By Jordan Barr

Introduction

Two common crystal structures of materials with the formula AB are the cesium chloride (CsCl) structure and the rock salt (NaCl) structure. In the CsCl structure, the Cs atoms are in the simple cubic structure and the Cl atoms are in the middle of the Cs atoms cubic structure. In the rock salt structure, the two atom types form two interpenetrating face-centered cubic (fcc) lattices. In this study, ScAl is tested in both the CsCl and NaCl structures to predict the lattice constant of ScAl in these two forms and it is shown that ScAl prefers the CsCl structure over the rock salt structure.

Computational Details

The density functional theory (DFT) calculations were conducted using the plane-wave pseudopotential method as implemented in CASTEP [1]. Convergence testing gave total energies converged to 5 meV/atom using a cutoff energy of 600.0 eV and k-point mesh of 10x10x10. The PBE-GGA functional [2] was used in all calculations and the SCF tolerance was set to 2.0e-6 eV/atom. The aluminum pseudopotential consisted of 3 valence electrons and a cutoff of 2.00 Bohr and the scandium pseudopotential consisted of 11 valence electrons with a cutoff of 1.80 Bohr.

The CsCl structure basis is defined by the space group Pm3m with Cs at (0, 0, 0) and Cl at (0.5, 0.5, 0.5). The primitive vectors are given by a1 = a(1, 0, 0), a2 = a(0, 1, 0), and a3 = a(0, 0, 1). Figure 1 shows the CsCl structure.

Figure 1: ScAl in the CsCl structure. Purple atoms are Sc atoms and blue atoms are Al atoms.

For the rock salt structure, the space group is given by Fm3m and the basis consists of Na at (0, 0, 0) and Cl at (0.5, 0, 0) and the primitive vectors are a1 = a(0.5, 0.5, 0), a2 = a(0, 0.5, 0.5), and a3 = a(0.5, 0, 0.5). Figure 2 shows the NaCl structure.

Figure 2: ScAl in the NaCl structure. Purple atoms are Sc atoms and blue atoms are Al atoms.

To determine the lattice constant of ScAl in the CsCl and rock salt structures, the energy versus lattice parameter data was fit to the Birch-Murnaghan (BM) equation of state [3]:

where is the equilibrium lattice constant, is the equilibrium volume per atom, is the zero pressure bulk modulus, and is the derivative of the bulk modulus with respect to pressure at constant temperature. These four parameters are treated as fitting parameters.

Results

In figure 3 is plotted the DFT energy versus lattice parameter for the CsCl and NaCl structure and the BM fit to the data.

Figure 3: Energy versus lattice parameter for ScAl in the CsCl and NaCl structure. Solid lines represent the BM fit and the circles represent the raw DFT data.

From the BM fit for the CsCl structure of ScAl, the equilibrium lattice constant is calculated to be 3.403 Å and the bulk modulus is found to be 0.450 GPa. For the BM fit for the NaCl structure of ScAl, the equilibrium lattice constant is found to be 5.663 Å and the bulk modulus is calculated to be 0.111 GPa. From figure 3, it is indeed seen that the CsCl structure for ScAl is favored over the NaCl structure as its ground state energy is lower than that of NaCl. This follows the experimental phase of ScAl, which is found to be the Pm3m space group with a lattice constant of 3.388 Å [4]. The difference in lattice constant between the DFT calculated lattice parameter and the experimentally determined value can be attributed to multiple factors. One such factor can be the chosen pseudopotential; the use of other pseudopotential types such as LDA can be tested to see if a better comparison between computational and experimental values can be obtained.

[1] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne, “First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)

[2] John P. Perdew, Kieron Burke, and Matthias Ernzerhof, “Generalized Gradient Approximation Made Simple”, Phys. Rev. Lett. 77, 3865 – Published 28 October 1996; Erratum Phys. Rev. Lett. 78, 1396 (1997)

[3] D. Sholl and J. Steckel, Density Functional Theory: A Practical Introduction. (Wiley 2009)

[4] Schuster J.C., and Bauer J., The ternary systems Sc-Al-N and Y-Al-N, J. Less-Common Met., Vol. 109, 1985, p 345-350