Formation of various geometric patterns during crystal growth remains a subject of continuous interests of many researchers. It is important as an interesting example of far from equilibrium, stationary process and because of practical applications in the crystal growth technology. Kinetic Monte Carlo simulation method of analysis is based on a simple model of particles jumping between well-defined crystal site positions. The main task in the extended kinetic Monte Carlo (kMC) study is to find proper parameters of the model, i.e. coupling constants between particles, energy barriers for each type of particle jump such that describe main characteristics of the studied system. When correctness of the model is already verified by comparison with experimental data it can be used in further investigations. The simplicity of the model and small number of control parameters allows studying systems of large number of particles, performing long time simulations and examining the system behavior in various conditions. The kMC simulation results for two different systems are compared : GaN(0001) surface model in N-rich conditions represented by lattice of Ga atoms only and full, two component model of 4H SiC(0001). Within certain parameters steps move uniformly and stay straight during growth or sublimation processes of both simulated crystals. However within a wide parameter range different instability phenomena are observed. Step bunching and step meandering are the most characteristic ones. Both processes were studied as a function of time, temperature and misorientation angle. Detailed analysis of the character of structures and their evolution dependence　on the growth parameters was done. For some parameters step bunching happens rapidly and as a result single bunch of all steps is formed. For other range of parameters two, four step arrangements develop or step train structure is observed. Phase diagram for all observed structure was found. The comparison of the kMC results with simple analytic models of step dynamics confirms that bunching and meandering instabilities are caused by the step movement during the process of crystal growth or sublimation. All surface patterns, emerging in the simulations, have their corresponding cases in the experimental results. The numerical models allow to observe how they develop and to study their time evolution.

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