Shape and scaling of moving step bunches

We study step bunching under conditions of attachment/detachment limited kinetics in the presence of a deposition or sublimation flux, which leads to bunch motion. Analysis of the discrete step dynamics reveals that the bunch velocity is inversely proportional to the bunch size for general step-step interactions. The shape of steadily moving bunches is studied within a continuum theory, and analytic expressions for the bunch profile are derived. Scaling laws obtained previously for non-moving bunches are recovered asymptotically, but singularities of the static theory are removed and strong corrections to scaling are found. The size of the largest terrace between two bunches is identified as a central scaling parameter. Our theory applies to a large class of bunching instabilities, including sublimation with attachment asymmetry and surface electromigration in the presence of sublimation or growth.