Author Archives: njk22

Hopping Diffusion Barrier for Silver on the 100 Facet

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

Surface Energies for the 100 and 111 Surfaces of Silver

Introduction

We will investigate some properties of crystalline silver: the relative surface energies for the 100 and 111 cleaved faces. In this post, we will use DFT calculations using CASTEP [1] software to calculate the energy of bulk silver as well as relaxing a cleaved surface. These properties can reveal the preferred behavior in surface Ag formation. Note that silver is an FCC crystal with a lattice constant of 4.09 Å [2] and that this experimental result will be used in the calculations.

Surface Energy Calculation

First, in order to calculate the surface energy for a particular surface, we need to refer to the following equation found in “Density Functional Theory : A Practical Introduction”:

$\sigma_{surface} = \frac{1}{2A}(E_{slab}-nE_{bulk})$

So for both 100 and 111 surfaces, we need to calculate the energy of bulk Ag.

Using CASTEP with the GGA PBE functional and OTFG ultrasoft pseudopotentials, we run through different energy cutoffs to find an appropriate energy cutoff for energy convergence. Ag has the electron configuration of 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s1, and the pseudopotential treats 4s2 4p6 4d10 5s1 as the valence electrons. The convergence tolerances were chosen somewhat arbitrarily to be small: energy at 2.0e-5 eV/atom, force at 0.05 eV/Å, stress at 0.1 GPa, and displacement at 0.002 Å. For all the rest of the options, the defaults were used.

Energy convergence with respect to ENCUT

Change in energy from the previous ENCUT energy calculation

Seeing this, it is sufficient for our purposes to use a cutoff of 900 eV.

Next, we check for k-point energy convergence at our cutoff energy of 900 eV.

k-point convergence with energy as a function of k-points

Here we see that 88 k-points is sufficient.

From this we find that $E_{bulk} =$ -4006.239 eV/atom.

Next, we need to relax the surface of 111 Ag. We choose a 7 atom thick slab, and a 10 Å thick vacuum. The ENCUT was set to be 900 eV once again and an nxnX1 grid of k-points was selected to keep the k-point density equivalent in the plane of the surface, where n is varied as energy is converged with respect to k-points. It should be noted that fewer k-points are needed in the direction of the vacuum which is why only 1 k-point is needed in the direction of the vacuum. Both sides were relaxed while the central layers were fixed to simulate bulk. 3 atomic layers were fixed and the “top” 2 on each side were allowed to relax.

The exact same procedure is repeated for the 100 surface with a 900 eV basis cutoff energy, and an nxnx1 grid of k-points.

The (111) unit cell. Here the yellow highlighted atoms are held fixed while the outer layers are allowed to relax.

Energy convergence on the (111) surface with respect to k-points.

The change in energy from the previous k-point Energy calculation.

The (100) unit cell. Here the yellow highlighted atoms are held fixed while the outer layers are allowed to relax.

Energy convergence on the (100) surface with respect to k-points.

The change in energy from the previous k-point energy calculation.

The following Energies were found: $E_{111} =$ -28043.265 and $E_{100} =$ -28043.196. This gave the following results for the surface energy densities:

$\sigma (111) =$ 0.0244 eV/Å^2

$\sigma (100) =$ 0.0286 eV/Å^2

Conclusion

The results are in agreement with the known result [4] that says that the (111) surface is energetically favorable to the (100) surface of Ag. The energy of the (111) surface is found to be 15% lower than the (100) surface. In previous comparisons against multiple functionals [4], the percentage difference was lower. This result shows that the (111) surface is the preferred face for crystal growth.

With better computational ability, these results should be checked against varying layers for the surface calculations as well as a higher ENCUT for both the bulk calculation and the surface calculations.

References

“First principles methods using CASTEP” Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005) S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne
https://periodictable.com/Elements/047/data.pr.html
Density Functional Theory: A Practical Introduction. (2009) David S. Sholl, Janice A. Steckel
Patra, A., Bates, J. E., Sun, J., & Perdew, J. P. (2017). Properties of real metallic surfaces: Effects of density functional semilocality and van der Waals nonlocality. Proceedings of the National Academy of Sciences of the United States of America, 114(44), E9188–E9196. https://doi.org/10.1073/pnas.1713320114

Geometry Optimization of an Ag Crystal using CASTEP

Introduction

This project applies a method for confirming the lattice constant for crystalline silver (Ag). The starting assumption is that the crystal has an FCC structure. This is to reduce the scope of this investigation purely to the value of the lattice constant. The structure is allowed to vary its volume, but the shape (FCC structure/ratio) is held constant. The DFT calculations were run using CASTEP[1], OTFG ultrasoft pseudopotentials, the default GGA PBE Functional, and Koelling-Harmon “relativistic treatment.” Ag has the electron configuration of 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s1, and the pseudopotential treats 4s2 4p6 4d10 5s1 as the valence electrons. The Convergence tolerances were chosen somewhat arbitrarily to be small: Energy at 1.0e-5 eV/atom, Force at 0.01 eV/Å, stress at 0.05 GPa, and displacement at 0.001 Å. For all the rest of the options, the defaults were used.

Figure 1: The Ag structure, FCC single atom basis.

Results

1. Initial Guess

An initial automatic optimization using geometry optimization was run to get an initial value to use in further testing. The experimentally known value for the lattice constant of Ag is 4.09 Å.[2] Knowing this, the first chosen starting guess was 4 Å. This first calculation was run with the default energy cutoff of 517 eV and a 7x7x7 k-point grid (24 k-points). It achieved the convergence criteria after 4 steps and the lattice constant found was 4.201 Å. This is approximately a 2% deviation from the true value.

2. Convergence Steps in Initial attempt

2. k-points

Using a fixed lattice parameter 4.20 Å, several different values for number of k points were tested against the energy to see which is the optimal choice. These calculations were run with the default energy cutoff of 517 eV. The following results were found.

3. System energy as a function of k-points

4. Sampling size vs. number of k-points

From this data, it is clear that a grid size of 9x9x9 should be chosen.

3. Energy Cutoff

Next, calculations were run keeping the geometry fixed and k-points in a 9x9x9 grid with the energy cutoff value varied between 400 and 750 eV. It is clear that at 600 eV, there is fairly good convergence. Any cutoff greater than 600 eV is sufficient. The convergence criteria for each of these values was 1.0e-5 eV/atom.

5. Energy Cutoff Analysis

4. Final Calculations

Lastly, calculations were run to optimize the lattice parameter. The lattice parameter was fixed at various values and compared against the energy. The lattice parameter that minimizes the energy is most likely to be correct. An energy cutoff of 610 eV was used and a k-point grid of 9x9x9.

6. Energy vs. Lattice parameter

The energy reaches a minimum at 4.1 Å, which is an improved estimate in terms of accuracy to the true experimental result of 4.09 Å.

References

“First principles methods using CASTEP” Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005) S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne
https://periodictable.com/Elements/047/data.pr.html
Density Functional Theory: A Practical Introduction. (2009) David S. Sholl, Janice A. Steckel

Density Functional Theory and Practice Course

Blog where students post their results

Author Archives: njk22

Hopping Diffusion Barrier for Silver on the 100 Facet

Abstract

Introduction

Method

Conclusion

References

Surface Energies for the 100 and 111 Surfaces of Silver

Introduction

Surface Energy Calculation

Conclusion

References

Geometry Optimization of an Ag Crystal using CASTEP

Introduction

Results

1. Initial Guess

2. k-points

3. Energy Cutoff

4. Final Calculations

References