Morphologies from slippery ballistic deposition model: A bottom-up approach for nanofabrication

We report pattern formation using a slippery ballistic deposition (SBD) model where growth germinates from a single site or from sites distributed periodically on a lattice. By changing the sticking probability $p_s$ and choosing systems with different lattice constants and symmetries, we demonstrate that a variety of patterns can be generated. These patterns can be further used as scaffolds for nanofabrication. We also demonstrate that by choosing a lateral sticking probability $p_l$ at the base that is different than $p_s$, one can control both the early and late time morphologies originating from a seed. Furthermore, we indicate a possible generalization of preparing patterns to higher dimensions that in principle can have potential technological applications for preparing grooves and scaffolds of specific shapes and periodicities.

Figure 1

1D ballistic growth in the bulk for lattice sizes $L=512$, 1024, 2048, 4096, and 8192. The inset shows the scaled plots.

Figure 2

The height fluctuation $\xi (L,t\to \infty )$ (black circles) as a function of lattice size $L=32$–$8192$. The red line is a fit to the data using lattice sizes $L=32$–$2048$ described by the equation $\xi =(0.66\pm 0.05)L^{0.449\pm 0.008}$. The green line is described by $\xi =(0.52\pm 0.02)L^{0.484\pm 0.003}$ using the points $L=256$–$3072$. The blue line is described by $\xi =(0.46\pm 0.04)L^{0.499\pm 0.005}$ using the points $L=3072$–$8192$. The inset shows the corresponding log-log plot and the straight line (blue) corresponds to the slope $0.499\pm 0.005$.

Figure 3

Scaled height fluctuations $\xi (L,t)/L^{\alpha }$ as a function of time for lattice sizes $L$ and $2L$ using the value for $\alpha$ obtained using Eq. (3) as listed in Table ; (a) $L$ $=$ 64 and 128, (b) $L$ $=$ 256 and 512, (c) $L$ $=$ 2048 and 4096, and (d) $L$ $=$ 4096 and 8192. The inset demonstrates the best data collapse with rescaled time axis $t/L^z$. This value of the $z$ exponent that corresponds to this “best” data collapse is listed in Table .

Figure 4

Comparison of height fluctuations in the 1D BD model (black circle) and the fluctuations in the actual depths of the corresponding columns generated using the BD algorithm without the voids (red squares). In the same figure we plot height fluctuations in the RD model (green diamonds). We note that the “depth” fluctuation in the BD model does not saturate and is identical to the height fluctuation in the RD model.

Figure 5

Rendition of the slippery ballistic model. The approaching particle (empty blue square) missed the opportunity to stick at sites 1, 2, 3, and 4 and finally lands at an unoccupied position (hashed blue square).

Figure 6

1D ballistic growth from a single site for various sticking probabilities: (a) $p_s=0.01$, (b) $p_s=0.1$, (c) $p_s=0.25$, (d) $p_s=0.50$, and (e) $p_s=1.0$.

Figure 7

1D ballistic growth from a single site for various sticking probabilities: (a) $p_s=0.01$, (b) $p_s=0.1$, (c) $p_s=0.25$, (d) $p_s=0.50$, and (e) $p_s=1.0$. These snapshots are obtained by superimposing 100 patterns.

Figure 8

Definitions of width $\langle w\rangle$, height $\langle h\rangle$, and the angle $\theta$ subtended by the cone for 1D SBD model. The angle $\theta \to {90}^0$ when $p_s\to 0$ and saturates at $\theta \simeq {57}^0$ as $p_s\to 1$.

Figure 9

Variation of mean height $\langle h\rangle$ as a function of the mean width $\langle w\rangle$ for 1D SBD for different sticking probabilities $p_s$. The black circles, red squares, green diamonds, and blue triangles correspond to $p_s=0.10$, 0.25, 0.50, and 1.00, respectively. The inset shows the variation of the angle $\theta =\tan {}^{-1}\left(\frac{\langle h\rangle }{\langle w\rangle }\right)$ subtended by the cone as a function of the sticking probability $p_s$.

Figure 10

Ballistic growth from a seed. Different options are shown for $t=1,2$, and 3. Each color in $t=2$ and 3 should be connected to the corresponding parent configuration.

Figure 11

Variation of the angle as a function of the growing width for $p_s=1.0$.

Figure 12

Illustration of how different morphologies could be obtained by adjusting the sticking probability in SBD onto nanoparticle arrays.

Figure 13

Patterns resulting from 1D SBD model using various values of $p_s$ and distances between nucleation sites.

Figure 14

Variation of the shape as a function of different $p_s$ when the lateral sticking probability $p_l=1.0$; (a) $p_s=0.5$, (b) $p_s=0.1$, and (c) $p_s=0.01$.

Figure 15

2D ballistic growth on a flat surface having uniformly distributed seed particles. The corresponding density variation as a function of the vertical distance is shown in Fig. 16.

Figure 16

Variation of surface density $\sigma (z)$ as a function of distance $z$ from the origin for several values of the density of the impurity sites. Black circles, red squares, green diamonds, blue triangles, orange plus, and magenta stars represent ${\rho }_{\text{imp}}=0.00004$, ${\rho }_{\text{imp}}=0.0001$, ${\rho }_{\text{imp}}=0.004$, ${\rho }_{\text{imp}}=0.0625$, ${\rho }_{\text{imp}}=0.25$, and ${\rho }_{\text{imp}}=1.0$ (bulk), respectively. Notice that each curve asymptotically approaches the same (bulk) density.