Step-flow growth of a nanowire in the vapor-liquid-solid and vapor-solid-solid processes

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.