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 ${\lambda }_{\max }$ 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, $\Lambda$, is determined when the exponential growth ends. The period depends on the strength $F_u$ of a current noise in diffusion as $\Lambda \sim {\lambda }_{\max }|ln{F}_u|$: such a logarithmic dependence is confirmed for the first time. A nonmonotonic $V_{\mathrm{p}}$ dependence of $\Lambda$ 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.

Figure 1

A simplified model for Ga-deposited Si(111) surface. Ga deposition expands the $6.3\times 6.3$ area, and the phase boundary emits the excess Si atoms, which make the step grow.

Figure 2

Schematic diagrams of the PM simulation. (a) Initial configuration of the system. (b) Initial profile of the fields $\varphi$ and $u$. (c) Profile of $\varphi$ and $u$ after some advancement of the source. Computation with Eqs. (8) and (13) is performed between $y=0$ and the position of the linear source at $y=y_{\mathrm{p}0}+V_{\mathrm{p}}t(=L_y)$.

Figure 3

(a) Morphology diagram with the noise strength $F_u={10}^{-5}$ and the source intensity $c_0=0.5$. Empty symbols indicate that the step fails to follow the source. Dendrite implies pattern with side branches. (b) Irregular pattern with $V_{\mathrm{p}}=0.005,{\epsilon }_4=0$. (c) Comblike pattern with $V_{\mathrm{p}}=0.005,{\epsilon }_4=0.05$. These step patterns are superposition of the step position [defined as the contour $\varphi (x,y)=0$] at different times.

Figure 4

Change of the period $\Lambda$ with the source velocity $V_{\mathrm{p}}$ (solid marks). The fixed parameters are $c_0=0.5,d_0=0.05,F_u={10}^{-5}$, and $L_x=800$. The data are averaged over 20 independent runs. The velocity of a free dendrite without noise, in a channel of width $\Lambda$ with $u(t=0)=0.5$, is plotted with empty marks (see Sec. 3d).

Figure 5

Time evolution of a comblike step with ${\epsilon }_4=0.05$ and $V_{\mathrm{p}}=0.005$ under different intensity of the noise (a) $F_u={10}^{-7}$, (b) $F_u={10}^{-5}$, and (c) $F_u={10}^{-3}$. Time increment of the successive curves is $2\times {10}^4$.

Figure 6

The period $\Lambda$ and the strength of the noise $F_u$ with $c_0=0.5,V_{\mathrm{p}}=0.005$, and ${\epsilon }_4=0.05$.

Figure 7

Change of the period $\Lambda$ with the density at the source $c_0$, for $V_{\mathrm{p}}=0.01$ in a system of the size $L_x=800$. The noise and the anisotropy are ${\epsilon }_4=0.05$ and $F_u={10}^{-5}$, respectively. The dashed line is from Eq. (19).

Figure 8

The comblike pattern guided by the source (left) and a needle pattern in a channel (right).

Figure 9

Comblike steps of the same period growing with (a) $V_{\mathrm{p}}=0.005$ and (b) $V_{\mathrm{p}}=0.017$. The red line is the position $\varphi (x,y)=0$, and the source is at the right end.