WIAS Preprint No. 1434, (2009)

From bell shapes to pyramids: A continuum model for self-assembled quantum dot growth


  • Korzec, Maciek D.
  • Evans, Peter L.

2010 Mathematics Subject Classification

  • 74A50 41A60


  • Anisotropic surface energy, self-assembly of quantum dots, small-slope approximation, stationary solutions, coarsening, linear stability




A continuum model for the growth of self-assembled quantum dots that incorporates surface diffusion, an elastically deformable substrate, wetting interactions and anisotropic surface energy is presented. Using a small slope approximation a thin film equation for the surface profile that describes facetted growth is derived. A linear stability analysis shows that anisotropy acts to destabilize the surface. It lowers the critical height of flat films and there exists an anisotropy strength above which all thicknesses are unstable. A numerical algorithm based on spectral differentiation is presented and simulation are carried out. These clearly show faceting of the growing islands and a logarithmically slow coarsening behavior.

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