This Project aims to predict the lattice constant of ScAl with CASTEP calculation.
A. Project Description
In this study, we focus on predicting the lattice constant of ScAl based on the CsCl structure, and figuring out a converged energy cutoff and k-points in the CASTEP energy calculation. Atomic electron configuration for Al is 3s2 3p1, for Sc is 3s2 3p6 3d1 4s2. The energy calculation in CASTEP can provide a reasonable crystal structure for ScAl. The energy calculation in this study is based on the exchange correlation Perdew-Burke-Ernzerhof (PBE) density functionale, which is from the class of Generalized gradient approximation (GGA) functional. The relationships between energy and lattice parameters, energy cutoff, and k-points are discussed below.
B. Crystal Model
ScAl has CsCl structure, where Scandium (Sc) locates at the corner and Aluminum (Al) in the center of the unit cell. This structure belongs to the cubic system, the lattice parameter a=b=c, α=β=γ=90 degrees.
The cell vectors are along x, y, direction of the unit cell, with orthogonal \(a_i\) (1,0,0), \(a_j\)(0,1,0), and \(a_k\)(0,0,1) respectively. The equivalent fractional coordinate of the Sc is (0,0,0) whereas the Al is (1/2, 1/2, 1/2). The real coordinates of Sc and Al in the unit cell depend on the lattice parameter a, with Sc at (0,0,0), (a,0,0), (0,a,0), (0,0,a), (a,a,0), (a,0,a), (0,a,a), and (a,a,a); Al at (a/2,a/2,a/2). Figure 1 shows an example of Sc and Al positions with the lattice parameter a=b=c=3.379 Å.
Figure 1. Simple cubic structure of ScAl, where Aluminum locates in the center and Scandium in the corner of the unit cell.
C. Determine the Lattice Parameters of ScAl
In order to predict a reasonable lattice parameter of ScAl, the energy of the unit cell is calculated with the variation on lattice parameters. Given the structure is from the cubic system, the lattice parameters a, b, and C are equal and will be referred to “a” as below. Before the calculation, the lattice parameter a is estimated based on the atomic radius of Al (1.43 angstrom) and Sc (2.30 angstrom). To get a well packed structure along the body diagonal in (111) face, the lattice parameter should be smaller than 2(r(Al)+r(Sc))/√3, which is 4.3 Å.
As a starting point the energy of the ScAl structure was calculated witha lattice parameter of 4.3Å using the CASTEP calculation ( Energy cutoff 500 eV, k-points 10*10*10). With the decreased lattice parameter from the starting point, the free energy of the unit cell reached a minimum to some point and then increased with the lattice parameter decreased further (Figure 2). This shape of curve is caused by the relative atom positions, either too far or too close, generateing higher energy (less stable structure) than the minimum energy (the most stable structure by calculation). The lattice parameter \(a_0\)= 3.379 Å corresponds the minimum cohesive energy (Figure 2).
Figure 2. Cohesive energy for simple cubic ScAl, using 10*10*10 kpoints and 500eV energy cutoff, as a function of lattice parameter
D. The Energy Cutoff
Multiple calculations for ScAl structure were completed with a variation of energy cutoff \(E_{cut}\) from 200eV to 800eV. Lattice parameter and kpoints remained the same at 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration respectively. An increase in the energy cutoff increases the number of plane-waves and improves the accuracy of ion cores, but costs longer computation time. Repeated calculations with higher energy cutoff aim to converge to a decent final free energy (Figure 3&4). Convergence is reached with an energy cutoff of 500eV providing an accuracy of the absolute energy better than 0.001eV (Figure 4).
Figure 3. Both Al and Sc atomic energy converged at 500eV with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.
Figure 4. Energy per cell and cohesive energy of ScAl as a function of energy cutoff with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.
The cohesive energy of a solid material is the energy to separate the condensed material into isolated free atoms.
\begin{equation}E_{coh}=(E_{total}-E_{atom})/N\end{equation}
with \(E_{total}\)is the total energy of the unit cell, \(E_{atom}\) is the total energy of the atom, and N as the number of atoms in the unit cell. Table 1 and Figure 4 indicate that the cohesive energy decreased with higher energy cutoff, converged to an accuracy of 0.01eV at energy cutoff of 500eV.
Table 1 Results from computing the total energy, atomic energy, and cohesive energy of simple cubic ScAl with 8*8*8 kpoints in different energy cutoffs
E. k-points
As k-point varied but energy cutoff (500 eV) and lattice parameter (3.379 Å) unchanged in the following calculation, the total energy of the cell converged with more kpoints. The cell volume and atomic energy remained the same, because the cell volume only depends on lattice parameter and atomic energy depends on the energy cut-off.
Figure 5. Energy per cell and cohesive energy of ScAl as a function of M*M*M k-points, with a lattice parameter 3.379 Å and energy cutoff 500eV.
The used k points are reduced by symmetry operations using Monkhorst-Pack approach with M*M*M k points in ScAl cubic structure. Table 2 and Figure 5 shows the number of k poins in irreducible Brillouin zone (IBZ) and correspondent total energy and binding energy. Both the odd (2n+1) and even (2n+2) values of M have the same number of k points in IBZ, but even values (2n+2) of M converges better than odd values in regard to the same computational time. In our cases, both the 7*7*7 and 8*8*8 k points required 8.08 seconds to finish the calculation, but 8*8*8 converging better because all k points are inside of the IBZ (Sholl and Steckel, 2011). In ScAl structure, 12*12*12 k points are enough to get accurate energy.
Table 2 Energy calculation with k points varies, energy cutoff 500eV, lattice parameter 3.379 Å.
Conclusion:
The total energy of simple cubic ScAl minimized at lattice parameter a=3.379 Å. Energy per cell converges when energy cutoff is 500eV, k-points are 12*12*12.
Reference
[1] Sholl, D. and Steckel, J.A., 2011. Density functional theory: a practical introduction. John Wiley & Sons.
[2] BIOVA, 2014. CASTEP guide , Material Studio. http://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/pdfs/castep.htm
