by Angela Nguyen
1. Introduction
The purpose of this project is to determine the crystal structure and lattice constant of platinum (Pt) through the usage of Density Functional Theory (DFT) with a plane wave basis set as implemented by the CAmbridge Serial Total Energy Package (CASTEP) module [1] in Materials Studio. Three different crystal structures for Pt will be investigated (simple cubic (sc), face centered cubic (fcc), and hexagonal close-packed (hcp)) and their optimal lattice constants will be calculated. In order to determine which crystal is preferred, the total energy for each optimized crystal structure will be compared, where the crystal structure with the lowest total energy will be the preferred crystal structure. From the DFT calculations, it was determined the optimal lattice constants for the sc and fcc conventional crystal structure were 2.64 Å and 3.97 Å respectively. In terms of the fcc primitive cell, the lattice constant was determined to be 2.80 Å. For the hcp crystal structure, the optimal \(\frac{c}{a}\) was determined to be 1.8 in which the optimal lattice constants were a = 2.72 Å and c = 4.90 Å. The total energy calculated for the sc, fcc, and hcp crystal structure were -8.51 eV/atom, -8.98 eV/atom and -8.91 eV/atom. From this, it was determined that the optimal crystal structure for Pt is the fcc with a lattice parameter of 3.97 Å.
2. DFT Parameters
Listed below are the parameters used for the optimization of each crystal structure.
| Exchange-Correlation Functional Type | Generalized Gradient Approximations [2] |
| Exchange-Correlation Functional | Perdew Burke Ernzerhof [3] |
| Relativistic Treatment | Koelling-Harmon [4] |
| Psuedopotential | "On the fly" generated (OTFG) ultrasoft [5] |
| K point Grid | Monkhorst-Pack [6] |
| Spin | Unrestricted |
The cutoff radius for the stated pseudopotential is 2.40 Bohr (1.27 Å) with 32 valence electrons in the following configuration 4f14 5s2 5p6 5d9 6s1.
3. Methodology
Using Materials Studio, three different crystal structures for Pt were built and optimized. For the sc and fcc crystal structure, the same optimization method was used as only one lattice parameter is needed to define the structure. The hcp crystal structure required a different optimization method as two lattice parameters are needed to define the crystal structure. In order to systematically vary the two lattice parameters, a ratio \(\frac{c}{a}\) was defined to vary the parameters accordingly.
3.1 sc and fcc Crystal Structure Optimization
3.1.1 Lattice Constant
First, the lattice constant to define each crystal structure was optimized. The lattice constant (\(a\)) was varied between 2.0 – 3.0 Å for the sc crystal structure and between 3.5 – 4.5 Å for the fcc crystal structure. The total energy was recorded for each different lattice constant. To normalize the data, the cohesive energy of each \(a\) was calculated using the atomic psuedopotential energy:
\begin{equation}E_{cos}=\frac{E_{tot}-E_{atomic}}{N_{atoms}}\end{equation}
where \(E_{cos}\) is the cohesive energy, \(E_{atomic}\) is the psuedo atomic energy, and \(N_{atoms}\) is the number of atoms in the cell.
The data was then plotted and fitted to the Birch-Murnaghan (BM) equation of state [7]:
\begin{equation}E_{tot}(a)=E_{0}+\frac{9V_{tot}B_{tot}}{16}{[(\frac{a_{0}}{a})^{2}-1]^{3}}B_{0}^{‘}+[(\frac{a_{0}}{a})^{2}-1]^{2}[6-4(\frac{a_{0}}{a})^{2}] \end{equation}
where \(E_{tot} \) is the total energy, \(V_{0}\) is the equilibrium volume per atom, \(B_{0}\) is the bulk modulus at zero pressure, \(B_{0}^{‘}\) is the partial derivative of B with respect to pressure at constant temperature, \(a\) is the lattice constant, and \(a_{0}\) is the lattice constant corresponding to the minimum energy. To fit the data, \(E_{0}\), \(V_{0}\), \(B_{0}\), \(B_{0}^{‘}\), and \(a_{0}\) were left as fitting parameters which varied depending on the crystal structure. Once the data was fitted, \(a_{0}\) would then correspond to the optimal lattice constant for each crystal structure.
3.1.2 K Points
The k point mesh was varied for each structure in order to ensure convergence of the last optimization of each structure using \(a_{0}\) as the lattice constant. Because the sc and fcc crystal structure have the same lattice constants in all three dimensions, the k point mesh had the form of M x M x M. For each variation of the mesh, the number of irreducible points for was determined and the energy was then recorded. The energy was then corrected using Equation 1.This data was then plotted to determine which k point mesh produced a cohesive energy value that was within 0.01 eV of the energy of the largest k point mesh used, while having the lowest amount of irreducible k points. This mesh would then be used for the final computation of determining the total energy of the system.
