DFT and Lattice Parameters
For many metals (simple cubic, body-centered cubic, and face-centered cubic) using DFT to calculate the lattice parameter for a metal or crystal is rather straightforward as there is only one parameter to vary. One may begin by assuming the total energy of the system is a Taylor expansion of the lattice parameter, a, as below:
\begin{equation}E_{tot}(a) = E_{tot}(a_{0}) + \alpha (a-a_{0}) + \beta (a-a_{0})^2\end{equation}
Following Sholl (1), we may reduce this equation to:
\begin{equation}E_{tot}(a) = E_{tot}(a_{0}) + \beta (a-a_{0})^2\end{equation}
Thus, the lattice parameter for many metals and crystals can be determined by sampling various values of a. The energy of the system can then be calculated at each value of a and should look like a quadratic with a minimum at the true value of the lattice parameter.
Lattice Parameters for hcp Crystals
For hexagonal-close packed (hcp) structures, determining the lattice parameters is not so straightforward. For sc, fcc, and bcc metals there is only one parameter to optimize the energy. When considering an hcp metal, there are two lattice parameters on which the total energy depends: a, and c.
It is still possible, however, to manage this multi-variable minimization using our technique from above. First, we fix the ratio c/a to some value, r and sample various values of a at which to calculate the energy. With c/a fixed we will also know the value of c and we may construct our energy curves at varying values of r. The curve with the lowest total energy at its minima will be considered the “theoretical values” of the lattice parameters.
Lattice Parameters of Hf
Hafnium, element 72 on the periodic table, is a d-block transition metal with an hcp crystal structure. We wish to use DFT, as outlined above, to determine the equilibrium (ground state) lattice parameters of Hf.
Below are results obtained from CASTEP single point calculations for r = 1.40, 1.48, 1.58, 1.72, and 1.85 with an energy cut-off of 290 eV and a k-point grid of 8x8x6.
From this plot we can determine that the optimum value of r is about 1.58. Using this, a refined set of calculations may be performed at r = c/a=1.58 for various a to calculate an accurate estimate for both a and c. The results of these calculations are found below.
Fig 2. Refined determination of a.
In the above plot the blue points are CASTEP results and ther orange line is the harmonic approximation of the total system energy around the true lattice parameter. From this data we can observe a couple of things. First, we can say that the lattice parameters of Hf are approximately:
$$a_{0} = 3.22$$ $$c_{0} = 5.10$$
We may also see that the behavior of the energy is not truly quadratic with respect to the lattice parameter(s). The harmonic potential is a very good approximation for a near a0, but for a > a0 the harmonic potential is an overestimate of the energy and an underestimate for a < a0. This is because at larger separations (a > a0) the energy decreases as it should approach the dissociation energy of Hf (a → ∞) while at shorter separations (a < a0) the energy increases due to strong repulsive forces.
Fig. 3 Optimized Hf Cell
WebElements, an online reference for chemical elements, reports the lattice constants for Hafnium as (2):
$$a_{0}^{ref} = 3.20$$
$$c_{0}^{ref} = 5.05$$
Thus our calculations agree quite well with available data and we are satisfied.
Convergence
The above results were found using a relatively small energy cut-off (290 eV) and k-point grid (9x9x6) to allow for quick calculations that give an idea of the behavior or the energy with respect to the lattice constant. We now wish to see if we were converged with respect to the energy cut-off and k-point grid.
For the energy cut-off:
Fig 4. Energy Cut Off Convergence
While it appears that a rather high energy cut-off (~600 eV) is needed for convergence, it is important to realize that the energy differences, even between Ecut=250-270 are within chemical accuracy (~0.04 eV) and therefore using an Ecut of 290 should produce rather reliable results (as we have confirmed with literature values). Chemical accuracy is a standard used by computational chemists as a benchmark for making reliably accurate chemical predictions. Essentially, if energies are within ~0.04 eV then we can make confident predictions; this is exactly the case for our Ecut convergence.
Similarly for the k-points (where we have fixed the ratio kx/kz = ky/kz = 4/3) we find the following. Here our convention for “# of KPoints” is just to add the # of KPoints in each direction (e.g. 8x8x6 = 22 KPoints in the plot)
Fig. 5 K-Point Convergence
Again, we see that not very many k-points are needed (~20) before our energies are within chemical accuracy. This means that our use of an 8x8x6 k-point grid is sufficient for our needs.
Single Point Calculations
For the single point calculations performed for Hf with varying lattice constants the following (ultra-fine) calculation settings were used:
Exchange Correlation Functional Type: Generalized Gradient Approximation (GGA)
Exchange Correlation Functional: PBE
Plane-Wave Energy Cut-Off: 290.0 eV
K-Point Grid: 9x9x6
Pseudopotentials: Ultrasoft
Electronic Energy Convergence Criteria: 5.0 10-7 eV/atom
Geometry Optimization
For the geometry optimization of Hf, the following (ultra-fine) calculation settings were used:
Exchange Correlation Functional Type: Generalized Gradient Approximation (GGA)
Exchange Correlation Functional: PBE
Plane-Wave Energy Cut-Off: 290.0 eV
K-Point Grid: 9x9x6
Pseudopotentials: Ultrasoft
Electronic Energy Convergence Criteria: 5.0 10-7 eV/atom
Ionic Energy Convergence Criteria: 5.0 10-6 eV/atom
Ionic Force Convergence Criteria: 0.01 eV/Å
Future Work:
In the future it would be of interest to investigate the effect of the type of pseudo-potential used (e.g. ultrasoft, soft, etc.) as well as the psuedo-potential cut-off for treating the core and valence electrons. Also of interest would be how our results change using different XC Functionals.
References:
(1) Sholl, D. & Steckel, J. A. Density Functional Theory: A Practical Introduction. (John Wiley & Sons, 2011).
(2) https://www.webelements.com/hafnium/crystal_structure.html
(3) First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005) S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne
