Low-coverage surface diffusion in complex periodic energy landscapes: Analytical solution for systems with symmetric hops and application to intercalation in topological insulators

This is the first of two papers that introduce a general expression for the tracer diffusivity in complex, periodic energy landscapes with $M$ distinct hop rates in one-, two-, and three-dimensional diluted systems (low-coverage, single-tracer limit). The present report focuses on the analysis of diffusion in systems where the end sites of the hops are located symmetrically with respect to the hop origins (symmetric hops), as encountered in many ideal surfaces and bulk materials. For diffusion in two dimensions, a number of formulas are presented for complex combinations of the different hops 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 the 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, ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Considering the possibility of 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. The companion report (M. A. Gosálvez et al., Phys. Rev. B, submitted] extends the present study to the description of systems that contain asymmetric hops.

Figure 1

(a) Example of a complex potential energy landscape for a diffusing particle. Four adsorption site types are indicated: $f, h, t$ and $b$. (b) A possible diffusion track (random walk) involving 14 performed jumps at 10 different hop rates ${\nu }_{ij}$. (c–f) Possible hops from each site type ($h, b, f$, and $t$, respectively). Gray-shaded landscapes in (b)–(f) are used to highlight the colored arrows/hops. (g) Energy paths for hops starting/ending in $h$ sites. A longer arrow assigned to a hop rate (${\nu }_{th}, {\nu }_{ht}$, etc.) indicates a higher rate.

Figure 2

Schematic of three typical surfaces with (a) triangular symmetry [e.g., fcc(111) and hcp(0001)], (b) rectangular symmetry [e.g., fcc(110)], and (c) square symmetry [e.g., fcc(100)]. We refer to typical adsorption sites as $f$ (for fcc hollow), $h$ (for hcp hollow), $t$ (for on top), $b$ (for bridge/short bridge), and $B$ (for long bridge).

Figure 3

Fundamental paths for a system with four site types. FW, forward path; BW, backward path. (a–d) All four equivalent paths for $\mathrm{FW}=1231$, 1241, 1341, and 12341, respectively, and $\mathrm{BW}=1321$, 1421, 1431, and 14321, respectively. (e, f) All eight equivalent paths for $\mathrm{FW}=12431$ and 13241, respectively, and $\mathrm{BW}=13421$ and 14231, respectively. (g) Demonstration of the equality ${\nu }_{12}{\nu }_{23}\cdots {\nu }_{S1}={\nu }_{1S}{\nu }_{SS-1}\cdots {\nu }_{21}$ for $S=4$.

Figure 4

Illustration of the two simulation procedures used to perform KMC simulations of diffusion in this study: (a) KMC-1; (b) KMC-2. (c–e) Estimations of the attempt frequencies in this study. (f) Interpretation of the attempt-frequency dependence on the barrier size and path length.

Figure 5

(a) Crystal structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, showing the Bi and Se atoms as large and small spheres, respectively, using different colors according to the stacking sequence: red for $a$, blue for $b$, and black for $c$. (b) Several crystallographic cells are highlighted (solid, dashed, and dotted lines) for cross-reference. (b) Schematic supercell structure used for the calculation of diffusion barriers on the terrace, in the vDW gap, and across steps, as indicated. (c) Top view of possible adsorption sites at the vdW gap: $a, b$, and $c$ are locations aligned vertically with each stack, respectively. (d) Example of diffusion path geometries and energies determined through our DFT calculations for the Ag adatom.

Figure 6

Calculated diffusion length $\mathrm{\Lambda }$ (in logarithmic scale) as a function of temperature for Cu, Ag, and Rb adatoms: (a) on the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$(0001) surface and (b) in the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ vdW gap. Diffusion time: 1 min.

Figure 7

(a) Geometrical alignment between terrace and vdW gap at a morphological step. (b–d) Schematic of the low-coverage diffusivity (in ${\mathrm{cm}}^2$ ${\mathrm{s}}^{-1}$) of Rb, Cu, and Ag on the terrace and in the vdW gap of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ at room temperature. Larger arrows denote larger diffusivities. Our DFT-calculated diffusion barriers (in eV) are shown in boldface for (left) terrace diffusion, (center left) terrace reentry, (center right) vdW-gap penetration, and (right) vdW-gap diffusion.