Modeling one-dimensional island growth with mass-dependent detachment rates

TL;DR

This paper models one-dimensional island growth considering mass-dependent detachment rates, revealing how different detachment behaviors influence island size distributions and their dependence on temperature and particle density.

Contribution

It introduces a class of exactly solvable zero-range process models with mass-dependent detachment rates, linking detachment dynamics to island size distribution shapes.

Findings

Island size distributions can be monomodal or decreasing depending on parameters.

Rapidly increasing detachment rates produce peaked distributions near the average island size.

Slowly increasing detachment rates lead to broader distributions with fat tails.

Abstract

We study one-dimensional models of particle diffusion and attachment/detachment from islands where the detachment rates gamma(m) of particles at the cluster edges increase with cluster mass m. They are expected to mimic the effects of lattice mismatch with the substrate and/or long-range repulsive interactions that work against the formation of long islands. Short-range attraction is represented by an overall factor epsilon<<1 in the detachment rates relatively to isolated particle hopping rates [epsilon ~ exp(-E/T), with binding energy E and temperature T]. We consider various gamma(m), from rapidly increasing forms such as gamma(m) ~ m to slowly increasing ones, such as gamma(m) ~ [m/(m+1)]^b. A mapping onto a column problem shows that these systems are zero-range processes, whose steady states properties are exactly calculated under the assumption of independent column heights in the Master equation. Simulation provides island size distributions which confirm analytic reductions and are useful whenever the analytical tools cannot provide results in closed form. The shape of island size distributions can be changed from monomodal to monotonically decreasing by tuning the temperature or changing the particle density rho. Small values of the scaling variable X=epsilon^{-1}rho/(1-rho) favour the monotonically decreasing ones. However, for large X, rapidly increasing gamma(m) lead to distributions with peaks very close to <m> and rapidly decreasing tails, while slowly increasing gamma(m) provide peaks close to <m>/2$ and fat right tails.