Self-shadowing in ballistic fan formation from point seeds

We utilize the model of ballistic aggregation to investigate the growth of particles on a point seed under various oblique particle fluxes to model nanorod growth at early deposition times. In particular, we show that the angular nature of the particle flux leads to a self-shadowing behavior, where features of an individual aggregate can shadow other features of the same aggregate. We quantify the shape of the aggregates by the growth exponent $p$, where the radius of the aggregate $R$ is related to the height of the aggregate $z$ as $R\sim z^p$. We show that the self-shadowing effect is the dominant factor that controls the evolution of nanorod size during oblique angle deposition. The simulation predictions are consistent with recent experimental results.

Figure 1

Illustration of the growth of an aggregate on a point seed. The incident flux makes an angle $\theta$ with the suitably defined normal, and rotation occurs along the $\varphi$ direction.

Figure 2

Cross-section images of aggregate growth on a point seed with no surface diffusion. The images in (a)–(d) depict aggregate growth under oblique fluxes of angles 0°, 20°, 40°, and 60°, respectively.

Figure 3

(Color online) Aggregate profile under normally incident flux. The first order prediction (Ref. 3) for the aggregate shape is given by $r=r_0\sqrt{\cos \phi }$, which shows reasonable agreement with simulation results, although there is a small underestimate near the center of the aggregate.

Figure 4

Plot of aggregate fan-out angle $\phi$ versus oblique flux angle $\theta$ with no surface diffusion. A minimum fan-out angle is realized at $\theta \approx 20°$ because at this oblique flux angle, the particle flux is best aligned with column growth atop the aggregate.

Figure 5

Diagram of growth atop an aggregate for normal incidence particle flux. The particle flux makes an angle $\alpha$ with the local surface normal, and the local columnar structure grows at an angle $\beta$ with respect to the local surface normal. It follows that the local columnar growth makes an angle $\alpha -\beta$ with the incident particle flux.

Figure 6

(Color online) Cross-section image of aggregate density under a 20° oblique flux (a) without surface diffusion and (b) with surface diffusion. The profile in (b) shows characteristics of fingering, or small, high density growth volumes along the sides of the aggregate.

Figure 7

(Color online) Graph of aggregate radius $R$ versus height above seed $z$ used to determine the exponent $p$. Two simulation runs are shown for oblique fluxes of $\theta =25°$ and $\theta =70°$ observing $p=0.78\pm 0.04$ and $p=0.50\pm 0.05$, respectively.

Figure 8

Plot of aggregate fan-out angle $\phi$ versus oblique flux angle $\theta$ with surface diffusion.

Figure 9

(Color online) Plot of growth exponent $p(R\sim z^p)$ versus oblique flux angle $\theta$ with surface diffusion, along with selected experimental results (Refs. 14, 15, 16, 17, 40). A value of $p=1$ corresponds to a conical structure.