Reconciling Nano- and Micro-Scale VLS Growth by Including Multi-Scale Supersaturation: A Growth Model Applied to Lateral Ge Films on Si

Extensive work on metal-catalyst-assisted vapor-liquid-solid (VLS) growth has been focused on nanometer-scale wires (nanowires) and there have been several studies modelling the effect of diameter, growth time, surface diffusion, etc. on these nanowires. In this context of nanowires, since the catalyst is so small, one could and researchers have ignored the effect of the requirement to supersaturate the catalyst. In this work, an analytical growth model is presented to explain the effect of supersaturation on the size of micrometer-scale catalysts and resulting films grown via the VLS mechanism by chemical vapor deposition by introducing the idea of "multi-scale" supersaturation (supersaturation as a function of catalyst size). The proposed model is derived from a materials balance relation which reduces to the nanowire equation (without the need for considering catalyst supersaturation) for sufficiently small radii. By including the supersaturation term, the model predicts the existence of the maximum size range of film at certain growth conditions which is not present in the pre-existing models. The model could also explain the "induction time" required for the catalyst to reach a supersaturation state to begin VLS growth. More importantly, it agrees with the available experimental data of VLS lateral heteroepitaxy of micrometer-scale Ge films on Si substrate which suggests that 0.0379 mol/cm 3 of Ge atoms are required to supersaturate the Au catalyst at the growth conditions studied (375 °C). This new perspective on the multi-scale supersaturation effect unravels the role of the supersaturation term that had been hidden for more than half a century.