title: Natural scaling of size distributions in homogeneous and heterogeneous rate equations with size-linear capture rates.
abstract: We obtain exact solutions of the rate equations for homogeneous and heterogeneous irreversible growth models with linear size dependences of the capture rates. In the limit of high ratios of diffusion constant over deposition rate, both solutions yield simple analytical scaling functions with the correct normalizations. These are given by the cumulative distribution function and the probability density function of the gamma-distribution in homogeneous and heterogeneous cases, respectively. Our size distributions depend on the value of the capture rate a in the reaction of joining two mobile monomers A1 (A1 + A1 → A2) or the monomer attachment to the reactive defect B (A1 + B → AB). In homogeneous cases, the size distribution is monotonically decreasing regardless of a. In heterogeneous growth, the distribution is monotonically decreasing when a ≤ 1 and monomodal when a > 1. The obtained solutions describe fairly well the experimental data on the length distributions of Al, Ga, In, and Mn adatom chains on Si(100)-2 × 1 surfaces.
authors.affiliation: St. Petersburg Academic University, Khlopina 8/3, 194021 St. Petersburg, Russia.
authors.forename: V G
authors.initials: VG
authors.lastname: Dubrovskii
authors.affiliation: St. Petersburg Academic University, Khlopina 8/3, 194021 St. Petersburg, Russia.
authors.forename: Yu S
authors.initials: YS
authors.lastname: Berdnikov