The goal of this post is to identify if ScAl, which has AB stoichiometry, exists in the CsCl structure (figure 1) or NaCl Structure (figure 2 and figure 3). To determine this primitive cells of ScAl were produced for both types of structures. The plan is to plot the cohesive energy of the structure as a function of the volume per ScAl dimer. From these plots the optimal lattice parameters for each structure can be determined. Then by comparing the cohesive energies of the two structures with optimal lattice parameters we can determine which structure is preferred by ScAl. All energy calculations were carried out using plane-wave based DFT. The GGA based PBE functional was used to treat the exchange-correlation effects. The ion and core were treated using ultrasoft pseudopotentials generated on the fly (OTFG ultrasoft) with Koelling-Harmon relativistic treatments. Pseudo atomic calculations for Sc treated the 3s2 3p6 3d1 4s2 electrons as valence electrons and the electrons in lower levels were treated as part of the frozen core. Pseudo atomic calculations for Al treated the 3s2 3p1 electrons as the valence electrons.

Figure 1 : Primitive unit cell of ScAl in the CsCl structure. This a simple cubic structure with a two atom basis.

Figure 2 : (a) Conventional unit cell of ScAl in the NaCl structure. This a face centered cubic structure with a two atom basis. (b) Primitive unit cell of ScAl in the NaCl structure.

However before we begin calculating the data points to populate the plots we described in the previous paragraph, we need to select appropriate cutoff energy and \(\vec{k}\) point mesh for our calculations. The constraints on our choice of cutoff energy and \(\vec{k}\) point mesh are (1) computational cost, (2) convergence of results to a desired tolerance and (3) ensuring that we have approximately the same \(\vec{k}\) point density for each structure.

To determine the cutoff energy we use for our calculations we first determine the approximate lattice parameter of CsCl structure primitive unit cell. For this we use a cutoff energy of 460 eV and \(\vec{k}\) point mesh of \(8\times8\times8\) . The cutoff energy and \(\vec{k}\) point mesh chosen here are the default settings for the “ultra-fine” quality energy calculation using the CASTEP tool implemented in Material Studio. The cohesive energy vs volume per  ScAl dimer plot (figure 4) indicates that cohesive energy is minimized when the lattice parameter is \(\sim 3.5\;\mathring{A}\).

Figure 3 : Plot of Energy per ScAl dimer vs lattice parameter for ScAl in the CsCl structure, used to determine the approximate value for the optimal lattice parameter.

Next we investigated the convergence of the total energy of a primitive unit cell of ScAl in the CsCl structure with respect to the cutoff energy used for the calculation. The results were plotted as shown in figure 5.  From this plot note that we get a convergence of \(\sim\;1\;m\,eV\) for a cutoff energy of \(500\;m\,eV\).

Figure 4 : Calculated total energy of a unit cell of ScAl in the CsCl structure (lattice constant = 3.5 \(\mathring{A}\)) vs the cutoff energy used for the calculation.

Next to obtain the most suitable \(\vec{k}\) point mesh, we plot the energy per ScAl dimer for ae a fixed cutoff energy of \(500\;eV\) while varying the number of \(\vec{k}\) points used to sample the first brillouin zone. Since all the reciprocal lattice vectors ( and real space lattice vectors) have the same length, we can specify the \(\vec{k}\) mesh by specifying the number of \(k\)-points used along each reciprocal lattice vector. Figure 6 shows this plot; and we can see that a \(8\times8\times8\) \(\vec{k}\) point is sufficient for the energy per ScAl dimer to have a convergence of \(\sim\;10\;m\,eV\). The resulting spacing between sampled \(k\) points is \(0.0357\;\mathring{A}^{-1}\). To ensure our subsequent calculations have the same degree of convergence, we will impose that the separation between two adjacent \(k\) points that are sampled along a reciprocal lattice vector is at most \(0.0357\;\mathring{A}^{-1}\).

Figure 5 : Plot of energy per ScAl dimer vs volume per ScAl dimer.

Now we are ready to calculate the energy per ScAl dimer and the corresponding volume per dimer, for both structures and various lattice parameters. Figure 6 shows the plot of energy per ScAl dimer vs volume per dimer, for ScAl in the CsCl and NaCl structure. From the plots in figure 6 it is clear that ScAl prefers CsCl structure over the NaCl structure. From the best fit line we obtain the optimal lattice parameter in the CsCl structure to be \(3.38\;\mathring{A}\). If ScAl were to be found in the NaCl structure the optimal lattice parameter would be \(4.00\;\mathring{A}\).

Experimentally ScAl has been verified to exist in CsCl structure with a lattice parameter of \(3.450\;\mathring{A}\) [1]. Our results verify this and estimate the lattice parameter within \(\sim\;2%\) of the experimentally determined lattice constant.

[1] O. Schob and E. Parthe. Ab Compounds with Sc Y and Rare Earth Metals. I. Scandium and
Yttrium Compounds with Crb and Cscl Structure. Acta Crystallographica, 19:214-&, 1965.