by Sharad Maheshwari
Introduction
The aim of the following work is to use Density Functional Theory to predict the lattice parameters of the bulk crystal lattice of Hafnium (Hf). Experimentally, Hf crystallizes in a hexagonal close-packed structure with the ratio of lattice parameters, c/a = 1.58 [1]. Herein, we would use the DFT calculations to evaluate the c/a ratio and compare it with the above experimental value.
Figure 1. Top, side and perspective view of the primitive unit cell of Hafnium (Hf) in hexagonal closed packing with two lattice parameter, c and a.
Methodology
We use the plane wave DFT calculations to evaluate the binding energy of a primitive lattice of Hf as shown in the equation below. By minimizing the binding energy as a function of lattice parameters, we will evaluate the ratio of lattice parameters (c/a).
Binding Energy = DFT Energy Calculated – 2*Atomic Energy of Hf
All the calculations were performed using CASTEP Simulation Package [2] in Material Studio. The exchange and correlation energies were calculated using the Perdew, Burke, and Ernzerhof (PBE) [3] functional described within the generalized gradient approximation (GGA) [4]. Electronic convergence tolerance of 2E-06 was used for all the calculations. The core electrons were treated using “on the fly” generated (OTFG) ultra-soft pseudo-potential with core radius 2.096 Bohrs (1.109 Angstrom) generated with panel of 26 valence electrons (4f14 5s2 5p6 5d2 6s2).
Cut-off Energy and K-Point Optimization
We use a plane wave basis set to perform our DFT calculations for the periodic lattice of Hf. In order to use the plane wave basis sets, it is essential to ensure the convergence of system energy with respect to the cut-off energy and k-point mesh. Therefore, we first perform cut-off energy and k-point optimization. For this, we have used the hcp lattice with a = 3.1946 Å and c = 5.0511 Å which are the default dimensions for the Hf lattice in Material Studio and is very close to the experimental values[5, 6].
Cut-Off energy
In Fig. 2, we have plotted the absolute change in the consecutive total energy (|ΔE|) as the cut-off energy is varied. As a convergence criterion, we assume that the total energy is converged if |ΔE| is less than 0.003 eV. From Fig. 2, we can say that the cut-off energy of 500 eV sufficiently converged the total energy of the system.
Figure 2. Convergence of absolute change in consecutive total DFT energy with respect to cut-off energy. The blue horizontal line indicates the convergence criterion used (0.003 eV) and the plot is further zoomed in to illustrate the convergence
K-Point
In Fig. 3, we have plotted the absolute change in the consecutive total energy (|ΔE|) as the number of irreducible k-points, corresponding to different k-point grid used as shown in Table 1., is changed. As a convergence criterion, we assume that the total energy is converged if |ΔE| is less than 0.003 eV. From Fig. 3, we can say that k-point mesh generated automatically using the Monkhorst Pack method [7] of 10X10X6, corresponding to 42 irreducible k-points samples the Brillouin zone sufficiently well and converges the total energy of the system.
Figure 3. Convergence of total DFT energy with respect to irreducible k-points corresponding to different k-point grid. The orange horizontal line indicates the convergence criterion used (0.003 eV).
| K-point grid | Irreducible k-points |
| 8X8X5 | 30 |
| 9X9X6 | 36 |
| 10X10X6 | 42 |
| 11X11X7 | 64 |
| 12X12X8 | 76 |
| 13X13X8 | 84 |
Hf Lattice Optimization
To optimize the hcp lattice of Hf, we minimize the binding energy of the system with respect to the two lattice constant, a and c. To carry out this two-parameter optimization, we keep the ratio of c/a constant as the value of a is varied and then the same procedure is repeated for different values of c/a.
In order to ascertain the minima for any specific ratio of c/a, the binding energy is evaluated at 3 distinct values of a. A parabola is then fit through these points and the value of a corresponding to the minima of the parabola is evaluated using the equation of the parabola. The binding energy is then calculated at this evaluated value of a to check if this indeed is the minima. The binding energy is also evaluated for the points in the vicinity of this minimum to ensure that we find the least energy for the given c/a ratio.
Results
The above procedure for lattice optimization was repeated for different values of c/a. Fig. 4 plots the calculated binding energy with respect to a for different c/a ratio. Table 1 lists the minimum energy obtained for different c/a ratio and corresponding value of a.
Figure 4. DFT Energy of the crystal lattice of Hf as a function of one of the lattice parameter, ‘a’ for different values of the ratio c/a. An “Overall” curve passes through minimum for different “c/a” ratios and thus minimizing w.r.t. different c/a
| c/a | Binding Energy (eV) | a (Å) |
| 1.581 | -17.487 | 3.204 |
| 1.481 | -17.649 | 3.282 |
| 1.781 | -17.693 | 3.082 |
| 1.381 | -17.648 | 3.355 |
| 1.681 | -17.541 | 3.141 |
A parabola is then plotted to pass through the minima obtained for different c/a ratio. Minimum of this parabola gives us the local minimum of binding energy with respect to c/a and thus, the optimum value for c/a.
The above optimization for binding energy minimization occurs at c/a = 1.581 and a = 3.204 Å (c =5.066 Å ). This value evaluated using the DFT calculation agree well with the experimental observation [6, 7].
Conclusion
Density Function Theory based calculations were used to ascertain the lattice parameters of the metal Hafnium. The binding energy of the primitive hcp lattice of Hf was minimized by altering the c/a ratio and the value of the lattice parameter a. The optimized lattice parameters obtained are reasonably coherent with the experimental results. This project thus helps illustrate that DFT can be used to predict lattice parameters reasonably well.
References
[1] D. Sholl, J.A. Steckel, Density Functional Theory: A Practical Introduction, Wiley2009.
[2] 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, 2005, pp. 567.
[3] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett., 77 (1996) 3865-3868.
[4] 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 theTHE Generalized Gradient Approximation for Exchange and Correlation, Phys. Rev. B, 46 (1992) 6671-6687.
[5] http://periodictable.com/Elements/072/data.html.
[6] https://www.webelements.com/hafnium/crystal_structure.html.
[7] H.J. Monkhorst, J.D. Pack, Special Points for Brillouin-Zone Integrations, Phys. Rev. B, 13 (1976) 5188-5192.
