by Wilson Yanez
Introduction
The aim of this project is to compare the methods of energy and enthalpy minimization to compute the lattice constant of platinum and to determine its preferred crystal structure between the simple cubic (SC), face centered cubic (FCC) and hexagonal closed packed (hcp) lattices.
Energy minimization consists of finding the minimum of the total energy of the system at 0 K with different values of lattice constants and in different crystal structures . Thus predicting the most favorable configuration that happens in nature which minimizes the energy.
In order to reduce the computational demand, we perform a binary search of the optimal lattice parameter. Once we have three different energies that differ by less than 1 eV, we perform a second order polynomial fit using these values and determine the value of the lattice constant that minimizes the energy. Finally, we compute the energy of the system at that point.
The second method is to use the geometry optimization package provided with the Materials Studio suite that minimizes the enthalpy (H=U+PV) of the system. This method computes the pressure of the material using its stress tensor and also computes the volume of the unit cell in order to determine the value of the enthalpy. The algorithm converges to a solution when the energy difference between the actual and the previous value of enthalpy are lower than the defined tolerance and the maximum stress in the material is lower than a user defined value.
All of the calculations in this work have been performed using the generalized gradient approximation (GGA) Perdew, Burke and Ernzerhof (PBE) functional and an on the fly generated (OTFG) pseudopotential [1,2]. The partial core correction is RC=1.679 and the electronic configuration is 4f14 5s2 5p6 5d9 6s1. The smearing scheme is Gaussian with a width of 100 meV and the energy cutoff is 272.1 ev. Unless otherwise specified, we use an 8x8x8 k points mesh with variable number of k points depending on the symmetry of the problem. We have neglected spin orbit coupling in this calculation since is a higher order relativistic effect.
In the case of the geometry optimization, the minimization has been performed by the two point steepest descent (TPSD) method with a maximum energy difference of 2E-6 eV/atom, a maximum force of 0.05 eV/Å, a maximum displacement of 0.2 pm and a maximum stress of 0.1 GPa.
Results
Simple Cubic
We define a simple cubic unit cell using the 221 PM-3M symmetry group as can be seen in figure 1. After we perform our energy minimization algorithm we obtain the results shown in figure 2. As we can see the minimal energy of the system was -13050.111 eV with a lattice constant of 2.65 Å
Figure 1: Simple cubic lattice and respective Brillouin zone
Figure 2: Energy minimization data for the simple cubic structure after 3 iterations. Just the final iteration is shown for clarity
For our second method we obtained an enthalpy of -13050.098 eV and a lattice constant of 2.66 Å
Face Centered Cubic
We define a face centered cubic unit cell using the 225 FM-3M symmetry group as can be seen in figure 3. After we perform our energy minimization algorithm we obtain the results shown in figure 4. As we can see the minimal energy of the system was -13050.528 eV with a lattice constant of 3.97 Å
Figure 3: Face centered cubic lattice and respective Brillouin zone
Figure 4: Energy minimization data for the face centered cubic structure after 5 iterations. Just the final iteration is shown for clarity
For our second method we obtained an enthalpy of -13050.4917 eV and a lattice constant of 4.02 Å
Hexagonal Closed Packed
We define an hexagonal closed packed unit cell using the 194 P63/MMC symmetry group as can be seen in figure 5. The minimization is performed by finding the a value with diferent c/a ratios as can be seen in figure 6. After that we perform a second order fit from the lowest energy values of the different c/a to find the correct minimum, as can be seen in figure 7. After all this process, we find that the minimal energy of the system was -13050.494 eV with a lattice constant of a=2.78 Å and c=4.73 Å with c/a=1.7
Figure 5: Hexagonal closed packed lattice and respective unit cell
Figure 6: Energy minimization data for closed packed structure. Just the final iteration of each c/a ratio is shown for clarity. The energy in the surface plot has been shifted by 13050 eV.
Figure 7: Interpolation of the c/a ratio considering the minimum of each data set.
