### Physical Review B Volume 57, p 14891

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

Da-Jiang Liu and John D. Weeks
### 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 deg 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*^{beta}, with *beta* = 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 new 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.

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