MBE growth of compound semiconductors: Part I. Stochastic modeling
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A stochastic model for the MBE growth kinetic study of compound semiconductors is developed based on the master equation approach, the solid-on-solid restriction, and the quasi-chemical approximation. The developed model is suitable for the zinc blende crystals with 001 as the growth direction. In the modeling, the diamond cubic structure and the two sublattice nature of the zinc blende crystal are taken into account. The stochastic model is extended to compound semiconductor alloys such as GaAlAsSb to make it suitable for the MBE kinetic studies of alloys. Up to four elements with two in each sublattice can be accommodated. The presence of two elements in the same sublattice was taken into account. A procedure for the evaluation of the model parameters based on the available thermodynamic and experimental data is discussed. Advantages and limitations of the stochastic model over the available theoretical models are discussed.
I. INTRODUCTION Kinetic Ising models with solid on solid (SOS) restriction have been employed for the study of crystal growth from melt and vapor.1"5 The surface kinetic processes considered in these models are the relaxation processes such as the adsorption and the evaporation, and the surface diffusion processes such as the intralayer diffusion and the interlayer diffusion. Analytically, these models were studied with either pure relaxation kinetics or pure diffusion kinetics. Saito and Krumbhaar6 studied the combined influence of the relaxation and the surface diffusion processes on the kinetics of crystal growth. Henceforth, let us call the stochastic model of Ref. 6 as the SK model. The SK model is based on the master equation approach with quasi-chemical approximation (QCA) and the SOS restriction. By comparing the results of the Monte Carlo (MC) simulations with that of the model, it was shown that the QCA scheme gives rise to reliable results on the kinetics of crystal growth for a chemical potential difference, A/x (between the vapor and the solid), greater than the critical value, A/i c , for the existence of the metastable state with infinite lifetime below even the roughening point. It was also shown that, in general, the surface diffusion enhances the growth rate and reduces the surface roughness. With the advent of the ultra high vacuum crystal growth technique of Molecular Beam Epitaxy (MBE) for the growth of compound semiconductors, there is a surge in the research activity to understand the MBE growth kinetics, experimentally7"13 and theoretically.14"20 The theoretical studies have been based on the MC simulations14"20 which usually have crystal size and computer time limitations. J. Mater. Res., Vol. 7, No. 5, May 1992 http://journals.cambridge.org
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In situ doping kinetics is impossible to investigate using the MC simulations because of the crystal size limitation. In the MC simulations, the number of atoms deposited during growth is about 20000 for a typical growth simulation. Even if the doping density in the epilayer is as high as 10 18 /cm 3 , o
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