Abstract
DFT was used to study the surface diffusion of a silver atom between two hollow sites on the Ag 100 surface. Diffusion by hopping was inspected using the GGA PBE functional and VASP to perform a transition state search. The diffusion barrier was found to be 0.576 eV which agrees well with similar studies.
Introduction
Materials not at absolute zero will have some diffusion of surface atoms. The energy barrier to diffusion is an important property that predicts rates of diffusion. In particular, these rates are useful for kinetic Monte Carlo (KMC) calculations. We will inspect a method for finding the minimum energy path of a silver atom diffusing by hopping on a silver (100) surface. In particular, we will look at the results using the GGA (generalized gradient approximations) PBE (Perdew–Burke-Ernzerhof) functional. These calculations are made using VASP (Vienna Ab initio Simulation Package).
Method
A 3×3 supercell of 100 is chosen to ensure minimal influence from periodicity. A slab thickness of 3 atomic layers is chosen with the bottom 2 layers fixed. These are chosen for computational efficiency. A vacuum slab of 12 Angstroms is chosen.
We use PAW (Projector augmented-wave) potentials [1]. Ag has the electron configuration of 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s1, and the pseudopotential treats 4d10 5s1 as the valence electrons. The convergence tolerances were chosen to be: energy at 1.0e-5 eV/atom, force at 0.05 eV/Å, stress at 0.1 GPa, and displacement at 0.002 Å. A basis cutoff energy of 900 eV is chosen.
Initial configuration of surface atom
Initial configuration of surface atom – top view
Side view of the transition state
Topside view of the transition state
Final configuration of surface atom
Final configuration of surface atom – top view
The initial and final states are geometry optimized first, then a 5 frame trajectory is made. An odd number of frames is chosen to avoid missing the peak of the energy barrier by symmetry.
Calculations were run for various k-point values against the barrier height to test for energy convergence.
Energy barrier peak as a function of k-points
Absolute value of the change in energy from the previous k-point calculation
We see here that by 242 k-points we have reached k-point convergence within 1.0e-2 eV Which is a reasonable level of accuracy for our purposes.
The calculation results at 242 k-points are shown below.
The energy of the particle at each point along the equispaced points. The x-axis spacing is to be read as the number of steps one sixth of the distance from the starting state to the final state.
We find the energy peak to be 0.576 eV, which is in reasonably good agreement with similar studies [4] which found a value of 0.53 eV: a difference of about 9%.
Conclusion
While the results using the parameters chosen gave good results, for additional accuracy, a thicker slab should be used. The slab thickness used was chosen to cut down on computation time. Additionally, convergence with respect to the cutoff energy should be performed beyond 900 eV to confirm energy convergence.
References
- P.E. Blöchl, “Projector augmented-wave method”, Phys. Rev. B 50, 17953 (1994).
G. Kresse, and J. Joubert, “From ultrasoft pseudopotentials to the projector augmented wave method”,
Phys. Rev. B 59, 1758 (1999). - Density Functional Theory: A Practical Introduction. (2009) David S. Sholl, Janice A. Steckel
- https://periodictable.com/Elements/047/data.pr.html
- “Anisotropy of Growth of the Close-Packed Surfaces of Silver” Yu, Byung Deok, Scheffler, Matthias, PhysRevLett.77.1095, p 1095-1098
