By Nathan McKee

Introduction

This post examines the diffusion of a platinum adatom on the (100) surface of platinum. Platinum has an fcc structure, so the (100) surface has a fourfold hollow site to which a platinum adatom can bind. An adatom in this position has multiple ways to diffuse to an adjacent fourfold hollow site, two of which are considered in this post. One is the simple hopping of the adatom to an adjacent site. The other is the atomic exchange of the adatom with an atom on the top layer of the surface, such that the surface atom travels to the adjacent site while it is replaced by the adatom. These processes are illustrated in Figure 1. DFT calculations were performed to find the transition states for both of these processes, and to estimate the frequency of each. The DFT calculations were carried out with the plane-wave based code CASTEP. The GGA PBE functional was used¹, as well as on-the-fly generated (OTFG) ultrasoft pseudopotentials². These pseudopotentials include 32 valence electrons per platinum atom, in the 4f14 5s2 5p6 5d9 6s1 configuration, with a cutoff radius of 1.27 Å. The convergence tolerance was set at 2.0*10^-4 eV per atom.

Figure 1: Illustrations of surface diffusion in the cases of a) hopping, and b) exchange. All atoms shown are Pt, with some being highlighted to indicate that they play the biggest part in the transition process.

Cell Construction

To represent the (100) surface of platinum, a vacuum slab was used, with a vacuum distance of 12 Å. The thickness of the slab was chosen to be 2 layers, so as to represent the surface sufficiently without surpassing the computational limits of the system. In choosing the x and y dimensions of the cell, the interaction between adatoms must be considered. The use of plane-wave DFT necessitates the construction of a periodic crystal, but the goal of this post is to examine the diffusion of only a single adatom. This requires the cell to be large enough that the adatoms in adjacent cells do not affect each other significantly. A 3 x 3 cell was chosen to this end. The resulting cell is shown in Figure 1.

Cutoff Energy and k Points

To ensure that the calculations converge properly, an analysis of the selection of k points and the cutoff energy was performed on the slab, including the adatom. The k points were chosen to be an NxNx1 Monkhorst-Pack grid³ of evenly spaced points in reciprocal space, as is conventional for slab models in which the “a” and “b” lattice constants are equal. This analysis revealed that using a 6x6x1 k point grid in conjunction with a cutoff energy of 350 eV yields the energy within 0.1 eV. While this uncertainty is substantial, it is less than the difference between the transition state energies of the diffusion processes being considered, as will be shown later.

Transition State Search

Before considering transition states, the product and the reactant had to be identified through geometry optimization. The bottom layer of the slab was held in place to represent the bulk platinum crystal, while the top layer and the adatom were allowed to relax and find the optimal configuration. The force tolerance was set to 0.05 eV/Å, and the energy tolerance was set to 2.0 * 10^-4 eV per atom. In general, one would expect to have to perform this geometry optimization for both the reactant and the product, but in the case the reactant and product are identical except for a constant translation. Therefore, only the configuration of the reactant was optimized, and the product was created by manually shifting the reactant atoms along the x axis.

With the reactant and product complete, the search for transition states begins by matching corresponding atoms in the reactant and product configurations. For the hopping transition, the adatoms in the reactant and product were matched with each other, while the surface atoms were matched according to position so that they barely moved. For the atomic exchange transition, the adatom in the reactant was matched with one of the surface atoms nearest to the hopping path in the product. Likewise, that surface atom in the reactant was matched with the adatom in the product.

The algorithm used in the transition state searches was the complete LST/QST approach. This algorithm begins with a linear synchronous transit (LST)⁴, followed by a conjugate gradient mnimization. The state found in this way is then used as the intermediate state in a quadratic synchronous transit (QST)⁴. If necessary, the last two steps are repeated until a transition state is found.

Results

These calculations yield a transition state for each diffusion process. When the reactant energies are subtracted from the transition state energies, the result is called the activation energy, ΔE. The transition rate can then be calculated using the Arrhenius equation⁵:

where k is the transition rate, and the prefactor nu is the atomic vibration frequency.

The activation energies obtained from the calculations are 1.04 ± 0.10 eV and 0.82 ± 0.10 eV for the hopping transition and the exchange transition, respectively. Estimating the vibrational frequency to be 5E-12 Hz, predictions can be made for the transition rates at any temperature. At 300 K, for instance, the rates are calculated to be:

k_hop = 1.7E-29 Hz

k_exch = 9.3E-26 Hz

From these results, it is apparent that a small difference in activation energy can have a large impact on transition rates. To predict transition rates more precisely, calculations with smaller uncertainties would need to be performed. Nonetheless, these calculations indicate with some confidence that the atomic exchange transition is more frequent than the hopping transition. This preference is supported experimentally⁶, albeit for a slightly different exchange process in which both atoms move along the same diagonal direction.

References

1. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
2. Gonze, X. & Finocchi, F. Pseudopotentials Plane Waves–Projector Augmented Waves: A Primer. Phys. Scr. 2004, 40 (2004).
3. Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 13, 5188–5192 (1976).
4. Halgren, T. A.; Lipscomb, W. N. Chem. Phys. Lett., 49, 225 (1977)
5. Arrhenius, S. A. (1889). “Über die Dissociationswärme und den Einfluß der Temperatur auf den Dissociationsgrad der Elektrolyte”. Z. Phys. Chem. 4: 96–116.
6. G. L. Kellogg and P. J. Feibelman, Phys. Rev. Lett. 64 (1990) 3147.