Lattice constant for ScAl in CsCl and NaCl structures

Lev Krainov

Introduction:

The goal of this post is to study crystal structure of ScAl. Two possible configurations were investigated: CsCl and NaCl crystal structures. For each structure lattice constant was found by seeking a minimum of a ground state energy. Data shows that ScAl prefers CsCl structure with lattice constant $a=3.378\AA$ over the NaCl structure.

Computational details:

Ground state energy computations were performed using DFT plane-wave pseudopotential method implemented in CASTEP[2]. With CASTEP, we use the GGA-PBE as an exchange-correlation functional [3]. We also employ On-the-fly generated (OTFG) ultrasoft pseudopotential was used to describe the interactions of ionic core and valance electrons with a core radius of 2.4Bohr(1.27 Å) [4]. Pseudo atomic calculation is performed for 3s2 3p6 4s2 3d1 orbitals of Sc and 3s2 3p1 orbitals of Al. SCF convergence tolerance was set to 1.0E-6eV/atom. The Koelling-Harmon relativistic treatment was used for Sc orbitals.

K-points convergence

First we investigate convergence for both geometries. Since we need to pick some lattice constant we performed CASTEP geometry optimization using BFGS hill-climbing algorithm[1] with 15 k-points and $E_{cut}=600eV$ . These values were determined to give unit cell size converged up to $0.001\AA$ tolerance. The resulting lattice parameters are $a=3.378\AA$ for CsCl and $a=5.656\AA$ for NaCl. Using these we investigate how the ground state energy converges with the number of k-points with fixed cutoff energy $E_{cut}=500eV$ .

Figure 1 shows that for both geometries at 17 k-points energy is convergent up to 0.001 eV. Energies are shifted by $E_f=-1385.315eV$ for CsCl structure and by $E_f = -1384.149eV$ for NaCl structure, which was set to the most accurate ground state energy obtained for each. Due to the symmetry of NaCl crystal for this structure odd number of k-points leads to half as many total points in the full 3D Brillouin zone compared to even number of k-points in each dimension.

Figure 1. Convergence of energy with respect to the number of k-points for CsCl(blue) and NaCl(orange) geometries.

Cutoff energy convergence

Next we achieve same level of convergence with respect to cutoff energy keeping number of k-points fixed at 17 for CsCl and at 15 for NaCl geometries. Figure 2 shows that $E_{cut}=600eV$ is enough to obtain energy up to 0.001eV.

Figure 2. Convergence of energy with respect to the cutoff energy for CsCl(blue) and NaCl(orange) geometries in logarithmic scale. The high value of the last point is explained by the error present in [latex]E_f[/latex].

Geometry optimization

Now we use found values of $E_{cut}$ and k-points to find ground state energy as a function of unit cell size and look for a minimum approximating it by a parabola. Also we run BFGS geometry optimization with the same values of $E_{cut}$ and k-points, shown as orange dot on Figure 3. For both geometries energy of a minimum of a parabola and energy given by BFGS agree up to 0.0001eV. But since for NaCl minimum is about two times wider this energy error introduces larger error for lattice constant of $\delta a\approx0.0035\AA$ while for ScAl the error is less than $0.001\AA$ .

Figure 3. Lattice constant optimization for CsCl(top) and NaCl(bottom) structures. Orange dot for both shows the result of CASTEP geometry optimization with the same number of k-points and cutoff energy.

Results

The results are shown in the table below. Error estimation for lattice constant was performed assuming 0.001eV error in energy and parabolic approximation. The resulting error is on the order of 0.01Å and probably could be improved by including more points in geometry optimization. Energies of ground states show that ScAl prefers CsCl structure with $a\approx3.38\AA$ which is in a good agreement with experimental value of 3.388Å[6].

Crystal structure	Ground state energy E0, eV	lattice parameter a, Å	Cutoff energy E_cut, eV	number of k points
CsCl	-1385.319(1)	3.377(13)	600	17
NaCl	-1384.168(1)	5.656(55)	600	15

[1] R. Fletcher; A new approach to variable metric algorithms, The Computer Journal, Volume 13, Issue 3, 1 January 1970, Pages 317–322

[2] 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)

[3] Perdew, J. P; Burke, K; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868

[4] CASTEP GUIDE, BIOVIA, UK, 2014. URL : http://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/pdfs/castep.htm.

[5] D D Koelling and B N Harmon 1977 J. Phys. C: Solid State Phys. 10 3107

[6]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

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