Weakly coupled two-dimensional correlations in finite-level epitaxy and chemisorption

We formulate a two-dimensional ordering model for a finite-level system of adatoms on a crystalline surface. The placement of steps in the two surface directions obeys Markovian disorder. Steps in the two directions are weakly correlated due to the restriction that the system occupies only a finite number of levels in the vertical dimension. In particular, we derive the surface atom pair correlation function and the diffracted intensity for the two-level system and obtain a closed-form solution. We discuss the effect of the weak correlation of the two surface directions. This is perhaps the simplest two-dimensional model that can describe the molecular-beam epitaxy measurements of Si/Si(111) reported by Gronwald and Henzler [Surf. Sci. 117, 180 (1982)]. The results of our calculation are also compared to our previous one-dimensional and fully correlated two-dimensional model calculations.