[Paper Review] Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

Jian‐Guo Liu, Jianfeng Lu+2 more|arXiv (Cornell University)|Mar 12, 2017

nanoparticles nucleation surface interactions3 citations

TL;DR

This paper proposes a continuum model based on a degenerate parabolic PDE with exponential mobility to describe asymmetric facet dynamics in homoepitaxial crystal surface relaxation. The model, derived from atomistic broken-bond and mesoscale step-diffusion theories, analytically demonstrates that top facets expand rapidly regardless of size, while bottom facets only evolve if exceeding a critical length, revealing a fundamental asymmetry in convex and concave facet evolution due to nonlinear surface diffusion kinetics.

ABSTRACT

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond model and a mesoscale model for steps. By constructing an explicit solution to the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the PDE via a regularized surface free energy.

Motivation & Objective

• To understand the origin of asymmetry in facet evolution during crystal surface relaxation in homoepitaxy.
• To develop a continuum model that captures the nonlinear, exponential dependence of adatom flux on step chemical potential.
• To demonstrate analytically that convex (top) and concave (bottom) facets evolve differently under the same physical conditions.
• To provide a theoretical explanation for counter-intuitive asymmetries observed in kinetic Monte Carlo simulations and experimental annealing of Si gratings.
• To lay the foundation for extending the model to 2+1 dimensions and including elastic interactions in future work.

Proposed method

• Formulates a singular-diffusion PDE as a nonlinear, weighted $H^{-1}$ gradient flow of a convex, singular surface free energy.
• Uses the subgradient formalism to construct an explicit analytical solution for the height profile in 1+1D periodic surface corrugations.
• Incorporates an exponential mobility function derived from the Arrhenius law and Gibbs-Thomson relation, reflecting strong step stiffness and non-linear adatom flux.
• Validates the analytical solution against numerical simulations using a regularized surface free energy to handle singularities.
• Relies on the continuum limit of the Burton-Cabrera-Frank (BCF) model and atomistic broken-bond models to justify the PDE's physical basis.
• Applies the framework to a periodic surface profile to isolate and analyze the dynamics of top and bottom facets.

Experimental results

Research questions

• RQ1 Why do top and bottom facets in a periodic crystal surface corrugation evolve at different rates despite symmetric initial conditions?
• RQ2 How does the exponential dependence of adatom flux on step chemical potential lead to asymmetry in facet dynamics?
• RQ3 Can the observed asymmetry in kinetic Monte Carlo simulations be explained by a continuum model with nonlinear surface diffusion?
• RQ4 What is the critical size threshold for bottom facet motion, and how does it emerge from the PDE structure?
• RQ5 How does the absence of linearization in the chemical potential relation affect the evolution compared to standard Fickian diffusion models?

Key findings

• Top facets expand rapidly and continuously, regardless of their initial size, due to the exponential mobility in the PDE.
• Bottom facets only evolve if their initial length exceeds a critical threshold, which emerges from the nonlinear structure of the PDE.
• The asymmetry arises directly from the exponential mobility term, which couples the flux to the exponential of the step chemical potential.
• The analytical solution confirms that the model captures the qualitative behavior observed in kinetic Monte Carlo simulations, particularly the counter-intuitive asymmetry.
• The model is consistent with the continuum limit of the BCF theory and atomistic broken-bond models, validating its physical grounding.
• Numerical simulations with regularized free energy confirm the analytical predictions, showing distinct dynamics for convex and concave regions.