Author : Shyam Deo

Introduction

Methods

Plane-wave Density Functional Theory (DFT) calculations were carried out using the Vienna Ab Initio Simulation Package (VASP), version 5.4.4. The electron-electron exchange and correlation energies were computed using the Perdew, Burke, and Ernzerhof functional (PBE) [1]. The projector augmented-wave (PAW) [2] method was used to represent the ion-core electron interactions. For Cu, the cutoff radius for the above psuedopotential is 2.3 Bohr (1.22 Å) with 11 valence electrons in the following configuration 3d10 4s1. For Ag, the cutoff radius for the above pseudopotential is 2.4 Bohr (1.27 Å) with 11 valence electrons in the following configuration 4d10 5s1. The structural convergence criteria were 0.05 eV Å^-1 for all unconstrained atoms, while the convergence criteria defining self-consistency of the electron density was 10^-5 eV.

Figure 1: (a) Cu (111) 3×3 unit cell with 5 layers and vacuum. (b) Convergence test for k-points. (c) Convergence test for ENCUT

A Cu FCC experimental bulk lattice constant of 3.615 Å was used for building Cu (111) surface [3]. A five layered 3 x 3 unit slab of Cu (111) was used for surface calculations where the bottom three layers were frozen during the optimization and the top two layers were unconstrained (Figure 1a). A plane wave energy cutoff of 450 eV and Monkhorst-Pack [4] k-point mesh of 5 x 5 x 1 was used for all surface slab calculations after proper convergence tests described in the next section. To minimize spurious interslab dipole interactions between the periodic slabs, a vacuum space of 15 Å was used and dipole corrections were added in the direction perpendicular to the surface. Transition state was located using the climbing image nudged elastic band (CI-NEB) method [5] (5 images) by relaxing the reaction tangent force below 0.05 eV Å^-1.Transition state was verified to contain a single imaginary frequency along the reaction coordinate. These vibration frequencies were calculated by the tag IBRION = 7 implemented in VASP. It determines the Hessian matrix (matrix of the second derivatives of the energy with respect to the atomic positions) using density functional perturbation theory [6-7].

Convergence Tests 

Convergence test for K-points mesh sampling the brilliouin zone was done at a plane wave energy cut off of 550 eV by varying the k-points against the single point energy of Cu (111) surface. The relative energies are plotted against the number of irreducible k-points as shown in Figure 1b. The energy varied by an amount 0.006 eV between 13 and 24 irreducible K-points. However, the energy fluctuated when the K-points were raised further. To reduce the computation cost, lower K-points were chosen due to limited availability of computational resources which should be a reasonable starting point because we are mostly concerned with a difference in energies between different systems and not the absolute numbers themselves. Then for ENCUT convergence, the cut-off energy for plane wave was varied from 350 eV to 550 eV as plotted in Figure 1c. The energy changed only by 0.017 eV between 450 eV and 550 eV. Finally, the k-points mesh of 5x5x1 with 13 irreducible K-points and a plane wave energy cut off of 450 eV was considered suitable for all subsequent surface calculations of adsorption energies.

Results and Discussions

Adsorption of Ag Atoms over Cu (111)

A Cu (111) surface has two types of three-fold binding sites available for adsorption by a silver atom – hollow-fcc and hollow-hcp sites shown in Figure 2a. The figure shows adjacent hollow-fcc and hollow-hcp sites. A hollow fcc site is a three fold site with no atom directly below  but with an atom in the layer next to it. A hollow hcp site is a three fold site with an atom directly below  and  with no atom in the layer next to it. We probed the adsorption of silver atom over 3×3 Cu (111) unit cell over these two adjacent sites.

The adsorption energy for Ag was calculated as:

E_ads,Ag = E_Ag* – E_Ag – E*

where E_ads,Ag is the adsorption energy of Ag, E_Ag* is the energy of Ag bound to the surface, E_Ag is the energy of single Ag atom and E* is the energy of the bare Cu(111) surface. The adsorption energies for the two adjacent three-fold sites are labelled in Figure 2b. Both sites show the same adsorption energy of 2.38 eV (Figure 2b and 2c). These two states obtained were then studied for diffusion between the two adjacent sites and the activation energy was probed for the same.

Figure 2: a) Cu (111) surface showing two adjacent hollow-fcc and hollow-hcp sites of adsorption, b) Ag atom adsorbed on a hollow-fcc site on Cu (111) and c) Ag atom adsorbed on a hollow-hcp site on Cu (111). Adsorption energies for the two states of adsorption are also shown for comparison.

