Two mechanisms forming a comblike step pattern induced by a moving linear adatom source

We carry out phase field simulations to study properties of the comblike step patterns induced by an adatom source. When an adatom source advances right in front of a step, step wandering is caused by the asymmetry of the surface diffusion field and small protrusions are formed. If the velocity of the source $V_{\mathrm{p}}$ is smaller than a critical value $V_{\mathrm{p}}^{\mathrm{c}}$, the protrusions follow the adatom source with coarsening of the step pattern, and a regular comblike pattern with finger-like protrusions is formed. With a sufficiently small $V_{\mathrm{p}}$, the gap of the supersaturation is large at the adatom source. Since the period of protrusions, $\mathrm{\Lambda }$, decreases with increasing $V_{\mathrm{p}}$, the coarsening of step pattern is irrelevant for the protrusions to catch up with the adatom source. Near $V_{\mathrm{p}}^{\mathrm{c}}$, the gap of the supersaturation at the adatom source is small. Taking account of the increase in $\mathrm{\Lambda }$ with increasing $V_{\mathrm{p}}$, the coarsening of the step pattern is essential for the protrusions to follow the adatom source.

Figure 1

Snapshots of the comblike pattern at (a) $t=3.8\times {10}^5$ and (b) $t=9\times {10}^4$. Parameters are ${\epsilon }_4=5\times {10}^{-2}, W_0=3, F_u={10}^{-5}$, (a) $V_{\mathrm{p}}=4\times {10}^{-3}$, and (b) $V_{\mathrm{p}}=1.7\times {10}^{-2}$. The region with $\varphi >0$ is blue (dark) and the line parallel to the $x$ axis near the tips represents the position of adatom source.

Figure 2

Relation between the period of protrusions ($\mathrm{\Lambda }$ or $\lambda$) and the source velocity $V_{\mathrm{p}}$. The data are averaged over 10 or more runs. Relation between the channel width $\mathrm{\Lambda }$ and the tip velocity $V_{\mathrm{p}}$ in dendritic growth in a channel is also shown. Dotted lines show $V_{\mathrm{p}}^{-1/2}$ fitting to the data.

Figure 3

Field of supersaturation and the step profile with the slow adatom source, $V_{\mathrm{p}}=4\times {10}^{-3}$. ${\epsilon }_4=5\times {10}^{-2}, W_0=3$, and $F_u={10}^{-5}$. Time is (a) $t=2.2\times {10}^4$, (b) $t=3.3\times {10}^4$, (c) $t=4.9\times {10}^4$, and (d) $t=1.45\times {10}^5$. Dotted lines represent isoconcentration lines separated by 0.05. The thick solid line represents the position of the adatom source.

Figure 4

Field of supersaturation $u$ and the step profile with the fast adatom source, $V_{\mathrm{p}}=1.7\times {10}^{-2}$. ${\epsilon }_4=5\times {10}^{-2}, W_0=3$, and $F_u={10}^{-5}$. Time is (a) $t=3\times {10}^3$, (b) $t=5\times {10}^3$, (c) $t=1.75\times {10}^4$, and (d) $t=4\times {10}^4$. Dotted lines represent isoconcentration lines of separated by 0.05. The thick solid line represents the position of the adatom source.

Figure 5

Dependence of distance between the highest tip and the adatom source, $d_{\text{tip-source}}$, on $V_{\mathrm{p}}$. The parameters are $V_{\mathrm{p}}=1.7\times {10}^{-2}, {\epsilon }_4=5\times {10}^{-2}, W_0=3$, and $F_u={10}^{-5}$. The system width is $L_x=800$. The data are averaged over 10 runs.

Figure 6

Effect of the noise strength $F_u$ on the period $\mathrm{\Lambda }$ with (a) $V_{\mathrm{p}}=1.7\times {10}^{-2}$ at $t=1.7\times {10}^5$ and (b) $V_{\mathrm{p}}=5\times {10}^{-3}$ at $t=1.2\times {10}^5$. The system width is $L_x=800$. The tip position of the highest protrusion is about (a) 2900 and (b) 680. The data are averaged over (a) 10 runs and (b) 20 runs.

Figure 7

Dependence of number of protrusions on $L_x$ with (a) $V_{\mathrm{p}}=4\times {10}^{-3}$ and (b) $V_{\text{p}}=1.7\times {10}^{-2}$. $F_u={10}^{-5}$ and $L_y=1000$ or 2000. Circled numbers represent the numbers of protrusions.

Figure 8

Snapshots of the comblike pattern at (a) $t=5\times {10}^4$ and (b) $t={10}^5$. Parameters are ${\epsilon }_4=5\times {10}^{-2}, W_0=3, F_u={10}^{-5}, V_{\mathrm{p}}=1.7\times {10}^{-2}$. The region with $\varphi >0$ is blue (dark) and the line parallel to the $x$ axis near the tips represents the position of adatom source.

Figure 9

Time evolution of $d_{\text{tip-source}}$. The parameters are $V_{\mathrm{p}}=1.7\times {10}^{-2}, {\epsilon }_4=5\times {10}^{-2}, W_0=3$, and $F_u={10}^{-5}$. The maximum system size is $L_x\times L_y=240\times 1800$.