Being motivated by a comb-like step pattern observed during Ga deposition on Si(111) surfaces[1], we theoretically study growth of a step which is guided by a line adatom source moving in front of the step. The pattern formation in a lattice model[2] and in a phase-field model is discussed.

The line source of adatoms located initially next to a straight step moves at a constant speed V_p away from the step. It releases adatoms of the density c =1/2, and makes the step grow. Many intrusions are soon formed by Mullins-Sekerka instability. In growth towards [11] direction, the comb-like step pattern appears with small V_p after coarsening of the pattern via competition of the intrusions. Periodic tips of the comb-like pattern follow the source steadily. With large V_p, a fractal pattern is formed by noise.

For the comb pattern, the structure becomes finer with increasing V_p. The period of initial intrusions is given by the capillary length d₀ and the diffusion length l_D=D_s/V_p as (d₀l_D)^1/2. The period λ of the steady comb-like pattern is controlled by the noise strength F_u. The period becomes logarithmically shorter with increasing the noise:  λ^_~ (d₀l_D)^1/2|lnF_u|. Such noise dependence suggests a peculiar mechanism to terminate the coarsening.

Fig.1: Time evolution of an irregular step pattern(upper) and a periodic comb-like pattern(lower).

[1] H. Hibino, H. Kageshima and M. Uwaha: Surf. Sci., 602 (2008) 2421.

[2] S. Kondo, M. Sato, M. Uwaha and H. Hibino: Phys. Rev. B 84, 045420 (2011). 

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