p) for LW2 and TE2.