Scaling laws for step bunching on vicinal surfaces: the role of the dynamical and chemical effects

TL;DR

This paper develops a comprehensive step-flow model that incorporates dynamical and chemical effects to explain step bunching on vicinal surfaces, revealing new scaling laws and validating them through continuum and analytical approaches.

Contribution

It introduces a thermodynamically consistent model including dynamical and chemical effects, deriving new scaling laws for step bunching without extraneous mechanisms.

Findings

Power-law coarsening of surface profile with H ~ t^{1/2}

Minimal interstep distance scales as N^{-2/3}

Effective iES barrier interpretation of combined effects

Abstract

We study the evolution of step bunches on vicinal surfaces using a thermodynamically consistent step-flow model that (i) circumvents the quasistatic approximation that prevails in the literature by accounting for the dynamics of adatom diffusion on terraces and attachment-detachment at steps (referred to as the dynamical effect), and (ii) generalizes the expression of the step chemical potential by incorporating the necessary coupling between the diffusion fields on adjacent terraces (referred to as the chemical effect). Having previously shown that these effects can explain the onset of step bunching without recourse to the inverse Ehrlich-Schwoebel (iES) barrier or other extraneous mechanisms, we are here interested in the evolution of step bunches beyond the linear-stability regime. In particular, the numerical resolution of the step-flow problem yields a robust power-law coarsening of the surface profile, with the bunch height growing in time as $H\sim t^{1/2}$ and the minimal interstep distance as a function of the number of steps in the bunch cell obeying $\ell_{min}\sim N^{-2/3}$. Although these exponents have previously been reported, this is the first time such scaling laws are obtained in the absence of an iES barrier or adatom electromigration. In order to validate our simulations, we take the continuum limit of the discrete step-flow system, leading to a novel nonlinear evolution equation for the surface height. We investigate the existence of self-similar solutions of this equation and confirm the 1/2 coarsening exponent obtained numerically for $H$. We highlight the influence of the combined dynamical-chemical effect and show that it can be interpreted as an effective iES barrier in the standard BCF theory. Finally, we use a Pad\'e approximant to derive an analytical expression for the velocity of steadily moving step bunches and compare it to numerical simulations.