Linear theory of unstable growth on rough surfaces

TL;DR

This paper develops a linear continuum theory to analyze unstable growth on rough surfaces, identifying key length scales that influence surface roughness evolution and comparing theoretical predictions with experimental data.

Contribution

It introduces a linear model incorporating substrate roughness, terrace size, and Ehrlich-Schwoebel length to explain surface growth instability, highlighting the role of noise and nonlinear effects.

Findings

Surface width exhibits a minimum at a specific coverage.

Linear theory captures overall roughness evolution features.

Deviations suggest nonlinearities significantly influence growth.

Abstract

Unstable homoepitaxy on rough substrates is treated within a linear continuum theory. The time dependence of the surface width $W(t)$ is governed by three length scales: The characteristic scale $l_0$ of the substrate roughness, the terrace size $l_D$ and the Ehrlich-Schwoebel length $l_{ES}$. If $l_{ES} \ll l_D$ (weak step edge barriers) and $l_0 \ll l_m \sim l_D \sqrt{l_D/l_{ES}}$, then $W(t)$ displays a minimum at a coverage $\theta_{\rm min} \sim (l_D/l_{ES})^2$, where the initial surface width is reduced by a factor $l_0/l_m$. The r\^{o}le of deposition and diffusion noise is analyzed. The results are applied to recent experiments on the growth of InAs buffer layers [M.F. Gyure {\em et al.}, Phys. Rev. Lett. {\bf 81}, 4931 (1998)]. The overall features of the observed roughness evolution are captured by the linear theory, but the detailed time dependence shows distinct deviations which suggest a significant influence of nonlinearities.