Nanocomposites and nanomaterials

Modeling of crystal growth by statistical methods
Politanskyi R.L.1, Vistak M.V.2
1 Department of RadioEngeeniring and CyberSecurity , Yuriy Fedkovych Chernivtsi National University. Kotsyubynsky, 2, Chernivtsi-58012, Ukraine. 
E-mail:polrusl@i.ua
2 Faculty of Pharmacy/Department of Biophysics, Danylo Halytsky Lviv National University, Pekarska 69, Lviv-79010, Ukraine.

The paper analyzes the growth models of crystalline heteroepitaxial films with a small lattice mismatch parameter. The most characteristic and common type of materials that form these films are Si-Ge films for which the mismatch parameter can vary from 20 to 0% depending on the concentration of germanium (Si1-xGex).
Most theoretical approaches in the study of the problem of growth of two-component epitaxial structures can be divided into the following groups:
•	equilibrium growth theory based on the critical layer thickness
•	theories of non-equilibrium growth;
•	theory of plastic relaxation.
The most widely used theories of growth of a layer with a constant lattice, which is not very different from the constant lattice of the substrate, are the theory of van de Merve [1], Matthews and Blacksley, which are one-dimensional but were extended to two-dimensional by Jesser and Kuchelmann-Wilsedorf. These theories and experiments that confirm them reflect the following general patterns of growth of Ge1-xSix systems: the growth of the epitaxial layer is coherent, ie, first fills one atomic plane over the entire surface, then another, etc. the transition surface is considered flat and atomically smooth, ie the growth of islands is not considered, the energy of the transition layers is described by the sinusoidal potential. Under such conditions, it is possible to apply more modern theories that have been developed to polymeric materials [2], will prevail in this approach, the theory of platinum relaxation of hot epitaxial layers.

1.	Levi A.C., Kortla M. Theory and simulation of crystal growth // J. Phys.: Condens. Matter. - 1997. -9, - P. 299-344.
2.	Borodin A., Gorin V. Lectures on Integrable Probability. 	arXiv:1212.3351.