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Department of Mathematics, MIT October 27, 2004
Advances in fabrication of small devices have stimulated interest in low-temperature kinetic processes on crystal surfaces. In most experimental situations, nanoscale solid structures decay in time with a lifetime that typically is a large power of the feature size and increases with decreasing temperature. Strategies for skirting the lifetime limitations involve processing at ever-lower temperatures for ever-smaller feature sizes. At temperatures below the roughening transition crystal surfaces evolve via the motion of interacting steps at the nanoscale, and may develop macroscopically flat parts known as facets. The study of surface evolution at such temperatures is an area of active research. The subject of this talk is a description of the morphological relaxation of three-dimensional crystal surfaces below the roughening temperature by use of continuum equations derived from discrete step-flow models. For isotropic diffusion of point defects (``adatoms'') across each terrace and attachment-detachment of atoms at each step, the geometry of steps causes diffusion-induced lateral flows of adatoms parallel to steps that can be distinctly different from flows transverse to steps. A partial differential equation (PDE) is derived for the height profile that apparently unifies via scaling arguments experimental observations of decaying biperiodic surface corrugations via an interplay of step energetics, kinetics, and geometry. For axisymmetric crystals with a facet, the facet evolution is treated as a free-boundary problem recognizing that there is a region of rapid variations of the slope, a boundary layer, near the facet. For long times, singular perturbation theory is applied for self-similar shapes close to the facet to derive from the PDE simple scaling laws with the step energy parameters for the boundary layer width, maximum slope and facet radius. These scaling results compare favorably with kinetic simulations. |
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