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**Simulation of misfit dislocations**

In the simplest examples of hetero-epitaxy, adsobate and substrate
crystallize in the same lattice types, but with slightly different lattice constants,
i.e. inter-atomic distances.

In the early stages of such hetero-epitaxial growth the adsorbate is
*coherent* with the substrate.
In such a so-called *pseudomorphic* film, the crystal topology is
that of a perfect crystal, i.e. each particle has the same coordination
number and its nearest and next-nearest neighbors form the same
geometrical figure with only slightly modified distances.

As the thickness of the adsorbate film increases the elastic
energy of the film rises until it is energetically favorable to form
*misfit dislocations* in order to relieve the strain.
The lattice structure is locally disturbed, allowing the adsorbate
atoms to maintain a spacing which is closer to their undisturbed bulk
lattice structure.

Depending on the sign and magnitude of the relative lattice misfit
between adsorbate and substrate, dislocations can be found right at
the adsorbate/substrate interface, or emerge and remain in the
growing film. The following image shows misfit dislocations in the
former case, i.e. for fairly large, positive mismatch.

(grey scale values correspond to the average distance to nearest neighbors)

In our work we have studied the emergence of misfit dislocations
qualitatively in terms of (1+1)-dimensional models with
continuous particle positions.
Assuming
simple, classical pair-wise interactions, such as Lennard-Jones or
Morse potential, we determine for each configuration the rates
of, e.g., diffusion processes. This is done by means of the
so-called *Molecular Static* method. The off-lattice KMC
simulation method was essentially formulated by D. Wolf and
co-workers, see for instance
[A. Schindler, PhD thesis, Duisburg, 1999].

Strain effects are not introduced *by hand*, but result directly from
the particle interactions. Effectively long range effects emerge from the
elastic deformation of the substrate, for instance.

We determine, for instance,
the critical layer thickness for the formation of dislocations
as a function of the misfit. Currently we study the influence
of *buried* dislocations on diffusion properties on the surface
(M. Walther, PhD thesis work in
Würzburg). We find, for instance,
correlations between the location of buried
misfit dislocations and the formation of mounds in later stages of the growth
process.

The following (selected) publications deal with the subject of misfit dislocations:

F. Much, M. Ahr, M. Biehl and W. Kinzel

Kinetic Monte Carlo Simulations of dislocations in heteroepitaxial growth

Europhys. Lett. **56**(6), 791 (preprint version)

M. Biehl, F. Much, and C. Vey

Off-lattice Kinetic Monte Carlo simulations of
strained heteroepitaxial growth

preprint version of an invited contribution
to an MFO Mini-Workshop (Oberwolfach, 2004),

in: Multiscale Modeling in Epitaxial Growth,

ed. A. Voigt, Int. Series of Numerical Mathematics
149 (Birkhaeuser, 2005), 41-57