Diffusion of two-dimensional epitaxial clusters on metal (100) surfaces: Facile versus nucleation-mediated behavior and their merging for larger sizes

Abstract

For diffusion of two-dimensional homoepitaxial clusters of N atoms on metal(100) surfaces mediated by edge atom hopping, macroscale continuum theory suggests that the diffusion coefficient scales like D N ~ Nβ with β = 3/2. However, we find quite different and diverse behavior in multiple size regimes. These include: (i) facile diffusion for small sizes N < 9; (ii) slow nucleation-mediated diffusion with small β < 1 for "perfect" sizes N = N p = L 2 or L(L+1), for L = 3, 4,… having unique ground state shapes, for moderate sizes 9 ≤ N ≤ O(10 2 ); the same also applies for N = N p +3, N p + 4,… (iii) facile diffusion but with large β > 2 for N = N p + 1 and N p + 2 also for moderate sizes 9 ≤ N ≤ O(10 2 ); (iv) merging of the above distinct branches and subsequent anomalous scaling with 1 ○ β < 3/2, reflecting the quasi-facetted structure of clusters, for larger N = O(10 2 ) to N = O(10 3 ); and (v) classic scaling with β = 3/2 for very large N = O(10 3 ) and above. The specified size ranges apply for typical model parameters. We focus on the moderate size regime where show that diffusivity cycles quasi-periodically from the slowest branch for N p + 3 (not N p ) to the fastest branch for N p + 1. Behavior is quantified by Kinetic Monte Carlo simulation of an appropriate stochastic lattice-gas model. However, precise analysis must account for a strong enhancement of diffusivity for short time increments due to back-correlation in the cluster motion. Further understanding of this enhancement, of anomalous size scaling behavior, and of the merging of various branches, is facilitated by combinatorial analysis of the number of the ground state and low-lying excited state cluster configurations, and also of kink populations.

Modeling of Diffusivity for 2D Vacancy Nanopits and Comparison with 2D Adatom Nanoislands on Metal(100) Surfaces Including Analysis for Ag(100)

“…Previous analysis of data for adatom islands for φ = 0.20 eV and δ = 0 at 300 K (using the data shown in Figure 4) revealed that the effective scaling exponent, βeff, did achieve this asymptotic value of βmacro = 1.5 for N ≥ 2300. This is consistent with the expectation that such asymptotic behavior should be achieved for linear island sizes well above the characteristic separation of kinks on close packed step edges, 24 Lk = ½ exp[½φ/(kBT)] ≈ 24 in units of 'a', i.e., for N well above (Lk) 2 ≈ 580. From Figure 4, convergence of the effective size scaling exponent to this asymptotic value for vacancy pits is somewhat slower with βeff ≈ 1.32 for 2025 ≤ N ≤ 3250.…”

“…An assessment of the merging point is based on an assessment of when the distinct identity of different branches is lost, which in turn relies on determination of when clusters have a significant probability of being in an excited state configuration (above the T = 0 K ground state). 23,24 An appropriate combinatorial analysis in Appendix C gives results consistent with the above observations of disrupted behavior occurring earlier for smaller φ.…”

“…See below. For adatom islands with sizes below N = O(10 2 ), as in recent studies, 23,24 we just show: the locally maximum diffusivity for facile sizes N = Np + 1; the moderate diffusivity for perfect sizes N = Np; and locally minimum diffusivity for sizes N = Np + 3. It is clear that vacancy pits diffuse significantly slower than adatom islands when δ = 0 for a broad range of sizes N ≤ O(10 3 ), but that "slow" merging is apparent for larger N. Again, the latter is expected from macroscopic continuum formulations.…”

“…This contrasts behavior for adatom islands where facile sizes N = Np + 1 have higher diffusivity than perfect sizes N = Np, and where these two classes of sizes have different effective diffusion barriers (at least for the moderate size regime). 23,24 We mention that identification of these oscillations either for islands or pits in experimental data would be inhibited by uncertainties in both diffusion coefficients and cluster size. Finally, we provide a more comprehensive analysis of the variation of diffusivity with the strength of the kink rounding barrier, δ, for φ = 0.24 eV at T = 300 K. Figure 6 shows ln[DN/(a 2 he)] versus δ for both vacancy pits and adatom islands for selected sizes N = Np and Np + 1 choosing Np = 64.…”

“…[2][3][4]22 Returning to our analysis of cluster diffusion on metal(100) surfaces, we note that recent theoretical studies of adatom island diffusion revealed surprisingly rich behavior for moderate sizes N ≤ O(10 2 ) for realistic model parameters. 23,24 This includes distinct edge-nucleation-mediated and facile branches of diffusion with quite different sizescaling, and also a cyclical variation of DN with N. The distinct branches merge and oscillations disappear for larger sizes N ≥ O(10 2 ), but unconventional size scaling persists until much larger sizes N = O(10 3 ) for typical model parameters. In the current study, we will explore the existence of analogous behavior for vacancy pits, and compare behavior with previous results for adatom islands.…”