Capture-Zone Scaling in Island Nucleation: Universal Fluctuation Behavior

In island nucleation and growth, the distribution of capture zones (in essence proximity cells) can be described by a simple expression generalizing the Wigner surmise (power-law rise, Gaussian decay) from random matrix theory that accounts for spacing distributions in a host of fluctuation phenomena. Its single adjustable parameter, the power-law exponent, can be simply related to the critical nucleus of growth models and the substrate dimensionality. We compare with extensive published kinetic Monte Carlo data and limited experimental data. A phenomenological theory elucidates the result.

Figure 1

Plots of the GWS of Eq. (1) $P_n(s)$, $n=1,2,3,4$, of relevance in this Letter, indicated by long dashes alternating with $n$ short dashes, also $P_{3/2}(s)$, indicated by the long dashed, short dashed, dotted curve. The thin (blue) solid curve shows the gamma distribution ${\Pi }_7(s)$, discussed later.

Figure 2

(a) Schematic for 1D (vertical). Black rectangles correspond to 1D islands. Horizontal lines mark the midpoints between the edges of two neighboring islands, with the CZs defined as the resulting proximity cells (edge cells [1, 20]). An alternative definition, implicit in the point-island approximation, uses the midpoint between the centers of islands, indicated by dashed lines and leading to Voronoi cells. For islands (nearly) the same size, the two (nearly) coincide; otherwise, edge cells have a narrower distribution [21]. (b) 2D illustration of the islands (approximated as circular) and the Voronoi polygons that bound their CZ, from Ref. 6b.

Figure 3

CZ size distributions for critical-nucleus sizes $i=1$ and $d=1$. Symbols are from Fig. 12 of Ref. 4, for various pairs of values of ($D/F$ in units of ${10}^5$, coverage in monolayers) ⊡ (5, 0.11), ⊙ (5, 1.19), △ (5, 12.65), $\times$ (50, 0.11), $+$ (50, 1.19). The thin curve is the theory prediction (requiring an integration and fitting a free parameter) of Ref. 4. The thick (red) curve is the simpler $P_4(s)$. The inset shows similar results for distribution of gaps between point islands from Fig. 11 of Ref. 4, with the added thick (red) curve giving $P_{3/2}(s)$; its self-convolution [4] [viz. $2{\int }_0^{2s}{P}_{3/2}(s^{\prime })P_{3/2}(2s-s^{\prime })d{s}^{\prime }$] $\approx P_4(s)$, the CZ distribution. Generalization to $d=2$ is unclear.

Figure 4

(a) Symbols are numerical data from Fig. 2(b) of Ref. 5, giving the CZ size distribution for nucleation of islands with $i=0$ in 2D. The thick (blue) curve is $P_1(s)$. (b) Same as panel (a), but symbols for $i=1$ are from Fig. 2(d) of Ref. 5, and the thick curve is $P_2(s)$. In both panels the thin curve is the theory of Ref. 5.