Quantitative theory of current-induced step bunching on Si(111)

Abstract

We use a one-dimensional step model to study quantitatively the growth of step bunches on Si(111) surfaces induced by a direct heating current. Parameters in the model are fixed from experimental measurements near 900 $\ifmmode^\circ\else\textdegree\fi{}$C under the assumption that there is local mass transport through surface diffusion and that step motion is limited by the attachment rate of adatoms to step edges. The direct heating current is treated as an external driving force acting on each adatom. Numerical calculations show both qualitative and quantitative agreement with experiment. A force in the step down direction will destabilize the uniform step train towards step bunching. The average size of the step bunches grows with electromigration time $t$ as ${t}^{\ensuremath{\beta}}$, with $\ensuremath{\beta}\ensuremath{\approx}0.5$, in agreement with experiment and with an analytical treatment of the steady states. The model is extended to include the effect of direct hopping of adatoms between different terraces. Monte Carlo simulations of a solid-on-solid model, using physically motivated assumptions about the dynamics of surface diffusion and attachment at step edges, are carried out to study two-dimensional features that are left out of the present step model and to test its validity. These simulations give much better agreement with experiment than previous work. We find a step bending instability when the driving force is along the step edge direction. This instability causes the formation of step bunches and antisteps that is similar to that observed in experiment.