Continuum model description of thin film growth morphology

TL;DR

This study applies a continuum model to describe the surface morphology of hetero-grown GeSi films, revealing linear scaling behavior and the significance of non-linear terms, specifically the Kuramoto-Sivashinsky equation, in growth dynamics.

Contribution

It demonstrates that the surface evolution of GeSi films can be modeled by the Kuramoto-Sivashinsky equation, incorporating non-linear effects and instability mechanisms.

Findings

Surface roughness scales linearly with lateral size at small scales.

The growth dynamics are governed by the Kuramoto-Sivashinsky equation.

Non-linear terms become significant at scales larger than 1 mm.

Abstract

We examine the applicability of the continuum model to describe the surface morphology of a hetero-growth system: compositionally-graded, relaxed GeSi films on (001) Si substrates. Surface roughness versus lateral dimension was analyzed for samples what were grown under different conditions. We find that all samples belong to the same growth class, in which the surface roughness scales linearly with lateral size at small scales and appears to saturate at large scales. For length scales ranging from 1 nm to 100 $\mu$m, the scaling behavior can be described by a linear continuum model consisting of a surface diffusion term and a Laplacian term. However, in-depth analysis on non-universal amplitudes indicates the breaking of up-down symmetry, suggesting the presence of non-linear terms in the microscopic model. We argue that the leading non-linear term has the form of $\lambda _1(\nabla h)^2$, but its effect on scaling exponents will not be evident for length scales less than 1 mm. Therefore, the growth dynamics of this system is described by the Kuramoto-Sivashinsky equation, consisting of the two linear terms plus $\lambda _1(\nabla h)^2$, driven by a Gaussian noise. We also discuss the negative coefficient in the Laplacian term as an instability mechanism responsible for large scale film morphology on the final surface.