Author: Jeremy Hu

Introduction

Silver (Ag) has been used for multiple chemical engineering applications, most notably in the field of catalysis. In fact, Ag catalysts are used in a wide field of applications including electrocatalysis and oxidation/reduction reactions due to their high activity and stability [1]. Understanding the crystal structure of Ag may aid in explaining the structure and behavior of Ag in various applications.  Experimental studies on the crystal structure of Ag suggest that Ag prefers the fcc crystal structure with a lattice constant of a = 4.079 Å [2]. To offer a comparison to the experimental data on Ag, plane-wave density functional theory (DFT) was used to characterize and optimize three crystal structures of Ag using CASTEP in Materials Studio [3]. The structure and energy of Ag in simple cubic (sc), face-centered cubic (fcc), and hexagonal close-packed (hcp) crystal structures were analyzed [4].

Methods

Electronic Methods

DFT analysis of Ag was run using the plane-wave basis set in CASTEP in Materials Studio using the following parameters:

Table 1. Parameters for DFT Calculations in CASTEP

Exchange correlation functional type	Generalized Gradient Approximation (GGA) [5]
Exchange correlation Functional	Perdew–Burke-Ernzerhof (PBE) [5]
Pseudopotentials	On-the-fly generated (OTFG) ultrasoft
Relativistic treatment	Koelling-Harmon
Core radius	1.6 a.u.
Valence electron configuration	4s2 4p6 4d10 5s1 (19 valence electrons)
SCF Tolerance	2E-06 eV/atom

First, the energy of the sc, fcc, and hcp crystal structure of Ag was iterated with respect to the energy cutoff (ENCUT) and the irreducible number of k-points. The energy cutoffs and irreducible number of k-points were considered converged when the energy did not vary significantly from the energy at the infinite energy cutoff or irreducible k-point limit (i.e., the energy at the highest ENCUT or number of k-points). For each convergence check, the relative energy was reported with respect to the energy value at the highest number of irreducible k-points or ENCUT. For the initial convergence check for the minimum number of irreducible k-points, the default ENCUT in CASTEP of 489.8 eV was used. The assumption that the default ENCUT was high enough to ensure convergence was confirmed during the ENCUT convergence check.

After a minimum ENCUT and irreducible number of k-points were identified (i.e., the values at which the energy was considered converged), lattice parameters were varied for three different crystal structures of Ag (e.g., sc, fcc, and hcp) and the relative energy was plotted as a function of the lattice parameters. The energy minimum of each plot corresponds to the preferred (i.e., energetically favorable) lattice parameter.

Model

Crystal structures of Ag were tested with the following parameters:

Table 2. Iteration ranges of parameters for three possible crystal structures of Ag

Lattice type	Lattice parameter (a) range	Irreducible k-points (Monkhorst [6])	Energy cutoff (eV)
Simple cubic (sc)	a: 2.0 - 3.2 Å	1 - 120	200 - 600
Face-centered cubic (fcc)	a: 3.5 - 4.5 Å	1 - 182	200 - 600
Hexagonal close-packed (hcp)	c/a: 1.5 - 1.7	15 - 343	200 - 600

Results and Discussion

1. K-point convergence

The energy of Ag in the sc structure was converged after 20 irreducible k-points, with the relative energy reaching within 0.01 eV/atom (Fig. 1).

Figure 1. Relative energy (eV/atom) as a function of irreducible k-points for the simple cubic (sc) structure of Ag at a = 2.7 Å and ENCUT = 489.8 eV.

Next, the energy of the fcc structure of Ag was converged after 28 irreducible k-points, where the relative energy was less than 0.01 eV/atom than the energy at the infinite k-point limit (Fig. 2).

Figure 2. Relative energy (eV/atom) as a function of irreducible k-points for the face-centered cubic (fcc) structure of Ag at a = 4.1 Å and ENCUT = 489.8 eV.

Finally, the energy of the hcp structure was converged after 216 irreducible k-points, with a relative energy of 0.007 eV/atom (Fig. 3).

Figure 3. Relative energy (eV/atom) as a function of irreducible k-points for the hexagonal close-packed (hcp) structure of Ag at a = 2.8 Å, c/a = 1.7, and ENCUT = 489.8 eV.

For the purposes of brevity, it is assumed that the irreducible k-points necessary for convergence at the chosen arbitrary lattice constants is valid for lattice constants close to the chosen values.

