Quantitative study of the decay of intensity oscillations in transient layer-by-layer growth

An analytical formula for the diffraction from an initial transient layer-by-layer growth front has been derived. The approach utilizes recently developed dynamic scaling models which describe the growth-induced roughening evolution at the late stage of growth. The results can be applied to both out-of-phase and non-out-of-phase diffraction conditions. At the out-of-phase conditions, the Bragg peak intensity is given by ${\mathit{I}}_{\mathrm{Bragg}}$∝${\mathit{e}}^{\mathrm{-}\mathrm{\pi }2}$ ${\mathit{w}}^2$[1+cos(2π〈h〉)], which oscillates with the growth of the film thickness 〈h〉 but decays with the increase of the interface width w due to the growth-induced roughening. The derived diffraction formula is consistent with the peak intensity oscillation obtained from a transient layer-by-layer process observed in low-temperature molecular-beam epitaxy growth of Si/Si(111).