Crystal surfaces in and out of equilibrium: A modern view

The last two decades of progress in the theory of crystal surfaces in and out of equilibrium is reviewed. Various instabilities that occur during growth and sublimation, or that are caused by elasticity, electromigration, etc., are addressed. For several geometries and nonequilibrium circumstances, a systematic derivation provides various continuum nonlinear evolution equations for driven stepped (or vicinal) surfaces. The resulting equations are sometimes different from the phenomenological equations derived from symmetry arguments such as those of Kardar, Parisi, and Zhang. Some of the evolution equations are met in other nonlinear dissipative systems, while others remain unrevealed. The novel and original classes of equations are referred to as ``nonstandard.'' This nonstandard form suggests nontrivial dynamics, where phenomenology and symmetries, often used to infer evolution equations, fail to produce the correct form. This review focuses on step meandering and bunching, which are the two main forms of instabilities encountered on vicinal surfaces. Standard and nonstandard evolution scenarios are presented using a combination of physical arguments, symmetries, and systematic analysis. Other features, such as kinematic waves, some aspect of nucleation, and results of kinetic Monte Carlo simulations are also presented. The current state of experiments and confrontation with theories are discussed. Challenging open issues raised by recent progress, which constitute essential future lines of inquiries, are outlined.