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Low coverage surface diffusion in complex energy landscapes: Analytical solution and application to intercalation in topological insulators

A general expression is introduced for the tracer diffusivity in complex periodic energy landscapes with more than one distinct hop rate in two- and three-dimensional diluted systems (low coverage, single-tracer limit). For diffusion in two dimensions, a number of formulas are presented for complex combinations of hop rates in systems with triangular, rectangular and square symmetry. The formulas provide values in excellent agreement with Kinetic Monte Carlo simulations, concluding that the diffusion coefficient can be directly determined from the proposed expressions without performing such simulations. Based on the diffusion barriers obtained from first principles calculations and a physically-meaningful estimate of the attempt frequencies, the proposed formulas are used to analyze the diffusion of Cu, Ag and Rb adatoms on the surface and within the van der Waals (vdW) gap of a model topological insulator, Bi$_{2}$Se$_{3}$. Considering the possibility for adsorbate intercalation from the terraces to the vdW gaps at morphological steps, we infer that, at low coverage and room temperature: (i) a majority of the Rb atoms bounce back at the steps and remain on the terraces, (ii) Cu atoms mostly intercalate into the vdW gap, the remaining fraction staying at the steps, and (iii) Ag atoms essentially accumulate at the steps and gradually intercalate into the vdW gap. These conclusions are in good qualitative agreement with previous experiments.