Dynamics and fluctuations during MBE on vicinal surfaces. I. Formalism and results of linear theory

Abstract

We develop a full nonlinear theory including fluctuations for the study of dynamics of vicinal surfaces during molecular beam epitaxy. We consider the situation where the surface grows through step flow. The model is based on the Burton-Cabrera-Frank one, in which kinetic attachments, elastic interactions, and statistical fluctuations, through Langevin forces, are incorporated. Green's functions techniques are used. The step dynamics are governed in the general case by nonlinear and nonlocal coupled equations. At equilibrium we recover known results and some of them are revisited. For example, we find that the step meander behaves at equilibrium as $w\ensuremath{\sim}{l}^{\ensuremath{\alpha}}[\mathrm{ln}(L){]}^{1/2}$ $(l$ is the mean interstep distance and $L$ the lateral step extent). The quantity $\ensuremath{\alpha}=1/2$ or 1 depending on whether the elastic interaction is $\mathrm{ln}(l)$ or ${1/l}^{2}.$ During step flow growth the steps repel each other via the diffusion field. This repulsion prevails over the elastic one. It leads to an exponent $1/4;$ $w\ensuremath{\sim}{l}^{1/4}.$ Because the diffusive repulsion is much bigger than the elastic one, nonequilibrium conditions should first result in a drastic reduction of the vicinal surface fluctuation (steps wandering and terrace width fluctuations). However, on further increase of the incoming flux $F,$ the steps become morphologically unstable. This instability is driven by adatom diffusion. It is of deterministic origin and must be distinguished from purely statistical fluctuations. At the instability threshold and in the linear regime, the roughness behaves as $w\ensuremath{\sim}{\ensuremath{\epsilon}}^{\ensuremath{-}1/2}{L}^{1/2}$ ($\ensuremath{\epsilon}$ is the distance from the instability threshold) for an isolated step and $w\ensuremath{\sim}{\ensuremath{\epsilon}}^{\ensuremath{-}1/4}[\mathrm{ln}(L){]}^{1/2}$ for a train of steps. The exponent $1/4$ is a direct consequence of step-step interaction. At the instability point nonlinear terms become relevant. The nonlinear regime is discussed in detail in the following paper.