Abstract
Consider the growth of a nanowire by a step-flow mechanism in the course of vapor-liquid-solid and vapor-solid-solid processes. The growth is initiated by the nucleation of a circular step at the nanowire-catalyst interface near the edge of the nanowire (the triple junction) and proceeds by the propagation toward the center by the Burton-Cabrera-Frank mechanism. Two cases are considered: (i) bulk transport, where the interfacial diffusion of adatoms and the step motion are coupled to the diffusion flux of atoms from the bulk of the catalyst particle, and (ii) surface transport, where atoms from the vapor phase are adsorbed at the surface of the catalyst particle and diffuse along the surface toward the triple line, whence they diffuse to the nanowire-catalyst interface. The attachment kinetics of adatoms at the step, the adsorption kinetics of atoms from the bulk phase, the exchange kinetics at the triple contact line, and the capillarity of the step are taken into account. In case (i) the problem is reduced to an integral equation for the diffusion flux of atoms from the bulk phase to the nanowire-catalyst interface. This equation is solved numerically, and the flux, interfacial concentration of adatoms, and the bulk concentration near the interface are determined. The step velocity is calculated as a function of the step radius and the kinetic parameters. As a result, the growth rate of a nanowire is computed as a function of its radius. In case (ii) analytical solutions for the surface and interfacial concentrations are obtained. In the absence of step capillarity, an analytical formula for the dependence of the nanowire growth rate on the nanowire radius is derived. It is shown in both cases (i) and (ii) that the nanowire growth rate decreases with increasing nanowire radius due to the decrease in the magnitude of the concentration gradients. However, in case (ii), in the limit of negligible desorption of adatoms into the gas phase, the nanowire growth rate is independent of the radius. It is also shown that in the presence of step capillarity (the Gibbs-Thomson effect) increases the nanowire growth rate.
| Original language | English (US) |
|---|---|
| Article number | 074301 |
| Journal | Journal of Applied Physics |
| Volume | 104 |
| Issue number | 7 |
| DOIs | |
| State | Published - 2008 |
Funding
This work was supported by NSF Grant No. DMI-0507053. FIG. 1. Nanowire with a hemispherical catalytic particle and a step at the NWC interface. FIG. 2. Flux f ( r ) from the bulk phase at the NWC interface for different step locations. Here σ = κ ¯ s = k ¯ = δ = 1.0 . FIG. 3. Interfacial adatom concentration u ( r ) for different step locations. The parameters are the same as in Fig. 2 . FIG. 4. Bulk concentration v 0 ( r ) at the interface for different step locations. The parameters are the same as in Fig. 2 . FIG. 5. Surface flux from the bulk phase, f ( r ) (a); interfacial concentration u ( r ) (b); and bulk concentration v 0 ( r ) at the interface (c) for different values of the step mobility: κ ¯ s = 0.5 (dashed lines), κ ¯ s = 1.0 (solid lines), and κ ¯ s = 2.0 (dashed-dotted lines). Other parameters: σ = k ¯ = δ = 1.0 . FIG. 6. Surface flux from the bulk phase, f ( r ) (a); the interfacial concentration u ( r ) (b); and the bulk concentration at the interface, v 0 ( r ) , for different values of the adsorption coefficient: k ¯ = 0.5 (dashed lines), k ¯ = 1.0 (solid lines), and k ¯ = 2.0 (dashed-dotted lines). Other parameters: κ ¯ s = δ = σ = 1.0 . FIG. 7. Surface flux from the bulk phase, f ( r ) (a); interfacial concentration u ( r ) (b); and bulk concentration at the interface, v 0 ( r ) (c), for different values of the parameter δ characterizing the diffusion coefficient ratio: δ = 0.5 (dashed lines), δ = 1.0 (solid lines), and δ = 2.0 (dashed-dotted lines). The other parameters are κ ¯ s = k ¯ = σ = 1.0 . FIG. 8. Step velocity as a function of its location for different parameter values. The solid curves correspond to σ = k ¯ = δ = 1.0 and κ ¯ s = 0.5 (2), κ ¯ s = 1.0 (1), and κ ¯ s = 2.0 (3). The dashed curves correspond to σ = κ ¯ s = δ = 1.0 and k ¯ = 0.5 (4) and k ¯ = 2.0 (5). The dashed-dotted curves correspond to σ = κ ¯ s = k ¯ = 1.0 and δ = 0.5 (6) and δ = 2.0 (7). FIG. 9. Step-flow velocity as a function of the step location for γ = 0 (dashed line; no Gibbs–Thomson effect) and γ = 0.1 (solid line). The other parameters are σ = k ¯ = κ ¯ s = δ = 1.0 . FIG. 10. Dimensionless nanowire growth rate G as a function of the dimensionless nanowire radius ζ = R 0 / R ¯ 0 for γ = 0 (solid line; no Gibbs–Thomson effect), γ = 0.1 (dashed line), and γ = 1.0 (dashed-dotted line). The other parameters are σ = k ¯ = κ ¯ s = δ = 1.0 . FIG. 11. Dimensionless nanowire growth rate G as a function of the step mobility κ ¯ s (solid line) and the adsorption coefficient k ¯ (dashed line). The other parameters are σ = ζ = δ = 1.0 . FIG. 12. Nanowire growing by adsorption of atoms at the surface of the catalyst from the gas phase and diffusion to the NWC interface through the triple line region. FIG. 13. Functions Φ 0 ( R ¯ ) (solid line) and Φ λ ( R ¯ ) (dashed line) defined by Eqs. (56) and (59) , respectively, characterizing the dependence of the nanowire growth rate on the nanowire radius. Here κ ¯ s = 1.0 , k ¯ − = 2.0 , δ = 5.0 , K = 0.5 , Δ = 2.0 , and λ ¯ c = 1.0 .
ASJC Scopus subject areas
- General Physics and Astronomy