by-Hepeng Ye
Crystallography is widely applied to study inorganic molecules, proteins, metals, etc. Such process is driven by entropy and enthalpy at the same time which makes it very tricky to control the crystallization and hard to predict what lattice will a given compound form at certain condition.
Platinum (Pt) metal is studied using Castep through energy minimization. Basic idea behind is that the lowest energy stable should be the most favored state, and that corresponding lattice should be the most likely lattice that we shall expect to observe from experiment. Now, lets pretend that we do not know what is the crystal lattice for Pt, and I will show you how to find out the more likely lattice.
Energy calculation involves GGA-PBE functional[1], and pseudo-potentials were set as default as OTFG ultrasoft[2]. And the ultrasoft pseudopotential for Pt is set to have core radii to be 2.403 Bohr radii (~1.27Å), vis using valence electrons in the 4f14 5s2 5p6 5d9 6s1 configuration.
Two lattices are studied, face center cubic(FCC) and hexagonal close packing(HCP), and the following part will present parameters optimization and energy minimization for both lattice.
For FCC, m-3m space group is used since it is the only possible space group, and by using lattice parameter (a) equals 4.0Å, energy cut-off convergence is determined.
Figure1. energy cut-off optimization for Pt FCC
As shown in the figure 1, energy difference converges as I use larger energy cut-off, and the difference between using 420ev cut-off and 480eV are very close. Base on this, I am confident to say that 480eV is a good stop point and larger cut-off may no longer be efficient for DFT calculation.
Then, another parameter optimized is the K-points. I used three cell sizes and all with 480eV cut-off energy. And plot below shows the energy per atom from irreducible K-points from 10 up to 120.
Figure 2. K-points optimization for Pt FCC at three different lattice parameters using optimized energy cut-off.
It is clear that as irreducible k-points used go beyond ~25, energy starts to stabilize, though still fluctuates in a tolerable range. And by considering the computation capability and energy accuracy, I use the irreducible k to be 56 (fourth point from left) as the optimized K-points for further calculation.
Energy cut-off is determined to be 480eV and K-points is 56, then the only parameter left for FCC structure is the cell length. I performed the a-optimization by randomly picking three a values, and do the calculation until energy is minimized for each, then I use these three energies in function of a-values to fit a parabola, and use that as an indication to look for another three a-values on the curve which are likely to give me the minimized energy.
Figure 3. Energy diagram verses lattice parameter using optimized K-points and energy cut-off.
Three iterations are performed and totally nine data points give a nice parabola. And the minimum energy (energy per atom) from the parameter a=3.975Å with corresponding energy to be -13051.00eV(per atom).
For platinum in HCP lattice, D3H-3 space group is used. And to make data from FCC and HCP calculation comparable with each other, energy cut-off is kept the same (480eV). But for D3H-3 space group, k-points need to be re-optimized since the real space and reciprocal space are both different from the fcc.
Figure 4. K-points optimization for Pt HCP lattice structure.
Starting with a equals 3.9Å and a/c ratio being 1.53. k-points are tested from 16 to 312.
Usually, more than 10 K-points should be enough, and it is true from the plot. There is a bump around 40 k-points and as a reason, I choose to use k equals 135 for calculation.
Since HCP has two lengths to be modified, one edge is defined as ‘a’ and another one as ‘c’. We know that in the crystal structure, size matters, and we know the density is a description of how many mass in a certain volume, so similar idea is used here that we analyze the energy of lattice at different pressure (isobaric condition), and for each pressure there should be a corresponding volume, which is a function of ‘a’ and ‘c’. By modifying the ratio of a/c, there should be an optimized (lowest) energy for that specific volume. Eventually, a plot of energy with respect to volumes will be plotted.
Figure 5. Energy diagram for Pt HCP at multiple lattice parameters’ ratios for each specific volume.
The plots above shows: at each volume(Å3), there are 10 a/c ratios evaluated from a/c equals 1.3 to 1.8 (most metal hcp fall into this range).
And by extracting the lowest energy from each volume, the minimum energy is get from the lowest point, to be -13050.24eV per atom.
FCC has minimum energy to be -13051.00 eV
HCP has minimum energy to be -13050.24 eV
So, FCC has lower energy, and should be the expected crystal structure for platinum.
As mentioned at the beginning, after showing the energy difference between these two possible lattice structures, what is the actual structure?
From Crystallography Open Database[3], I could infd the experimental result for Pt is FCC, with lattice parameter ‘a’ to be 3.944+/-0.004Å. The final lattice is fcc which is what we expect.
reference:
[1]:Setting up pseudopotentials- ultrasoft and norm-conserving pseudopotentials.
https://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/modules/castep/tskcastepsetelecpotentials.htm
[2]:J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple.
Phys. Rev. Lett. 77, 3865 (1996)
[3]: Entry 1011103, F m -3 m #225, Crystallography Open Database.
http://www.crystallography.net/cod/1011103.html
