Pattern formation in the instability of a vicinal surface by the drift of adatoms

We study the behavior of steps in a vicinal face with drift of adsorbed atoms (adatoms) by an external field. When the drift is in the downhill direction and its velocity exceeds critical values, v(x)(c) and v(y)(c), the vicinal face is linearly unstable to long-wavelength fluctuations parallel and/or perpendicular to the steps. By taking the continuum limit of the step-flow model, we derive an anisotropic Kuramoto-Sivashinsky equation with propagative terms, which describes the motion of an unstable vicinal face. Its numerical solution shows ripples or a zigzag pattern expected from the linear analysis. Nonlinearity becomes important in the late stage and, depending on the condition, various patterns are formed: regular step bunches, a hill and valley structure tilted from the initial step direction, mounds, and a chaotic pattern.