The groups of (linear) similarity and coincidence isometries of certain modules $\Gamma $ in $d$-dimensional Euclidean space, which naturally occur in quasicrystallography, are considered. It is shown that the structure of the factor group of similarity modulo coincidence isometries is the direct sum of cyclic groups of prime power orders that divide $d$. In particular, if the dimension $d$ is a prime number $p$, the factor group is an elementary abelian $p$-group. This generalizes previous results obtained for lattices to situations relevant in quasicrystallography.

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

Efficient algorithms for the simulation of strained heteroepitaxial growth with intermixing in 2+1 dimensions are presented. The first of these algorithms is an extension of the energy localization method [T. P. Schulze and P. Smereka, An energy localization principle and its application to fast kinetic Monte Carlo simulation of heteroepitaxial growth, J. Mech. Phys. Sol., 3 (2009), 521-538] from 1+1 to 2+1 dimensions. Two approximations of this basic algorithm are then introduced, one of which treats adatoms in a more efficient manner, while the other makes use of an approximation of the change in elastic energy in terms of local elastic energy density. In both cases, it is demonstrated that a reasonable level of fidelity is achieved. Results are presented showing how the film morphology is affected by misfit and deposition rate. In addition, simulations of stacked quantum dots are also presented.

We report kinetic Monte-Karlo (KMC) simulation of self-assembled synthesis of nanocrystals by physical vapor deposition (PVD), which is one of most flexible, efficient, and clean techniques to fabricate nanopatterns. In particular, self-assembled arrays of nanocrystals can be synthesized by PVD. However size, shape and density of self-assembled nanocrystals are highly sensitive to the process conditions such as duration of deposition, temperature, substrate material, etc. To efficiently synthesize nanocrystalline arrays by PVD, the process control factors should be understood in detail. KMC simulations of film deposition are an important tool for understanding the mechanisms of film deposition. In this paper, we report a KMC modeling that explicitly represents PVD synthesis of self-assembled nanocrystals. We study how varying critical process parameters such as deposition rate, duration, temperature, and substrate type affect the lateral 2D morphologies of self-assembled metallic islands on substrates, and compare our results with experimentally observed surface morphologies generated by PVD. Our simulations align well with experimental results reported in the literature.

We model diffusion-controlled crystal growth as an interference problem. The crystal layers grow by nucleation (initiation of crystallization centers) followed by attachment of molecules to the nucleus. A forming crystal layer completes by either spreading across the length of the crystal or by colliding with another spreading crystal layer. This model differs from the classical Johnson-Mehl-Kolmogorov model in that nucleation happens only on boundaries of a ‘seed’ crystal as opposed to nucleation from random points in a given region. Our results also differ from the limiting results found for this classical model. We use the invariant measure of an embedded Markov process to find the growth rate of the crystal in terms of the nucleation rates. Ergodic theorems are then used to derive explicit formulae for some stationary probabilities.