3.1.3 Energy Cutoff
The energy cutoff was varied between 200 and 600 eV for each structure in order to ensure convergence of each structure using \(a_{0}\) as the lattice constant. Again, the energy for each energy cutoff point was recorded and corrected using Equation 1. The data was then plotted to determine the lowest energy cutoff needed while still ensuring the calculation was converged. The minimum energy cutoff was chosen by determining which cutoff produced a total energy that was within 0.01 eV of the highest energy, while having the lowest energy cutoff.
3.2 hcp Crystal Structure Optimization
3.2.1 Lattice Constant
The optimization method used for the hcp crystal structure was similar to that of the sc and fcc crystal structure except for varying the lattice constants (\(a\) and \(c\)). Because two lattice constants are needed to define the structure, a ratio of \(\frac{c}{a}\) was used to vary the lattice constants appropriately, where \(\frac{c}{a}\) varied between 1.6 to 1.9, and \(a\) was varied between 2.5 – 5.0 Å. For each \(\frac{c}{a}\), a plot of cohesive energy versus \(a\) was produced and then fitted to the BM equation of state to determine the \(a_{0}\) for each ratio. The lowest \(a_{0}\) was then chosen as the optimal lattice constant for the hcp crystal structure in which the second lattice parameter was calculated accordingly to the chosen \(a_{0}\)’s respective \(\frac{c}{a}\).
3.2.2 K Points
For the HCP crystal structure, the k point mesh was chosen differently as the unit cell is not cubic. Instead, the mesh had the form M x M x N.
4. Results
Shown below will be the results for the optimization of the sc crystal structure for the lattice constant, k point mesh, and energy cutoff for brevity of this post. Afterwards, a table summarizing the final optimizing parameters for each crystal structure will be given.
Figure 1. Cohesive energy (eV/atom) vs. Lattice constant (Å). The blue circles represents the data points obtained from DFT calculations in Materials Studio. The blue line represents the fitted BM equation of state for the given data set. The red square represents \(a_{0}\), or the minimum of the BM equation of state, that will be used in future optimization calculations. The inset is the magnified portion of the plot around \(a_{0}\), which is 2.64 Å.
Figure 1 represents the plot of the total energy per atom as a function of lattice constant. As expected, the plot shows almost a quadratic relationship between the energy per atom and the lattice constant. The BM equation of state can be used to fit the data points and determine the \(a_{0}\) for the crystal structure. From the fit, \(a_{0}\) for the sc crystal structure is 2.64 Å.
Figure 2. Cohesive energy (eV/atom) vs. Energy cutoff (eV). The blue circles represents the data points obtained from DFT calculations in Materials Studio. The blue line is to help guide the eye across the plot. The black dashed lines represent the +/- 0.01 eV tolerance used to determine the optimal energy cutoff. The red square represents optimal energy cutoff that will be used in future calculations. The inset is the magnified portion of the plot around the optimal energy cutoff, which is 400 eV.
Figure 2 displays a plot of the cohesive energy per atom as a function of the number of irreducible k points as determined by Materials Studio. From the plot, it can be seen that the number of irreducible k points significantly affects the total energy determined from the CASTEP module. If the wrong k point mesh is chosen, then the energy will not be converged and yield an invalid result. From the above plot, the k point mesh chosen for the sc crystal structure was 9 x 9 x 9, corresponding to 365 irreducible k points.
Figure 3. Cohesive energy (eV/atom) vs. Number of irreducible k points. The blue circles represents the data points obtained from DFT calculations in Materials Studio. The blue line is to help guide the eye across the plot. The black dashed lines represent the +/- 0.01 eV tolerance used to determine the optimal number of irreducible k point mesh. The red square represents optimal k point mesh that will be used in future calculations. The inset is the magnified portion of the plot around the optimal k point mesh, which is 9 x 9 x 9.
Figure 3 shows a plot of the total energy per atom as a function of energy cutoff. Again, it is important to choose an appropriate energy cutoff to ensure that the energy given is converged to one’s chosen tolerance, which in this case is 0.03 eV. From the plot, the chosen energy cutoff for the SC crystal structure is 400 eV. To note, this energy cutoff was used for the other two crystal structures as to make sure the same plane waves were used for all calculations. Also, each crystal structure had the same optimal energy cutoff of 400 eV.
Figure 4. Cohesive energy (eV/atom) vs. Lattice constant (Å) for hpc at various \(frac{c}{a}\). The colored circles represent the data points obtained from DFT calculations in Materials Studio where each color is denoted by the legend on the right. The colored lines represent the fitted BM equation of state for each given data set. The red square represents \(a_{0}\), or the minimum of the BM equation of state, that will be used in future optimization calculations, which is 2.72 Å.
