A study of a kinetic rate equation model for simulations of MBE crystal growth: a comparison with Monte Carlo simulations

Abstract

We present the simulations of molecular beam epitaxy (MBE) growth using a rate equation (RE) model and its comparison with Monte-Carlo (MC) simulations. The advantage of the RE model is the higher speed of calculations, so a much shorter time is required for obtaining results. The RE model is described by a set of differential equations that calculate at each time interval the change of the N kj numbers of atoms and islands of each k size in each jth layer. This change is due to kinetic processes occurring on the surface during the growth. In the original model (R. Kariotis and H.G. Lagally, Surf. Sci., 216 (1989) 557) the probabilities of these processes were described by parameters (input parameters for equations) and the simulations of MBE growth were realized by an appropriate choice of them. To make this model applicable to real simulations, we have included the substrate-temperature dependence of all input parameters using an Arrhenius form. This form is used in MC simulations to calculate a migration of atoms on the surface with substrate-temperature dependence. Since the RE model is described by a set of differential equations it was important to first find the allowed temperature range for simulations. This range includes the substrate temperature for the 3D growth mode (low temperatures) and also for the 2D growth mode (epitaxial temperatures). Using an Arrhenius form for temperature dependence of the parameters in the RE model we were able to compare the obtained results with MC calculations. We have made MC simulations (Š. Németh, R. Harman and M. Veselý, Correlation between the stochastic simulation of molecular beam epitaxy growth and experiment, 9th Int. Conf. of Thin Films, 6–10 September, 1993, Vienna, Austria) using the same input parameters ( T = 775 K, E n = 0.3 eV, E s = 1.45 eV). Since the RE model is strongly substrate-size dependent (D. Papajová, W.E. Hagston and P. Harrison, Appl. Phys. A, 59 (1994) 215–222; D. Papajová, Š. Németh, W.E. Hagston, H. Sitter and M. Veselý, J. Appl. Phys. A, submitted) we have found very good agreement in 2D growth for smaller substrate sizes S (in the RE model) only, when this dependence does not influence the results.