by Vishal Jindal
1. Introduction
Platinum (symbol Pt and atomic number 78) is a noble metal. It is used in catalytic converters, electrodes, and many other laboratory equipments, dentistry and jewelry [6]. Experimentally, platinum (Pt) is found to exist in cubic close-packed (ccp) structure with a lattice constant of a = 3.9242 Å [1]. In this post, we are using Density Functional Theory [DFT] [2] calculations to determine the preferred crystal structure for platinum metal amongst simple cubic (sc), cubic close-packed (ccp), also known as face-centered cubic (fcc) and hexagonal close packing (hcp) crystal structures.
2. Methodology
We used a plane-wave basis set with ultrasoft pseudopotentials as implemented in the CASTEP code [3] to perform DFT calculations to get total energy for different crystal structures and lattice parameters. Materials Studio was used as a builder, visualizer, and user interface for the CASTEP calculations. The exchange and correlation energies were calculated using the Perdew, Burke, and Ernzerhof (PBE) [4] functional described within the generalized gradient approximation (GGA) [5]. SCF tolerance is taken to be 2.0e-6 eV/atom for all the calculations. The “on the fly” generated ultrasoft pseudopotential for Pt has a core radius of 2.4 Bohr (1.27 Angstroms) and was generated with 32 electrons in the valence panel with (4f14 5s2 5p6 5d9 6s1) as the electronic configuration.
3. K-Points and Cut-off Energy Optimization
Before varying the lattice parameter to find the most stable crystal structure, we need to make sure that the plane-wave basis sets give convergent results for energies with respect to mesh size/ k-points and cut-off energy (ENCUT). To perform k-points and ENCUT optimization we used the experimental value of lattice parameter, i.e. a = 3.9242 Å.
3.1 K – Point Optimization
3.1.1 Simple Cubic (sc)
To optimize the k-points, we calculated total energy of simple cubic (sc) lattice for different mesh sizes (keeping ENCUT fixed at material studio default i.e. 272.1 eV). Fig. 1 shows the variation of total energy when we change the mesh size. We choose 8 x 8 x 8 lattice having 20 k-points as total energy changed less than 0.001 eV as we further increased the mesh size.
Figure 1. k-point optimization for simple cubic crystal lattice
3.1.2 Face-centered cubic (fcc)
Figure 2. k-point optimization for face-centered cubic crystal lattice
3.1.3 Hexagonal Close Packed (hcp)
Figure 3. k-point optimization for hexagonal close-packed crystal lattice
3.2 Energy cut-off
Now, using the optimized number of k-points from simple cubic, we varied ENCUT from 250-450 eV. Again, using the same convergence criterion of |ΔE| less than 0.001 eV on increasing the cut-off energy. In Fig. 4, we found that the energy cut-off of 425 eV sufficiently converged the total energy of the system. Similarly for FCC and HCP, 425 eV was taken to be the cut-off energy after checking the convergence of total energy at 400 eV, 425 eV, and 450 eV.
Figure 4. Total energy vs energy cut-off [ENCUT]. Total energy tends to converge at 425 eV.
4. Results
To optimize the lattice parameters, we minimize the total energy per atom (eV/atom) with respect to lattice constant (a) using respective k-points and energy cut-off for SC and FCC crystal lattice.
4.1 Simple Cubic (SC)
Figure 5. Total energy vs lattice parameter plot for SC
4.2 Face Centered Cubic (FCC)
Figure 6. Total energy vs lattice parameter plot for FCC
4.3 Hexagonal Closed Pack (HCP)
Figure 7. Total energy vs lattice parameter (a) plot for different values of c/a ratio along with trendlines connecting the same c/a ratio (c/a = 1.5, 1.67, 1.85)
5. Conclusion
The table below summarizes the optimized lattice parameter for all three types of crystal lattices studied in our report.
| Lattice System | Lattice Parameter/s (Angstroms) | Total Energy per atom (eV/atom) | k-points (M x M x N) | Energy Cut-off (eV) |
|---|---|---|---|---|
| Simple Cubic (SC) | a = 2.66 | -13049.89 | 20 (8 x 8 x 8) | 425 |
| Face Centered Cubic (FCC) | a = 3.97 | -13050.96 | 110 (10 x 10 x 10) | 425 |
| Hexagonal Close Packed (HCP) | a = 2.8 c = 4.2 | -13050.90 | 76 (12 x 12 x 8) | 425 |
We can observe that among the three types of lattice structure listed above, the minimum total energy per atom is found to be for the Face Centered Cubic (FCC) crystal structure. Therefore, our DFT calculations show that the Platinum(Pt) crystal is most stable in the FCC lattice structure with a lattice parameter (a) of 3.97 Å. This is in line with the experimentally observed Face Centered Cubic (FCC) crystal structure of Platinum with a = 3.9272 Å.
6. References
[1] https://www.webelements.com/platinum/crystal_structure.html
[2] D. Sholl, J.A. Steckel, Density Functional Theory: A Practical Introduction, Wiley 2009
[3] J. Clark Stewart, D. Segall Matthew, J. Pickard Chris, J. Hasnip Phil, I.J. Probert Matt, K. Refson, C. Payne Mike, First principles methods using CASTEP, Zeitschrift für Kristallographie – Crystalline Materials, 220(5-6) pp. 567-570 (2005)
[4] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett., 77 (1996) 3865-3868.
[5] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Atoms, Molecules, Solids, And Surfaces – Applications of the Generalized Gradient Approximation for Exchange and Correlation, Phys. Rev. B, 46 (1992) 6671-6687.
[6] https://en.wikipedia.org/wiki/Platinum
