  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 as 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.