Lastly, Figure 4 is a plot summarizing the optimization of the two lattice constants for the hcp crystal structure. From the plot, it can be seen that optimal \(a_{0}\) occurs when \(\frac{c}{a}\) is 1.8. When \(\frac{c}{a}\) is increased to 1.9, the cohesive energy starts to increase again relative to \(a_{0}\).
Below are the optimized crystal structures for Pt as well as a table summarizing the parameters chosen and energy values for each crystal structure. When comparing the energies in the table, it can be seen that the FCC crystal structure has the lowest energy. Thus, the preferred crystal structure for Pt is the FCC crystal structure with a lattice constant of 3.97 Å.
Figure 5. Optimized crystal structures for Pt. a) sc crystal structure. b) fcc crystal structure. c) hcp crystal structure.
| Crystal Structure | Lattice Parameter [Å] | K Point Mesh | Energy Cutoff [eV] | Cohesive Energy [eV/atom] |
| sc | 2.64 | 9 x 9 x 9 | 400 | -8.51 |
| fcc | 3.97 | 11 x 11 x 11 | 400 | -8.98 |
| hcp | 2.72
4.90 | 10 x 10 x 6 | 400 | -8.91 |
5. Conclusions
From the above given data, it can be seen that the preferred crystal structure for Pt is FCC with a lattice constant of 3.97 Å. In order to check on how accurate the lattice constant is to experimental values, two different sources were used: 1) Materials Project (a computational database populated with thousands of inorganic compounds, molecules, etc) and 2) experimental work tabulated by Wiley. The table below lists the lattice constant values for the this project, the Materials Project, and the experimental work.
| DFT | Materials Project [8] | Experimental [9] |
| a | 3.97 | 3.98 | 3.912 |
From the table, it can be seen that the value obtained from DFT after optimization is in good agreement with that of Materials Project. This is to be expected, since both values were obtained computationally using the GGA functional. When comparing to the experimental value listed by Wiley, it can be seen that there is a larger difference in values for the lattice constant (0.05 Å) as opposed to the reference computational value. This difference can be attributed to multiple factors such as the computation calculations are done in vacuum and that all crystals were of uniform shape and size. Experimentally, this is not possible as it is difficult to run experiments in vacuum and the shapes of each Pt crystal cannot be controlled. Overall, it can be concluded that the preferred crystal structure for Pt is indeed FCC with a lattice constant of approximately 3.97 Å.
6. References
[1] Clark Stewart J et al., “First principles methods using CASTEP ,” Zeitschrift für Kristallographie – Crystalline Materials , vol. 220. p. 567, 2005.
[2] J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.
[3] J. P. Perdew et al., “Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation,” Phys. Rev. B, vol. 46, no. 11, pp. 6671–6687, Sep. 1992.
[4] D. D. Koelling and B. N. Harmon, “A technique for relativistic spin-polarised calculations,” J. Phys. C Solid State Phys., vol. 10, no. 16, pp. 3107–3114, Aug. 1977.
[5] D. Vanderbilt, “Soft self-consistent pseudopotentials in a generalized eigenvalue formalism,” Phys. Rev. B, vol. 41, no. 11, pp. 7892–7895, Apr. 1990.
[6] H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Phys. Rev. B, vol. 13, no. 12, pp. 5188–5192, Jun. 1976.
[7] F. Birch, “Finite Elastic Strain of Cubic Crystals,” Phys. Rev., vol. 71, no. 11, pp. 809–824, Jun. 1947.
[8] K. Persson, “Materials Data on Pt (SG:225) by Materials Project,” 2015.
[9] W. P. Davey, “Precision Measurements of the Lattice Constants of Twelve Common Metals,” Phys. Rev., vol. 25, no. 6, pp. 753–761, Jun. 1925.
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is the equilibrium lattice constant, 
is the equilibrium volume per atom, 
is the zero pressure bulk modulus, and 
is the derivative of the bulk modulus with respect to pressure at constant temperature. These four parameters are treated as fitting parameters.
over the NaCl structure.
. These values were determined to give unit cell size converged up to 
tolerance. The resulting lattice parameters are 
for NaCl. Using these we investigate how the ground state energy converges with the number of k-points with fixed cutoff energy 
.
for CsCl structure and by 
for NaCl structure, which was set to the most accurate ground state energy obtained for each. Due to the symmetry of NaCl crystal for this structure odd number of k-points leads to half as many total points in the full 3D Brillouin zone compared to even number of k-points in each dimension.
and k-points to find ground state energy as a function of unit cell size and look for a minimum approximating it by a parabola. Also we run BFGS geometry optimization with the same values of 
while for ScAl the error is less than 
which is in a good agreement with experimental value of 3.388Å[6].
