Author Archives: Nathan Jay Mckee

Comparing Two Transition Processes for Platinum Adatom Diffusion on Pt (100)

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

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

Predicting the Preferred Binding Site of Atomic O on Pt (111)

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By Nathan McKee

Introduction

This post examines the adsorption of atomic oxygen on the platinum (111) surface. Platinum has an fcc structure, so there are four high-symmetry adsorption sites on the (111) surface. These are the top, bridge, hcp hollow, and fcc hollow sites. DFT calculations were performed for atomic oxygen placed on a slab of platinum (111) in each of the high-symmetry sites, and their energies were compared to determine the preferred binding site. 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 6 valence electrons for oxygen in the 2s2 2p4 configuration with a cutoff radius of 0.58 Å, and for platinum they include 32 valence electrons 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^-5 eV per atom.

Cell Construction

The unit cell used for the calculations was constructed to form a p(2 x 2) 0.25 monolayer (ML) surface coverage, as shown in figure 1. This means that the unit cell contains a 2 x 2 arrangement of Pt (111) unit cells with one oxygen atom on top. This results in having one oxygen atom adsorbed to the surface for every four platinum atoms on the surface of the slab. In other words, a quarter of a monolayer of oxygen covers the surface in an ordered pattern.

Figure 1: The p(2 x 2) 0.25 ML surface coverage is shown, with the supercell represented by the solid black line. Adapted from Sholl & Steckel³.

The thickness of the platinum slab was chosen to be 3 layers. While more layers would result in a more accurate calculation, three layers was estimated to be sufficient for identifying the preferred binding site of oxygen. In addition, including more than 3 layers became prohibitively expensive for the calculations.

The length of the vacuum in between slabs was chosen to be 10 Å. This distance was chosen to be sufficiently large to determine energies accurately enough to identify the preferred binding site. Larger vacuum gaps may provide more accurate calculations, but would require a larger cutoff energy and more computational time. It should also be noted that a self-consistent dipole correction in the z direction (normal to the slab) was implemented in the calculation to prevent different slabs from interacting with each other and altering the calculated energy.

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. The k points were chosen to be an NxNx1 Monkhorst-Pack grid⁴ of evenly spaced points in reciprocal space, as is conventional for slab models in which the “a” and “b” lattice constants are equal.

Figures 2 and 3 show how an energy calculation converges as the number of irreducible k points is increased and as the cutoff energy is raised. In both cases, an arbitrary constant (157046 eV) was added to the calculated energies so that the values would be close to zero and the convergence could be seen more easily. The calculations for k point convergence were performed with the oxygen on the top site, 2.01 Å away from the surface plane, with a cutoff energy of 650 eV. The calculations for the cutoff energy convergence were performed for the same site with a 6x6x1 k-point grid.

Figure 2: The energy calculation converges as the number of irreducible k points increases. The y axis has been shifted by +157046 eV to better show the differences. A 6x6x1 k point grid, with 18 irreducible k points, reproduces the results of a calculation with 41 irreducible k points within 0.01 eV.

Figure 3: The energy calculation converges as the cutoff energy increases. The y axis has been shifted by +157046 eV to better show the differences. A cutoff energy of 650 eV produces the same result as a cutoff energy of 750 eV, within 0.003 eV.

Using these graphs as a guide, a 6x6x1 k point grid was used in conjunction with a cutoff energy of 650 eV for further calculations. A variation of ~0.01 eV is sufficient for measuring energy differences between binding sites, which were later calculated to be on the order of ~0.5 eV.

Results

At first, calculations were made with a static slab, placing the oxygen atom on a particular site and manually setting the adsorbate’s distance from the surface. Energy calculations were made in this way, varying the vertical position of the adsorbate in order to minimize the energy. Thus an approximation of the minimum energy was made for each of the four high-symmetry binding sites. The results of these calculations are shown in figures 4-7. Note that all the y axes have been shifted by the same amount, allowing for easy comparisons of the energy. The results suggest that the fcc hollow site is the most preferred binding site, followed in order by the hcp hollow site, the bridge site, then the top site.

Figure 4: The equilibrium z-position of the oxygen atom on the top site is found through energy minimization. The y axis has been shifted by +157046 eV. A quadratic fit indicates a minimum energy of 2.21 eV.

Figure 5: The equilibrium z-position of the oxygen atom on the bridge site is found through energy minimization. The y axis has been shifted by +157046 eV. A quadratic fit indicates a minimum energy of 1.50 eV.

Figure 6: The equilibrium z-position of the oxygen atom on the hcp hollow site is found through energy minimization. The y axis has been shifted by +157046 eV. A quadratic fit indicates a minimum energy of 1.40 eV.

Figure 7: The equilibrium z-position of the oxygen atom on the fcc hollow site is found through energy minimization. The y axis has been shifted by +157046 eV. A quadratic fit indicates a minimum energy of 0.93 eV.

