By Nate Klassen
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
