Distributional fixed point equations for island nucleation in one dimension: a retrospective approach for capture zone scaling

TL;DR

This paper develops a novel retrospective approach using fixed point equations to analyze island nucleation and capture zone scaling in one-dimensional submonolayer deposition, aligning well with Monte Carlo simulations.

Contribution

It introduces a new retrospective framework with fixed point equations for gap and capture zone distributions, incorporating fragmentation bias and extending traditional fragmentation theory.

Findings

Fixed point equations accurately model gap distributions.

Mean field approximation justified by fragmentation bias considerations.

Results agree with Monte Carlo simulations and traditional models.

Abstract

The distributions of inter-island gaps and captures zones for islands nucleated on a one-dimensional substrate during submonolayer deposition are considered using a novel retrospective view. This provides an alternative perspective on why scaling occurs in this continuously evolving system. Distributional fixed point equations for the gaps are derived both with and without a mean field approximation for nearest neighbour gap size correlation. Solutions to the equations show that correct consideration of fragmentation bias justifies the mean field approach which can be extended to provide closed-from equations for the capture zones. Our results compare favourably to Monte Carlo data for both point and extended islands using a range of critical island size $i=0,1,2,3$. We also find satisfactory agreement with theoretical models based on more traditional fragmentation theory approaches.