Stochastic equations of motion for epitaxial growth

Abstract

We report an analytic derivation of the Langevin equations of motion for the surface of a solid that evolves under typical epitaxial-growth conditions. Our treatment begins with a master-equation description of the microscopic dynamics of a solid-on-solid model and presumes that all surface processes obey Arrhenius-type rate laws. Our basic model takes account of atomic deposition from a low-density vapor, thermal desorption, and surface diffusion. Refinements to the model include the effects of hot-atom knockout processes and asymmetric energy barriers near step edges. A regularization scheme is described that permits a (nonrigorous) passage to the continuum limit when the surface is rough. The resulting stochastic differential equation for the surface-height profile generically leads to the behavior at long length and time scales first described by Kardar, Parisi, and Zhang [Phys. Rev. Lett. 56, 889 (1986)] (due to desorption). If evaporation is negligible, the asymptotic behavior is characteristic of a linear model introduced by Edwards and Wilkinson [Proc. R. Soc. London, Ser. A 381, 17 (1982)] (due to asymmetric step barriers and/or knockout events). If the latter are absent as well, the surface roughness is determined by an equation independently analyzed by Villain [J. Phys. I 1, 19 (1991)] and Lai and Das Sarma [Phys. Rev. Lett. 66, 2348 (1991)] (which includes only deposition and site-to-site hopping). The consequences of reflection-symmetry breaking in the basic microscopic processes are discussed in connection with step-barrier asymmetry and Metropolis kinetic algorithms.