A vicinal surface model for epitaxial growth with logarithmic free energy

TL;DR

This paper introduces a continuum PDE model for vicinal surface evolution during epitaxial growth, proving global solutions and analyzing long-term behavior with analytical and numerical methods.

Contribution

It presents a novel nonlinear PDE model derived from step flow dynamics, establishing existence, uniqueness, positivity, and long-term convergence of solutions.

Findings

Proved global existence and uniqueness of solutions.

Demonstrated convergence to a constant surface slope.

Validated results through analytical and numerical analysis.

Abstract

We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, $u_t = -u^2(u^3+\alpha u)_{hhhh}$, gives the evolution for the surface slope $u$ as a function of the local height $h$ in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of $u$ converging to a constant that only depends on the initial data is also investigated both analytically and numerically.