Size Distributions of Anisotropic Nanostructures with Backward Diffusion and Nucleation Delay

Size distributions of different nanostructures, including length distributions of highly anisotropic semiconductor nanowires and nanomembranes, determine statistical properties within large ensembles of such structures. This is paramount from a fundamental viewpoint and for device applications. We study the Becker–Döring equations for heterogeneous growth with size-linear forward and backward rate constants that describe surface diffusion fluxes from the bottom to the top of the structures and in the reverse direction. Additionally, we account for a nucleation delay in the very first monolayer on a substrate. The strength of backward diffusion determines two principally different growth regimes: either limited growth with a length saturation or infinite growth, where the average length increases exponentially with time. We obtain three-parametric analytical solutions for the size distribution in both regimes, given by a modified Polya distribution for limited growth or a modified β-distribution for infinite growth. We analyze the size distribution shapes versus the control parameters and their scaling properties in the continuum limit and fit the length distributions of different III–V nanowires. These results should be useful for understanding and controlling the size distributions of different nanostructures and shed more light on the statistical properties at the nanoscale.