Author: Stephen Holoviak
Introduction:
Diffusion of adatoms on a surface is an important phenomena in many processes; from thin film deposition (1) to catalysis. This study is an attempt to understand the activation energy of an Ag atom diffusing between the adjacent threefold sites on the Cu(111) surface.
Figure 1: Threefold sites on Cu(111) surface.
Method:
All calculations were carried out in the Dmol3(2) package in materials studio.
Exchange-correlation functional used was LDA-PWC (3).
Kpoint density: 1x1x1, all meshes were distributed via the Monkhorst-pack method (4). If more computational resources were available, more k-points would be added.
SCF Tolerance: 10E-4 [eV]
Geometry Optimization Tolerance: 10E-4[Ha] = 0.0272[eV]
Orbital Cutoff: 3.4[Å] with an all electron core treatment and a double numeric (DN) basis set (5).
Convergence Testing:
The k-point mesh and SCF tolerance used in these calculations lead to unstable results that do not provide very accurate energies. Unfortunately, the computational resources available for this project were extremely limited and in order for the calculations to reach any result at all these extremely coarse parameters had to be used. The instability was confirmed by comparing the calculations against partial results with a SCF tolerance of 10E-6[eV] and a 4x4x1 k-point mesh. The system energies between the coarse calculations and the finer calculations varied by nearly 0.19[eV].
Calculations:
The reactant and product states of the reaction were defined as the hcp and fcc sites respectively.
A 3-layered 1×1 slab of Cu(111) with a vacuum spacing of 10[Å] was used for the calculations. An Ag atom was placed at the hcp site and the geometry of the Ag and all Cu layers were optimized in order to define the reactant state. An Ag atom was placed at the fcc site and the geometry of the system was optimized in order to define the product state. With more computational resources, this cell should probably be increased in size, with more unit cells and more layers, fixing the geometry of the bottom layers to represent the bulk material.
Figure 2: Reactant and product vacuum slabs used for transition state searching.
The transition state of the reaction was found using the complete LST/QST method as implemented in Dmol3 (6).
The energy of each image of the reaction was defined as follows:
Figure 3: Complete LST/QST transition state search.
The transition state was found to be the bridge site between the two threefold sites. While the exact numbers of the frequencies and energy at the transition state will not be very accurate, there was one imaginary frequency along the reaction coordinate that supports its identity as a transition state.
Figure 4: Reactant state, transition state, and product state structures.
The energy barrier found for the hcp to fcc transition was 0.051[eV], but the level of convergence for these calculations is on the same order of magnitude as this barrier, so the exact barrier cannot be determined from this study. The height of this barrier reported in literature is 0.023[eV] (7), giving the same order of magnitude as this study.
The energy of the two threefold sites was found to be slightly different, with the fcc site being 0.003[eV] more stable, however the calculations from this study cannot tell which one is actually more stable.
Conclusion:
Due to computational resource limitations, an exact value for the activation energy of Ag diffusing between adjacent threefold sites on Cu(111) could not be found, however the order of magnitude was determined to be 10E-3[eV]. The location of the transition state was found to be the bridge site in between the threefold sites, this location is supported by the presence of an imaginary frequency along the reaction coordinate. It was also found that the energy of the threefold sites is different, due to the presence(hcp) or absence(fcc) of a Cu atom directly below the site in the next layer, but it could not be determined which of these sites is more energetically favorable.
References:
(1) M. Ohring (2001) “Materials Science of thin Films”, Section 7.4: “Kinetic Processes in Nucleation and Growth”, 386-400
(2) B. Delley (2000) “From molecules to solids with the DMol3 approach” J. Chem. Phys. 113
(3) J. Perdew, Y. Wang (1992). “Accurate and simple analytic representation of the electron-gas correlation energy”. Phys. Rev. B. 45 (23): 13244–13249.
(4) H. Monkhorst, J. Pack (1976) “Special points for Brillouin-zone integrations.” Physical Review B, 13(12): 5188-5192
(5) B. Delley (1990). “An All-Electron Numerical Method for Solving the Local Density Functional for Polyatomic Molecules”. J. Chem. Phys. 92: 508–517.
(6) T. Halgren, W. Lipscomb (1977) “The synchronous-transit method for determining reaction pathways and locating molecular transition states.” Chem. Phys. Letters 49: 225-232
(7) A. Kotri, E. El Koraychy, M. Mazroui, Y. Boughaleb (2017) “Static investigation of adsorption and hetero‐diffusion of copper, silver, and gold adatoms on the (111) surface” Surface and Interface Analysis, 49 (8)
