Vicinal surfaces can exhibit a number of different instabilities and step pattern formation that are important in directed growth and nanofabrication. This dissertation attempts to present some theoretical progress made in understanding and predicting the evolution of surface morphology under direct current heating.

We study current-induced instabilities found on both Si(111) and Si(001) surfaces with a physically suggestive two-region diffusion model, motivated by the idea of surface reconstruction or rebonding that often occurs on semiconductor surfaces. The model not only gives a coherent and unified view of the seemingly different instabilities on both surfaces, but also provides a physical way of interpreting the boundary conditions in classic sharp step models. In particular, we find that the effective kinetic coefficient can be negative.

The studies of instabilities enlighten us to pursue a systematic study of the general linear kinetics boundary conditions in sharp step models. We construct a one dimensional discrete hopping model that takes into account both the asymmetry in the hopping rates near a step and the finite probability of incorporation into the solid at the step site. By appropriate extrapolation, we relate the kinetic coefficient and permeability rate in general sharp step models to the physically suggestive parameters of the hopping models. The derivation shows in general the kinetic rate parameters can be negative when diffusion is faster near the step than on terraces.

The subsequent step pattern formation resulting from current-induced instabilities are also discussed. The velocity function formalism is applied to step bunching and in-phase wandering. The more intricate step wandering patterns are treated by a nonlinear evolution equation derived from a geometric representation of the two dimensional curves. The results from numerical calculations resemble the patterns observed in experiments. Two dimensional kinetic Monte Carlo simulations are implemented in a qualitative way, with an emphasis on the physical realization of the effective boundary conditions in terms of microscopic hopping rates. The simulations confirm both the theory of current-induced instabilities and the derivation of boundary conditions.