  We study current-induced step bunching and wandering instabilities with
subsequent pattern formations on vicinal surfaces. A novel two-region diffusion
model is developed, where we assume that there are different diffusion rates on
terraces and in a small region around a step, generally arising from local
differences in surface reconstruction. We determine the steady state solutions
for a uniform train of straight steps, from which step bunching and in-phase
wandering instabilities are deduced. The physically suggestive parameters of
the two-region model are then mapped to the effective parameters in the usual
sharp step models. Interestingly, a negative kinetic coefficient results when
the diffusion in the step region is faster than on terraces. A consistent
physical picture of current-induced instabilities on Si(111) is suggested based
on the results of linear stability analysis. In this picture the step wandering
instability is driven by step edge diffusion and is not of the Mullins-Sekerka
type. Step bunching and wandering patterns at longer times are determined
numerically by solving a set of coupled equations relating the velocity of a
step to local properties of the step and its neighbors. We use a geometric
representation of the step to derive a nonlinear evolution equation describing
step wandering, which can explain experimental results where the peaks of the
wandering steps align with the direction of the driving field.
