Instabilities in crystal growth by atomic or molecular beams

We present new techniques for generation of unstructured meshes for geometries specified by implicit functions An initial mesh is iteratively improved by solving for a force equilibrium in the element edges, and the boundary nodes are projected using the implicit geometry definition Our algorithm generalizes to any dimension and it typically produces meshes of very high quality We show a simplified version of the method in just one page of MATLAB code, and we describe how to improve and extend our implementation 
Prior to generating the mesh we compute a mesh size function to specify the desired size of the elements We have developed algorithms for automatic generation of size functions, adapted to the curvature and the feature size of the geometry We propose a new method for limiting the gradients in the size function by solving a non-linear partial differential equation We show that the solution to our gradient limiting equation is optimal for convex geometries, and we discuss efficient methods to solve it numerically 
The iterative nature of the algorithm makes it particularly useful for moving meshes, and we show how to combine it with the level set method for applications in fluid dynamics, shape optimization, and structural deformations It is also appropriate for numerical adaptation; where the previous mesh is used to represent the size function and as the initial mesh for the refinements Finally, we show how to generate meshes for regions in images by using implicit representations (Copies available exclusively from MIT Libraries, Rm 14-0551, Cambridge, MA 02139-4307 Ph 617-253-5668; Fax 617-253-1690)

Mesh generation for implicit geometries

Morphological stability and compositional uniformity of alloy thin films

In heteroepitaxial growth, the mismatch between the lattice constants in the film and the substrate causes misfit strain in the film, making a flat surface unstable to small perturbations. This morphological instability is called Asaro-Tiller-Grinfeld (ATG) instability. In practice, most devices are fabricated on vicinal surfaces which consist of steps and terraces. In this case, the misfit strain causes step bunching and traditional continuum models for the ATG instability do not apply directly. In this paper, we present a continuum model for this elastically driving step bunching. We validate our model by performing linear stability analysis and numerical simulation in the nonlinear regime. We also present a continuum description for the misfit elastic energy on vicinal surface which incorporates the underlying atomic features of the vicinal surface.

/pdf/misfit-elastic-energy-and-a-continuum-model-for-epitaxial-4onqusgubi.pdf

Michael Grinfeld

Papers

Two-dimensional islanding atop stressed solid helium and epitaxial films.