##### Department of Mathematics,

University of California San Diego

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### Special Recruitment Colloquium

## Dr. Dionisios Margetis

#### MIT

## Continuum theory of crystal surface relaxation below roughening

##### Abstract:

Advances in the fabrication of small devices have stimulated interest in low-temperature kinetic processes on crystal surfaces. In most experimental situations, nanoscale solid structures decay with time having 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 structure 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 mathematical description of surface evolution at such temperatures is an area of intensive research. \vskip .1in \noindent The subject of this talk is a continuum description of the morphological relaxation of crystal surfaces in $(2+1)$ dimensions below the roughening temperature by use of $PDEs$. For processes limited by the diffusion of point defects (``adatoms'') on terraces between steps and the attachment and detachment of atoms to and from steps, the surface height profile outside facets is described via a nonlinear, fourth-order $PDE$ that accounts for step line-tension energy $g1$ and step-step repulsive interaction energy $g3$. The $PDE$ is derived from the difference-differential equations for the motion of individual steps, and, alternatively, via a continuum surface free energy. Particular solutions to the $PDE$ are shown to plausibly unify experimental observations of decaying biperiodic surface profiles. To further test the $PDE$, the facet evolution of axisymmetric profiles is treated analytically as a free-boundary problem. For long times, axisymmetric shapes and $g3/g1< 1$, singular perturbation theory is applied for self-similar shapes close to the facet. Scaling laws with $g3/g1$ are derived for the boundary-layer width, maximum slope and facet radius; and a universal $ODE$ for the slope profile is derived and solved uniquely via applying effective boundary conditions at the facet edge. The scaling results compare favorably with numerical solutions of the difference-differential equations for the step positions.

Host: Bo Li and Michael Holst

### January 6, 2005

### 10:00 AM

### AP&M 6438

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