To follow up on these estimations, two full geometry optimizations were performed. These calculations allow the oxygen atom to move around, and allow the top layer of the platinum slab to deform. The previous calculations were used to place the oxygen atom at a z-position close to the energy minima to ensure that the geometry optimizations converged correctly. The first optimization started with the adsorbate close to the hcp hollow site, and the second optimization began with the adsorbate close to the fcc hollow site. These sites were chosen because they had the two lowest energies from the first round of estimates. In each case, the oxygen atom was placed about 0.1 Å away (horizontally) from the site. This practice breaks the symmetry in the ab plane to test whether the site is a local minimum. In both cases tested, the adsorbate returned to the high-symmetry site being tested, indicating that the hcp hollow and fcc hollow sites are both local minima.

For the geometry optimization of the hcp hollow site, the calculated energy minimum was -157044.86 eV. For the fcc hollow site, it was -157045.47 eV. The fcc hollow site energy is lower by a margin of ~0.61 eV, indicating that atomic O prefers to bind to the fcc hollow site instead of the hcp hollow site. This is in agreement with the estimates obtained manually, but this result is more definitive because the calculations include deformations of the top layer of the platinum slab.

Note that both energies obtained through geometry optimization are lower than the energies obtained by manually adjusting the adsorbate position. This is expected, as the geometry optimization allows for surface relaxation on the top layer of the platinum slab.

Conclusion

These results indicate that atomic oxygen prefers to bind on the fcc hollow site of the Pt (111) surface. This is in agreement with previous results⁵, which also use DFT to identify the fcc hollow site as the preferred binding site.

References

Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Gonze, X. & Finocchi, F. Pseudopotentials Plane Waves–Projector Augmented Waves: A Primer. Phys. Scr. 2004, 40 (2004).
Sholl, David S. & Steckel, Janice A. Density Functional Theory: A Practical Introduction. John Wiley & Sons, Inc. (2009).
Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 13, 5188–5192 (1976).
Gu, Z. and P.B. Balbuena, Absorption of Atomic Oxygen into Subsurfaces of Pt(100) and Pt(111): Density Functional Theory Study. The Journal of Physical Chemistry C, 2007. 111(27): p. 9877-9883.

Lattice constant predictions for ScAl assuming both the CsCl and NaCl structures

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By Nathan McKee

Introduction

This post examines the structure of ScAl, whether the material prefers the structure of CsCl or that of NaCl (shown in figures 1 and 2, respectively), and predicts the lattice constant in each case. For each structure, the lattice constant was varied and the ground state energy of the structure was calculated, repeating to find the lattice constant that minimizes the energy for each structure. DFT calculations were carried out with the plane-wave based code CASTEP. The GGA PBE functional was used¹, as well as OTFG ultrasoft pseudopotentials². These pseudopotentials include the 3s2 3p1 valence electrons and a cutoff radius of 1.5 Å for Al, and the 3s2 3p6 3d1 4s2 valence electrons and a cutoff radius of 1.6 Å for Sc. The convergence tolerance was set at 2.0*10^-6 eV per atom.

Figure 1: The CsCl structure, which is a simple cubic lattice with two atoms per unit cell.

Figure 2: The NaCl sructure, which is an FCC lattice with two atoms per primitive unit cell.

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. The k points were chosen to be an NxNxN grid of evenly spaced points in reciprocal space, with the same number of points in each direction being appropriate for a cubic cell. Figures 3 and 4 show how an energy calculation converges as N increases and as the cutoff energy is raised. In both cases, an arbitrary constant was added to the calculated energies so that the values would be close to zero and the convergence could be seen more easily. The calculations for k point convergence were performed for the NaCl structure with a cutoff energy of 410.9 eV. The calculations for the cutoff energy convergence were performed for the same structure with a 6x6x6 k-point grid.

Figure 3: The calculated energy converges as the size of the NxNxN grid of k points increases. The energies have been shifted so that the differences are easy to see.

Figure 4: The calculated energy of the structure converges as the cutoff energy increases. The energies have been shifted so that the differences are easy to see.

Using these graphs as a guide, a 6x6x6 k point grid was used in conjunction with a cutoff energy of 410.9 eV for further calculations. It should also be noted that an origin shift was implemented in the k point grid in order to increase the total number of k points in the Brillouin Zone (BZ) without changing the number of k points in the Irreducible Brillouin Zone (IBZ). The shift used was 0.01 Å^-1in the x direction, 0.005 Å^-1in the y direction, and 0.003 Å^-1in the z direction. These values were chosen so that the k points would not lie on any symmetry axes, while maintaining a small shift compared to the spacing between k points.

Results

Figures 5 and 6 show the energy minimization with respect to the lattice constant for each of the structures being considered. The vertical axes have been shifted so that the minimum energy value obtained lies on the horizontal axis. For the CsCl structure, a minimum energy of -1385.213651797 eV was obtained, with a lattice constant of 3.380 Å. For the NaCl structure, a minimum energy of -1384.047417 eV was obtained, with a lattice constant of 5.649 Å.

Figure 5: Energy vs lattice constant for ScAl in the structure of CsCl. The value of Emin is -1385.213651797 eV.

Figure 6: Energy vs lattice constant for ScAl in the structure of NaCl. The value of Emin is -1384.047417 eV.