Paper SS+AS+NS-MoM3
Analyzing Capture Zone Distributions (CZD) in Growth: Theory and Applications

Monday, October 28, 2013, 9:00 am, Room 201 A

In submonolayer epitaxial island growth, it is fruitful to consider the distribution of the area of capture zones, i.e. Voronoi (proximity) cells constructed from the island centers. For random nucleation centers (Poisson Voronoi diagrams) the CZD is expected to follow a Gamma distribution, but more generally we have argued [1], drawing from experiences analyzing the terrace-width distributions of vicinal surfaces, that the CZD is better described by the single-parameter generalized Wigner distribution (GWD). Painstaking simulations by Amar's and Evans's groups showed inadequacies in our mean field Fokker-Planck argument relating the characteristic GWD exponent β to the critical nucleus size (conventionally called i+1), i.e. the size of the smallest cluster assumed not to decay. We refine our derivation to retrieve their finding that β is nearly i + 2 [2]. While the GWD describes the CZD in the regime in which there is significant data in experiments (i.e. between half and twice the mean), it has shortcomings in the tails at both high and low areas. For large areas, the distribution may decay exponentially rather than in gaussian fashion. We discuss several treatments of this issue, emphasizing the fragmentation model we developed [3], which depends on two physically motivated scaling exponents.

We discuss applications of this formula and methodology to experiments involving Ge/Si(001), various organics on SiO₂, and para-hexaphenyl (6P) films on amorphous mica. We report a series of studies by Fanfoni et al. of InAs quantum dots on GaAs and very recent applications to metallic droplets by Millunchick's group, also on GaAs. (The former also shows that the more-often-probed island-size distribution is comparable the CZD at lower temperatures but not at higher temperatures when detachment--and consequent coarsening--becomes important.)

We have also used the GWD framework to elucidate kinetic Monte Carlo studies of homoepitaxial growth on Cu(100) with codeposited impurities of different sorts. Finally, we have applied this approach to the distribution of metro stations in Paris [3] and to the distribution of of the areas of French districts (arrondissements) [3,4] , counties in southeastern US states [4], and other such secondary administrative units.

*Supported by NSF-MRSEC at UMD, Grant DMR 05-20471 and NSF-CHE Grant 07-50334.

[2] Alberto Pimpinelli and TLE, Phys. Rev. Lett. 104, 149602 (2010)

[3] Diego Luis González and TLE, Phys. Rev. E 84, 051135 (2011)

[4] Rajesh Sathiyanarayanan, Ph.D. thesis, U. Maryland, 2009; RS et al., preprint.