Thickness-dependent energetics for Pb adatoms on low-index Pb nanofilm surfaces: First-principles calculations

Adsorption, interaction, and diffusion of adatoms on surfaces control growth and relaxation of epitaxial nanostructures and nanofilms. Previous reports of key diffusion barriers for Pb diffusion on low-index Pb surfaces are limited in scope and accuracy. Thus, we apply density functional theory (DFT) to calculate the adsorption and diffusion energetics for a Pb adatom on Pb(111), Pb(100), and Pb(110) nanofilms with different thicknesses. We find that these quantities exhibit damped oscillatory variation with increasing film thickness. For Pb(111) films, energetics along the minimum energy path for Pb adatom diffusion between adjacent fcc and hcp sites varies significantly with film thickness, its form differing from other metal-on-metal(111) systems. For Pb(111) and Pb(100) nanofilms, diffusion barriers obtained for both adatom hopping and exchange mechanism differ significantly from previous DFT results. Hopping is favored over exchange for Pb(111), and the opposite applies for Pb(100). For Pb(110) nanofilms, Pb adatom hopping over an in-channel bridge is most facile, then in-channel exchange, then cross-channel exchange, with cross-channel hopping least favorable. We also assess lateral Pb adatom interactions, and characterize island nucleation during deposition on Pb(111).

Figure 1

Surface energies $\gamma$ (upper panel) and stability indices $\mathrm{\Delta }\mu$ (lower panel) versus Pb film thickness $L$ from our PBEsol calculations. (a) and (b) Pb(111), (c) and (d) Pb(100), and (e) and (f) Pb(110). The curves for fixed and relaxed films are indicated by different colors and symbols.

Figure 2

Total (adsorption plus interaction) energy $E_1\mathrm{M}$ and effective pair interaction ${\omega }_{\mathrm{p}}1,\mathrm{eff}$ for 1 ML adsorbed on (a) and (b) Pb(111) fcc sites; (c) and (d) Pb(100) 4fh sites; (e) and (f) Pb(110) 4fh sites versus film thickness $L$ from our PBEsol calculations.

Figure 3

(a) Adsorption energies of Pb adatom at fcc and hcp sites versus Pb(111) film thickness $L$ from our PBEsol calculations. Inset illustrates fcc, hcp, and bridge sites of Pb(111) surface. (b) Adsorption energy difference $\mathrm{\Delta }{E}_{\mathrm{ad}}(\mathrm{hcp}-\mathrm{fcc})=E_{\mathrm{ad}}\left(\mathrm{fcc}\right)-E_{\mathrm{ad}}\left(\mathrm{hcp}\right)$. (c) Diffusion barriers of Pb adatom hopping from hcp site to fcc site from our cNEB calculations. For corresponding MEPs, see Fig. 4.

Figure 4

MEPs of Pb adatom hopping from hcp (left endpoint) to fcc (right endpoint) sites versus Pb(111) film thickness $L$ from our PBEsol calculations using the cNEB method. Dots represent cNEB images, and the corresponding curves are generated from a modified Bézier method [66] by fitting the energies of ten data points (dots) versus reaction coordinates. Similar statement for other MEP figures below.

Figure 5

Energetics for Pb (a) and (b) dimers; (c) and (d) trimers; (e) and (f) 1 ML adsorbed at fcc or hcp sites (as indicated in corresponding insets) on Pb(111) film surface versus thickness $L$ from our PBEsol calculations. For more details, see Tables S2–S6 [64].

Figure 6

(a) Adsorption energies of Pb adatom at 4fh and bridge sites versus Pb(100) film thickness $L$ from our PBEsol calculations. Inset illustrates 4fh and bridge sites of Pb(100) surface. (b) Hopping barriers from Eq. (9). Inset indicates the hopping process for adatom A. (c) Exchange barriers from Eq. (10). Inset indicates the two-atom exchange process: the initial adatom A kicks the surface atom S out of the surface lattice and takes the place of S; simultaneously S is pushed to an adjacent 4fh site along diagonal direction and becomes a new adatom.

Figure 7

(a) The cNEB MEP of two-atom exchange along the close-packed $\langle 110\rangle$ direction on a 5-ML Pb(100) film from our PBEsol calculations. Inset shows the geometries corresponding to two endpoints and saddle point on the MEP, as indicated by arrows. (b) Left: Initial geometry used to directly obtain the TS for two-atom in-channel exchange diffusion by the SR method: initial adatom A and surface atom S are aligned along $\langle 110\rangle$ direction at the two original 4fh sites near the vacancy, and the initial height of both is set to be $a/5$ relative to top surface atom. Right: After full relaxation of the left configuration, it is optimized to a configuration with the almost same energy and geometry as those of saddle point in (a).

Figure 8

Unit-cell size $n\times n$ dependence of diffusion barriers $E_{\mathrm{d}}$ for hopping and two-atom diagonal exchange from our PBEsol and PBE calculations for Pb adatom on a 5-ML Pb(100) film.

Figure 9

(a) Adsorption energies $E_{\mathrm{ad}}\left(4\mathrm{fh}\right), E_{\mathrm{ad}}\left(\mathrm{ibridge}\right)$, and $E_{\mathrm{ad}}\left(\mathrm{xbridge}\right)$ of Pb adatom at 4fh, in-channel bridge, and cross-channel bridge sites versus Pb(110) film thickness $L$ from our PBEsol calculations, respectively. Inset illustrates the 4fh, in-channel bridge, and cross-channel bridge sites of Pb(110) surface. (b) Adsorption energy difference $\mathrm{\Delta }{E}_{\mathrm{ad}}(4\mathrm{fh}-\mathrm{ibridge})$ between 4fh site and in-channel bridge site. (c) Adsorption energy difference $\mathrm{\Delta }{E}_{\mathrm{ad}}(4\mathrm{fh}-\mathrm{xbridge})$ between 4fh site and cross-channel bridge site.

Figure 10

The MEP of a two-atom in-channel exchange on a 5-ML Pb(110) film from our PBEsol cNEB calculation. Inset indicates the configurations of key images (00, 04, 06, 08, and 11) during the exchange of adatom A and surface atom S.

Figure 11

The MEP of a two-atom cross-channel exchange on a 5-ML Pb(110) film from our PBEsol cNEB calculation. Inset indicates the configurations of key images (00, 05, 07, 09, and 14) during the exchange of adatom A and surface atom S.

Figure 12

(a) Average interlayer spacing $\overline{d}$, and (b) beat period $\tilde{\mathrm{\Lambda }}$ versus relaxed Pb(111) film thickness $L$ from our PBEsol calculations. Horizontal green lines indicate the constant values of $d$ and $\mathrm{\Lambda }$ for the films fixed to bulk lattice.

Figure 13

Schematics for three types of exchange between an initial adatom (blue dot) and one surface atom (labeled as S) of (111) surface. A red arrow indicates that the initial adatom kicks the surface atom S out of the surface lattice and takes the place of the atom S; a green arrow indicates that the atom S is pushed to an adjacent hcp or fcc site and becomes a new adatom.

Figure 14

Schematic of initial geometry used to directly obtain the TS or saddle point for two-atom diagonal exchange diffusion on a fcc or bcc (100) surface by the SR method. After full relaxation, it is optimized to the geometry indicated in the middle image of the inset of Fig. 6.