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

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model