Exact solution of a mean-field approach of an irreversible aggregation with a time dependent rate deposition

Abstract

In this paper we propose a solution for the time evolution of the island density with irreversible aggregation and a time dependent input of particle in the space dimensions $d=1,2$. For this purpose we use the rate equation resulting from a generalized mean field approach. A well-known technique for growing surfaces at the atomic scale is molecular beam epitaxy (MBE). Another approach is the pulsed laser deposition method (PLD). The main difference between MBE and PLD is that in the case of MBE we have a continuous rate of deposition $F$ of adatoms on the surface whereas in the case of PLD the adatoms are deposited during a pulse of a laser which is very short in comparison to the time span $T$ between the pulses. The generalized mean field theory is a useful model for both MBE and PLD with the most simple approximation, point-like island. We show that the parameter $T$ distinguishes the MBE regime from the PLD regime. We solve the rate equation for the PLD regime. We consider the time evolution of the density of immobile islands. For large time $t\gg T$, the PLD regime dominates the MBE regime and we find that the density of immobile islands grows as $t^{1/2}$ whereas for MBE we find the known behavior of the density, $t^{1/3}$ for $d=2$ and $t^{1/4}$ for $d=1$. We illustrate this result with Monte-Carlo simulations for $d=1,2$.