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Modelling and simulation of crystal growth

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Note: a few talks (ppt) about some of the topics are available here

Introduction   (back to the top)
One important area within my research activities is the theory, modelling, and simulation of non-equilibrium systems. This concerns, in particular, models of growing crystals and nano-structured surfaces.
The technique of Molecular Beam Epitaxy (MBE) is a particular clear-cut realization of epitaxial crystal growth. In this method, one or several adsorbate materials are heated in an oven which is contained in an ultra-high vacuum (UHV) chamber. Evaporated particles form an atomic or molecular beam which is directed onto a substrate crystal. After arrival at the substrate, particales become adatoms which may diffuse on the surface and are finally incorporated into the growing crystal.
Growth in an MBE environment is clearly far from equilibrium: the system is constantly driven by the depostion flux and an extremely low pressure is maintained in the UHV chamber. This is different from other situations closer to equilibrium, where the growing film can exchange particles with a vapor, for instance.
MBE growth is therefore highly attractive from a theoretical point of view. The well-defined non-equilibrium situation provides a workshop in which to develop analytical approaches as well as novel computer simulation techniques suitable for more general non-equilibrium systems.

Lattice gas models and KMC simulations:   (back to the top)
In our research, we have mainly studied and developed models that allow for the simulation of growth by use of simplifying lattice gas models (click for a review). Examples include the following topics (click for details where available):

· Mound formation, slope selection, and coarsening dynamics

· (Non-) Universal scaling behavior in unstable growth

· Surface reconstructions of II-VI semiconductor materials

· Atomic Layer Epitaxy of II-VI(001) semiconductor surfaces


Off-lattice KMC simulations:   (back to the top)
For the modelling of hetero-epitaxial systems we employ off-lattice KMC simulations (click for a review) with continuous particle positions. The method combines the ideas of KMC with a determination of transition rates by means of Molecular Static. It was developed and first applied by Dietrich Wolf and co-workers in the context of misfit dislocation formation.
Our investigations concern the following topics:

· Misfit dislocations in strained hetero-epitaxial crystal growth

· Stranski-Krastanov-like growth: self-organized Quantum Dots

· The formation of nano-structured surface alloys


Review articles
  (back to the top)
The following two articles appeared as invited contributions in the Proceedings of an MFO Mini-Workshop in Oberwolfach (2004).

M. Biehl
Lattice gas models and Kinetic Monte Carlo simulations of epitaxial growth.
pages 3-18.   discusses the basic concepts and points at further references)

M. Biehl, F. Much, and C. Vey
Off-lattice Kinetic Monte Carlo simulations of strained heteroepitaxial growth
pages 41-57   (introduction and example applications )

in: A. Voigt (ed.), Multiscale Modeling in Epitaxial Growth
Birkhaeuser, International Series of Numerical Mathematics, Vol. 149 (2005)