By Nathan McKee

Introduction

This post examines the structure of ScAl, whether the material prefers the structure of CsCl or that of NaCl (shown in figures 1 and 2, respectively), and predicts the lattice constant in each case. For each structure, the lattice constant was varied and the ground state energy of the structure was calculated, repeating to find the lattice constant that minimizes the energy for each structure. DFT calculations were carried out with the plane-wave based code CASTEP. The GGA PBE functional was used¹, as well as OTFG ultrasoft pseudopotentials². These pseudopotentials include the 3s2 3p1 valence electrons and a cutoff radius of 1.5 Å for Al, and the 3s2 3p6 3d1 4s2 valence electrons and a cutoff radius of 1.6 Å for Sc. The convergence tolerance was set at 2.0*10^-6 eV per atom.

Figure 1: The CsCl structure, which is a simple cubic lattice with two atoms per unit cell.

Figure 2: The NaCl sructure, which is an FCC lattice with two atoms per primitive unit cell.

Cutoff Energy and k Points

To ensure that the calculations converge properly, an analysis of the selection of k points and the cutoff energy was performed. The k points were chosen to be an NxNxN grid of evenly spaced points in reciprocal space, with the same number of points in each direction being appropriate for a cubic cell. Figures 3 and 4 show how an energy calculation converges as N increases and as the cutoff energy is raised. In both cases, an arbitrary constant was added to the calculated energies so that the values would be close to zero and the convergence could be seen more easily. The calculations for k point convergence were performed for the NaCl structure with a cutoff energy of 410.9 eV. The calculations for the cutoff energy convergence were performed for the same structure with a 6x6x6 k-point grid.

Figure 3: The calculated energy converges as the size of the NxNxN grid of k points increases. The energies have been shifted so that the differences are easy to see.

Figure 4: The calculated energy of the structure converges as the cutoff energy increases. The energies have been shifted so that the differences are easy to see.

Using these graphs as a guide, a 6x6x6 k point grid was used in conjunction with a cutoff energy of 410.9 eV for further calculations. It should also be noted that an origin shift was implemented in the k point grid in order to increase the total number of k points in the Brillouin Zone (BZ) without changing the number of k points in the Irreducible Brillouin Zone (IBZ). The shift used was 0.01 Å^-1 in the x direction, 0.005 Å^-1 in the y direction, and 0.003 Å^-1 in the z direction. These values were chosen so that the k points would not lie on any symmetry axes, while maintaining a small shift compared to the spacing between k points.

Results

Figures 5 and 6 show the energy minimization with respect to the lattice constant for each of the structures being considered. The vertical axes have been shifted so that the minimum energy value obtained lies on the horizontal axis. For the CsCl structure, a minimum energy of  -1385.213651797 eV was obtained, with a lattice constant of 3.380 Å. For the NaCl structure, a minimum energy of -1384.047417 eV was obtained, with a lattice constant of 5.649 Å.

Figure 5: Energy vs lattice constant for ScAl in the structure of CsCl. The value of Emin is -1385.213651797 eV.

Figure 6: Energy vs lattice constant for ScAl in the structure of NaCl. The value of Emin is -1384.047417 eV.

Conclusion

These results indicate that ScAl prefers the CsCl structure over the NaCl structure, because the minimum calculated energy of the CsCl structure was lower. In addition, these results predict a lattice constant of 3.380 Å for ScAl in the CsCl structure.

References

1. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
2. Gonze, X. & Finocchi, F. Pseudopotentials Plane Waves–Projector Augmented Waves: A Primer. Phys. Scr. 2004, 40 (2004).