A variational approach to nonlinear dynamics of nanoscale surface modulations

TL;DR

This paper introduces a variational method to analyze the nonlinear evolution of nanoscale surface modulations on crystals, avoiding ad hoc regularizations and relying solely on surface energies and diffusion constants.

Contribution

It presents a novel variational formulation that simplifies the study of singular surface evolution equations without regularization.

Findings

Successfully models surface morphological equilibration.

Provides a new computational approach for nanoscale surface dynamics.

Avoids regularization procedures used in prior methods.

Abstract

In this paper, we propose a variational formulation to study the singular evolution equations that govern the dynamics of surface modulations on crystals below the roughening temperature. The basic idea of the formulation is to expand the surface shape in terms of a complete set of basis functions and to use a variational principle equivalent to the continuum evolution equations to obtain coupled nonlinear ordinary differential equations for the expansion coefficients. Unlike several earlier approaches that rely on ad hoc regularization procedures to handle the singularities in the evolution equations, the only inputs required in the present approach are the orientation dependent surface energies and the diffusion constants. The method is applied to study the morphological equilibration of patterned unidirectional and bidirectional sinusoidal modulations on semiconductor surfaces through surface diffusion.