For our second method we obtained an enthalpy of -13050.484 eV and lattice constants of a=2.81 Å, c= 4.80 Å and c/a=1.71
The calculation results can be summarized and in table 1 for the energy minimization method and in table 2 for the enthalpy minimization.
| Lattice | a Angstrom | Energy/atom eV | Mesh | # of K points | Energy cut off | c/a |
|---|---|---|---|---|---|---|
| FCC | 2.83 | -13012.476 | 8x8x8 | 60 | 272.1 | |
| 3.95 | -13050.518 | 8x8x8 | 60 | 272.1 | ||
| 4.24 | -13050.089 | 8x8x8 | 60 | 272.1 | ||
| 5.66 | -13046.457 | 8x8x8 | 60 | 272.1 | ||
| 3.54 | -13048.268 | 8x8x8 | 60 | 272.1 | ||
| 4.10 | -13050.415 | 8x8x8 | 60 | 272.1 | ||
| 3.82 | -13050.333 | 8x8x8 | 60 | 272.1 | ||
| 3.97 | -13050.528 | 8x8x8 | 60 | 272.1 | ||
| SC | 3 | -13048.966 | 8x8x8 | 20 | 272.1 | |
| 2 | -13033.115 | 8x8x8 | 20 | 272.1 | ||
| 2.5 | -13049.818 | 8x8x8 | 20 | 272.1 | ||
| 2.75 | -13049.944 | 8x8x8 | 20 | 272.1 | ||
| 2.65 | -13050.110 | 8x8x8 | 20 | 272.1 | ||
| HCP | 2.5 | -13047.791 | 8x8x4 | 20 | 272.1 | 1.6 |
| 3 | -13050.121 | 8x8x4 | 20 | 272.1 | 1.6 | |
| 2.75 | -13050.399 | 8x8x4 | 20 | 272.1 | 1.6 | |
| 2.85 | -13050.465 | 8x8x4 | 20 | 272.1 | 1.6 | |
| 2.5 | -13048.747 | 8x8x4 | 20 | 272.1 | 1.7 | |
| 3 | -13049.918 | 8x8x4 | 20 | 272.1 | 1.7 | |
| 2.75 | -13050.486 | 8x8x4 | 20 | 272.1 | 1.7 | |
| 2.81 | -13050.485 | 8x8x4 | 20 | 272.1 | 1.7 | |
| 2.78 | -13050.493 | 8x8x4 | 20 | 272.1 | 1.7 | |
| 2.5 | -13049.284 | 8x8x4 | 20 | 272.1 | 1.8 | |
| 3 | -13049.712 | 8x8x4 | 20 | 272.1 | 1.8 | |
| 2.75 | -13050.479 | 8x8x4 | 20 | 272.1 | 1.8 | |
| 2.77 | -13050.441 | 8x8x4 | 20 | 272.1 | 1.8 | |
| 2.58 | -13050.031 | 8x8x4 | 20 | 272.1 | 1.8 | |
| 2.72 | -13050.470 | 8x8x4 | 20 | 272.1 | 1.8 |
| Lattice | a Angstrom | Enthalpy/atom eV | Energy/atom eV | Mesh | # of K points | Energy cut off | c/a |
|---|---|---|---|---|---|---|---|
| SC | 2.664 | -13050.098 | -13050.100 | 8x8x8 | 20 | 272.1 | |
| FCC | 4.020 | -13050.516 | -13050.509 | 8x8x8 | 60 | 272.1 | |
| HCP | 2.8811 | -13050.484 | -13050.482 | 8x8x4 | 20 | 272.1 | 1.709 |
Conclusions
It can be seen that both methods were effective in order to determine the lattice constant of platinum. The preferred crystal structure is an FCC lattice with a lattice constant of 3.97±3 Å and a minimum energy of -13050.528 eV
It has been noticed that the c/a ratio found in this work is actually different from the theoretical value of 1.633 predicted for an ideal hcp structure. As we can see from the previous data. It is really important to also optimize this ratio. Otherwise, there might be a difference on the lattice parameter of up to 0.1 Å.
This calculation matches the preferred structure of platinum found in nature of a face centered cubic lattice. The computed lattice constant exhibits a 1.25% discrepancy with the actual value of 3.92 Å using the energy minimization method and of 2.6% for the enthalpy minimization method.
Even though both methods predict the correct phase for Pt, the enthalpy minimization algorithm seems to be overestimating the stress tensor contribution on the material. Thus increasing the error in the calculation of its lattice constant.
References:
[1] J. P. Perdew K. Burke, Y. Wang. Phys. Rev. B 57, 14999 (1998)
[2] Perdew, J. P; Burke, K; Ernzerhof, M. Phys. Rev. Lett. 77, 3865-3868 (1996)