Activation Energy for Diffusion

The two states with adsorption of Ag over hollow-fcc site and hollow-hcp site was used as two end points for NEB calculation. The former was used as the initial state while the later was taken as the final state after diffusion from the hollow-fcc site to the hollow-hcp site. The intermediate images for NEB run were obtained using linear interpolation to give five images along the straight line trajectory for diffusion between the two adjacent sites. The energy profile for images along the reaction coordinate obtained from BEB run is shown in figure 3. The transition state is also shown. The activation energy obtained is 0.0328 eV for diffusion from hollow fcc site to hollow hcp site while the activation energy for reverse diffusion obtained is 0.0305 eV. A bridge site adsorption of Ag was obtained as the transition state (one with the highest energy). This was confirmed for single imaginary frequency (with value -1.89 THz) along the reaction coordinate (from hollow-fcc to hollow-hcp and back vibration). The other two vibration frequencies obtained were 4.12 THz (vibration towards the plane of surface and away) and 1.01 THz (vibration parallel to the plane of adsorption). A single imaginary frequency along the reaction coordinate confirms the transition state for diffusion from hollow-fcc to the adjacent hollow-hcp site. All vibrational frequencies are reported in Figure 4 (a).

Figure 3: Energy profiles computed using NEB (5 images) for diffusion of Ag atom from hollow-fcc site to hollow-hcp site of adsorption. The TS state is labelled and the transition state structure is shown.

The frequencies of hopping was then calculated by the following expression as a function of temperature (Figure 4) where ΔE is the activation energy for hopping, ν1 to νN are  the vibrational frequencies associated with the minimum initial state (hollow-fcc site adsorption) and ν†1 to ν†N-1 are  the real vibrational frequencies associated with the transition state. The rate of hopping has been plotted in Figure 4b as a function of temperature.

Figure 4: (a) Vibrational frequencies in THz for Ag atom adsorption over hollow fcc site and transition state (bridge site)  on Cu (111) and (b) hopping rate for Ag atom on Cu (111) as a function of temperature (K)

Conclusion

NEB calculations for diffusion from the hollow-fcc site adsorption of Ag to the adjacent hollow-hcp site was studied and an activation barrier of 0.0328 eV was obtained. These two sites showed same adsorption energies of 2.38 eV. As a further study, diffusion coefficient can be calculated by ab initio methods from the calculated activation energies and adsorption energies as done earlier by Minkowski et al. for Cu adatoms over Cu (111) surfaces [8]. The calculated adsorption energies and activation barrier for Ag adatom matches reasonably with the reported values of 2.43 eV and 0.023 eV, respectively [9]. The coefficients for surface diffusion and rate for hopping can help understand aggregation of atoms on surfaces to form clustered particles of metals on the catalyst support.

References

1. Perdew, J.P. and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy. Physical Review B, 1992. 45(23): p. 13244-13249.
2. Blöchl, P.E., Projector augmented-wave method. Physical Review B, 1994. 50(24): p. 17953-17979.
3. Straumanis, M.E. and L.S. Yu, Lattice parameters, densities, expansion coefficients and perfection of structure of Cu and of Cu-In [alpha] phase. Acta Crystallographica Section A, 1969. 25(6): p. 676-682.
4. Monkhorst, H.J. and J.D. Pack, Special points for Brillouin-zone integrations. Physical Review B, 1976. 13(12): p. 5188-5192.
5. Nudged-elastic band used to find reaction coordinates based on the free energy. The Journal of Chemical Physics, 2014. 140(7): p. 074109.
6. S. Baroni, P. Giannozzi and A. Testa, Phys. Rev. Lett. 58, 1861 (1987).
7. X. Gonze, Phys. Rev. A 52, 1086 (1995); X. Gonze, Phys. Rev. A 52, 1096 (1995).
8. Mińkowski, M. and M.A. Załuska-Kotur, Diffusion of Cu adatoms and dimers on Cu(111) and Ag(111) surfaces. Surface Science, 2015. 642: p. 22-32.
9. Kotri, A., El koraychy, E., Mazroui, M., and Boughaleb, Y. ( 2017) Static investigation of adsorption and hetero‐diffusion of copper, silver, and gold adatoms on the (111) surface. Surf. Interface Anal., 49: 705– 711. doi: 10.1002/sia.6211.