Author: Naveen Agrawal
Introduction: Hafnium is a chemical element with atomic number 72 and electronic configuration [Xe] 4f145d26s2. Hafnium has attracted great technological interest in nuclear science because of its exceptional corrosion resistance and high thermal neutron capture cross-section. Hafnium has been found to form an HCP like crystal as shown in the figure below structure with c/a ratio of 1.58 experimentally[1]. This study involves the Density Functional Theory calculations to predict the optimum lattice parameter for Hf using the Density Functional Theory with plane-waves basis set program ‘CASTEP’ [4] available in the Materials Studio.
Methods: CASTEP is used to perform plane-wave based electronic structure calculations which in general requires the selection of certain input parameters such as KPOINTS and ENCUT to optimize the computational effort and the accuracy of electronic structure results. In addition, we choose GGA (Generalized Gradient Approximation) based PBE ( Perdew Burke Ernzerhof) exchange-correlation functional [2,3] and 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) [5]. We first optimize the KPOINTS for an energy cutoff of 500 eV and later do the convergence for energy cut-off (ENCUT) which determines the maximum energy plane-wave included in the solution. These convergence tests and further calculations were performed with the SCF (Self-consistent-field) cycle convergence criteria of 2E-06 eV per atom, using finer criteria improved the accuracy of total energy insignificantly w.r.t to the desired accuracy for the study (1 meV). Optimized ENCUT and KPOINTS are used consistently for determining the optimum lattice parameters through variation in lattice constant ‘a’ for a chosen c/a ratio. Later, we compare the respective minimums obtained at different c/a ratios to determine the most energetically favorable lattice parameters.
KPOINTS convergence: Hafnium crystal structure (HCP) and lattice parameters (c/a = 1.5811 and a =3.194 A) imported from Materials Studio library were used for initial convergence testing of KPOINTS and ENCUT. Selection of KPOINTS grid was made in such a way to keep the density of KPOINTS in reciprocal space is uniform. Specifically, for Hafnium the number of KPOINTS were kept in the inverse ratio of unit cell vector length or reciprocal cell vector lengths. For the considered lattice, the ratio of KPOINTS to the nearest integers in X and Y directions to Z directions were kept equal to 1.58 approximately. Based on the above criteria, following grids as shown in the table below were considered for testing.
KPOINT grid | Irreducible Kpoints | Total energy per atom in eV | Relative energy to the energy for highest KPOINTS in meV |
---|---|---|---|
5x5x3 | 10 | -7866.695 | 1.504 |
7x7x4 | 16 | -7866.694 | 2.952 |
9x9x5 | 36 | -7866.700 | -2.623 |
11x11x7 | 64 | -7866.696 | 1.004 |
14x14x9 | 120 | -7866.697 | 0.000 |
The figure above shows the convergence of total energy per atom in eV relative to same determined at the highest number of KPOINTS against irreducible KPOINTS available for the considered grid of KPOINTS. With the tolerance of 1meV, KPOINTS grid of 11X11X7 seems to be a reasonable choice.
ENCUT convergence: ENCUT( Energy cutoff) is the kinetic energy of the highest kinetic energy plane wave that needs to be considered to obtain a converged solution such that any higher energy cutoff would not lead to an energy difference for the tolerance considered (1meV).
The figure above shows the variation in the total energy per atom in eV for the different values of ENCUT considered. Based on the desired tolerance, an ENCUT of 500 eV was determined to be optimum.
Optimization of lattice parameters: As Hafnium is an HCP metal, it needs two lattice constants ‘c’ and ‘a’ to specify the crystal structure. I chose to systematically vary both for a chosen c/a to find the minimum energy configuration for the considered c/a. I also compared the minimums obtained for different c/a ratios to find the most favorable set of lattice parameters.
The figure above shows the variation in cohesive energy per atom relative to the global minimum obtained in eV for different sets of lattice parameters. The systematic variation in lattice constant ‘a’ for a chosen c/a led to different minimum energy configurations indicated by solid black curve. Based on the cohesive energies per atom minimums obtained at different c/ratios, c/a =1.583 with a =3.2011 Angstrom were determined to be optimum lattice parameters.
c/a | a in Angstrom | Cohesive energy per atom in eV | Relative cohesive energy per atom in meV (millielectronvolt) |
---|---|---|---|
1.590 | 3.1968 | -8.84590 | 0.0749 |
1.585 | 3.1997 | -8.84598 | 0.0019 |
1.583 | 3.2011 | -8.84598 | 0.0000 |
1.581 | 3.2024 | -8.84596 | 0.0124 |
1.578 | 3.2045 | -8.84591 | 0.0641 |
1.575 | 3.2065 | -8.84583 | 0.1453 |
1.570 | 3.2097 | -8.84562 | 0.3547 |
Conclusion: Optimum Lattice parameters obtained through the DFT calculations (a = 3.2011 Angstrom and c/a = 1.583) using the CASTEP code with the tolerance of 1meV in the total energy came in close agreement with the experiments ( c/a =1.58, a =3.1964 Angstrom)[1]. However, it should be noted the energetic difference between several minimums obtained are of a lesser order than the tolerance specified, therefore, stricter convergence criteria should be useful to resolve the energy scale to such order confidently.
References:
[1] http://periodictable.com/Elements/072/data.html
[2] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett., 77 (1996) 3865-3868.
[3] 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.
[4] 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.
[5] http://www.physics.rutgers.edu/~dhv/uspp/