2. Cutoff energy convergence

The energy of the fcc structure was determined at various cutoff energies from 200 eV to 600 eV. The minimum cutoff energy was 450 eV, where the corresponding energy had less than a 0.1% difference than the energy at the highest cutoff energy (Fig. 4). Since the cutoff energy should be kept constant for all the crystal structures for optimization, it was assumed that the same minimum cutoff energy would be appropriate for the other crystal structures.

Additionally, Fig. 4 confirms the assumption that the default ENCUT value of 489.8 eV used for the previous k-points convergence was sufficient to ensure convergence with respect to the ENCUT value.

Figure 4. Relative energy (eV/atom) as a function of cutoff energy for the face-centered cubic (fcc) structure of Ag at a constant 10 x 10 x 10 Monkhorst k-point grid (i.e., 110 irreducible k-points) and a = 4.1 Å.

3. Lattice parameter optimization

In the following lattice parameter optimization calculations, the lattice parameter (a) was varied for each structure. Each energy value was taken relative to the lowest calculated energy in the most stable structure.

For the sc structure of Ag, the energy minimum was observed at around a = 2.7 Å (Fig. 5), which corresponds to a minimum relative energy of 0.325 eV/atom.

Figure 5. Relative energy (eV/atom) as a function of lattice parameter (Å) for the simple cubic (sc) structure of Ag at a constant 10 x 10 x 10 Monkhorst k-point grid and ENCUT = 489.8 eV.

Since the hexagonal close-packed structure of Ag has two lattice parameters (i.e., a and c), the ratio of c/a was kept fixed from c/a = 1.5 – 1.7 for each iteration of a. The relative energy minimum was found to be 0.237 eV/atom at a = 2.8 Å and c/a = 1.7 (Fig. 6).

Figure 6. Energy (eV/atom) as a function of lattice parameter (Å) for the hexagonal close-packed (hcp) structure of Ag at a constant 12 x 12 x 6 Monkhorst k-point grid and ENCUT = 489.8 eV.

Finally, iterations were performed for the fcc structure of Ag, where the lowest energy out of all systems (i.e., the state where the relative energy was set to zero) was observed at a = 4.1 Å (Fig. 7).

Figure 7. Energy (eV/atom) as a function of lattice parameter (Å) for the face-centered cubic (fcc) structure of Ag at a constant 10 x 10 x 10 Monkhorst k-point grid and ENCUT = 489.8 eV.

Conclusion

The convergence checks for each crystal structure with respect to k-points and cutoff energy confirm that the values of irreducible k-points and ENCUT used in the crystal structure calculations were converged with respect to energy. The final parameter values for each crystal system is shown below in Table 3.

Table 3. Final lattice parameters and parameter values for Ag in the sc, fcc, and hcp crystal structures.

Lattice structure	Cutoff energy	Monkhorst k-point grid	Irreducible k-points	Optimized lattice parameter	Minimum relative energy
sc	489.8 eV	10 x 10 x 10	35	a = 2.7 Å	0.325 eV
fcc	489.8 eV	10 x 10 x 10	110	a = 4.1 Å	0 eV
hcp	489.8 eV	12 x 12 x 6	216	a = b = 2.8 Å c/a = 1.7	0.237 eV

Thus, our DFT calculations suggest that Ag prefers the fcc crystal structure with a lattice parameter of a = 4.1 Å since it has the minimum energy (i.e., the most thermodynamically favorable crystal structure).  These results are consistent with the experimentally determined results, which show that Ag exists as the fcc crystal structure with a lattice constant of a = 4.079 Å [2]. These results on the crystal structure of Ag are promising for understanding Ag-based catalysts, such as catalysts with Ag nanoparticles.

Citations

[1] G. Liao, J. Fang, Q. Li, S. Li, Z. Xu, B. Fang, “Ag-Based nanocomposites: synthesis and applications in catalysis,” Nanoscale, 2019, 11, 7062-7096
[2] W. Davey, “Precision Measurements of the Lattice Constants of Twelve Common Metals,” Physical Review, 1925, 25 (6), 753-761.
[3] Clark Stewart J et al., “First principles methods using CASTEP ,” Zeitschrift für Kristallographie – Crystalline Materials, 2005, 220, 567
[4] D. Sholl, J. Steckel, “Density Functional Theory,” Somerset: Wiley, 2008
[5] J. P. Perdew, K. Burke, M. Enzerhof, “Generalized Gradient Approximation Made Simple,” Phys, Rev. Lett., 1996, 77, 3865.
[6] H. J. Monkhorst, J.D. Pack, “Special points for Brillouin-zone integrations,” Physical Review B, 1976, 13, 5188-5192.