Period of a comblike pattern controlled by atom supply and noise

Pattern formation of a step on a growing crystal surface induced by a straight line source of atoms, which is escaping from the step at a velocity ${V}_{\mathrm{p}}$, is studied with the use of a phase field model. From a straight step, fluctuations of the most unstable wavelength ${\ensuremath{\lambda}}_{\mathrm{max}}$ grow. Competition of intrusions leads to coarsening of the pattern, and survived intrusions grow exponentially. With sufficient strength of the crystal anisotropy, a regular comblike pattern appears. This peculiar step pattern is similar to that observed on a Ga-deposited Si(111) surface. The final period of the intrusions, $\ensuremath{\Lambda}$, is determined when the exponential growth ends. The period depends on the strength ${F}_{u}$ of a current noise in diffusion as $\ensuremath{\Lambda}\ensuremath{\sim}{\ensuremath{\lambda}}_{\mathrm{max}}|ln{F}_{u}|$: such a logarithmic dependence is confirmed for the first time. A nonmonotonic ${V}_{\mathrm{p}}$ dependence of $\ensuremath{\Lambda}$ indicates that the comblike pattern with a small ${V}_{\mathrm{p}}$ is related to an unstable growth mode of the free needle growth in a channel. The pattern is stabilized by the guiding linear source.