In submonolayer epitaxial island growth, it is fruitful to consider the distribution of the area of capture zones [BM96,ETB06,PE07], 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 [K66,FN07], but more generally we have argued [PE07], drawing from experiences analyzing the terrace-width distributions of vicinal surfaces [EP99], that the CZD is better described by the single-parameter generalized Wigner distribution (GWD):  P(s) =a s^β exp(-bs²), where s is the CZ area divided by its average value; a and b are beta-dependent constants that assure normalization and unit mean of P(s). Painstaking simulations by Amar's [SSA09] and Evans's [LHE10]  groups showed inadequacies in our mean field Fokker-Planck argument relating 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 [PE10].  While the GWD describes the CZD in the regime in which there is significant data in experiments (0.5 < s < 2), it has shortcomings in the tails at both high and low s [LHE10,OA11].  For large s, P(s) may decay exponentially rather than in gaussian fashion.  We discuss several treatments of this issue, emphasizing the fragmentation model we developed [GE11], which depends on two physically motivated scaling exponents (one of which changes values around s = 1.7).  

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 [PTW11,TW12].  We report a series of studies by Fanfoni et al. of InAs quantum dots on GaAs [FP07,FA12] and very recent applications to metallic droplets [NM12], also on GaAs.  (The former [FA12] also shows that the more-often-probed island-size distribution [ISD,ETB06] is comparable to the CZD at lower temperatures but not at higher temperatures when detachment--and consequent coarsening--becomes important.)  At Maryland, for small admixtures of pentacenequinone (PnQ) with pentacene, our experimental colleagues and we studied the CZD at 0.3 ML coverage on SiO₂ [CG08].  As the fraction of PnQ was increased, the value of β dropped from 6.7 below 1% to 5.0 above 1%, indicative of the poorer packing possible when PnQ was present.  For thick 50 ML films, this sudden change around 1% is reflected in a sudden decrease grain size and a consequent decrease in linear mobility.  We also found for C₆₀ on ultrathin SiO₂ that β varies nonmonotonically with temperature, increasing from 2 at room temperature, peaking near 3 around 375K, then declining; other evidence suggests that surface defects are responsible for this behavior.  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 [HSPE11]. Finally, we have applied this approach to the distribution of metro stations in Paris [GE11] and of the areas of French districts (arrondissements) [GE11,S09,LCD93], counties in southeastern US states [S09], and other such secondary administrative units such as Polish powiaty and gminy [S09].

*Supported by UMD NSF-MRSEC Grant DMR 05-20471, NSF-CHE Grant 07-50334, and the Condensed Matter Theory Center, with ancillary support from CNAM (Center for Nanophysics and Advanced Materials).

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