We study a minimal stochastic model of step bunching during growth on a one-dimensional vicinal surface. The formation of bunches is controlled by the preferential attachment of atoms to descending steps (inverse Ehrlich-Schwoebel effect) and the ratio $d$ of the attachment rate to the terrace diffusion coefficient. For generic parameters $\left(d>0\right)$ the model exhibits a very slow crossover to a nontrivial asymptotic coarsening exponent $\beta \simeq 0.38$. In the limit of infinitely fast terrace diffusion $\left(d=0\right)$ linear coarsening $\left(\beta =1\right)$ is observed instead. The different coarsening behaviors are related to the fact that bunches attain a finite speed in the limit of large size when $d=0$, whereas the speed vanishes with increasing size when $d>0$. For $d=0$ an analytic description of the speed and profile of stationary bunches is developed, and a connection to the problem of front propagation into an unstable state is pointed out.

• Received 13 September 2004

DOI:https://doi.org/10.1103/PhysRevE.71.041605